Thread 16805540

The famous diagonal argument goes like this. Suppose you had a countable enumeration of some real numbers, then you could use the diagonal algorithm to produce a real number not in your enumeration. So far so good.

The fallacy occurs when you then claim that this somehow shows that the set of reals is a bigger infinite than the set of naturals. Clearly, the diagonal argument made in the paragraph above contained no statement about comparing infinities, so it is logically impossible to deduce a statement comparing infinities from the diagonal argument in the above paragraph.

You may complain "But not being able to find a countable enumeration is by definition the same as being an uncountable infinity". Well, you can make that definition if you want to but then you would be confusing epistemology for ontology. Just because a countable enumeration of all the reals can't feature in your proofs does not mean the reals are "uncountably large" (whatever that term is really supposed to mean) or that a countable enumeration does not "exist".

I'm posting this because I too once believed in uncountable infinities like most simple-minded modern mathematicians, but the realization of this basic flaw in the argument destroyed my faith in set theory and now I think the entire field is built on a fallacy. This might help liberate others out there who are struggling with the unreasonable demands made by the set theory religion. Thanks.

>>16805540 (OP)
The diagonal argument is that you can't list a list of infinite lists into a list without missing part of the list. If you start from the natural observation that an infinite list can't exist in the first place, you've missed the point altogether.

>>16805543
Did you read the post? I don't see what this is supposed to be addressing or where I said an infinite list can't exist.

>>16805544
If you believe that a complete infinite list can exist, you can't complete an infinite list of your own completed infinite lists.

>>16805540 (OP)
But you do agree that there is no bijective correspondence between 1,2,3,... and all real numbers, right?

>>16805540 (OP)
Disregarding your idiotic "proof"
Uncountable vs countable infinities are intuitive
You are just a midwit for not immediately seeing this in the first place
The proof just confirm what people with a brain intuitively understand

>>16805545
I already said that in the first paragraph of the op though

>>16805546
It depends on what you mean by "there is no". I can agree that such a bijection can't be written down in ordinary proofs, but that's not the same as saying it doesn't exist.

>>16805548
If you think they are intuitive, can you use your intuition to answer whether there are any infinities of size strictly between the reals and the naturals? This is a test of your claimed intuition. If you can't solve it, it means you don't have any intuition at all.

>>16805549
>I already said that in the first paragraph of the op though
So how's it a fallacy if you said it yourself?

>>16805553
Read the paragraphs after the first one.

>>16805554
>Clearly, the diagonal argument made in the paragraph above contained no statement about comparing infinities
It makes the statement that you can't list a list of infinite lists into a list without missing part of the list. If you start from the natural observation that an infinite list can't exist in the first place, you've missed the point altogether.

>>16805559
Okay, now read the third paragraph.

>>16805561
>The first thing I wrote was retarded
>The second thing I wrote was retarded
>PLEASE READ THE THIRD THING
No, you're a fucking idiot. It's as simple as believing that a complete infinite list can exist. If you believe that it can, you also can't complete an infinite list of your own completed infinite lists.

>>16805571
If you didn't even read the post, you're not here to discuss things in good faith, so there's no point in discussing this with you.

>>16805573
I give evil people two chances to prove they're not evil. You're quite obviously an evil person.

>>16805574
Ok, hope you get better or whatever.

>>16805575
>You may complain "But not being able to find a countable enumeration is by definition the same as being an uncountable infinity". Well, you can make that definition if you want to but then you would be confusing epistemology for ontology. Just because a countable enumeration of all the reals can't feature in your proofs does not mean the reals are "uncountably large" (whatever that term is really supposed to mean) or that a countable enumeration does not "exist".

Fake and gay

>>16805578
>NIGG-MASTER5000

Fuck you go die in a hole bitch

>>16805582
>Indian reply syntax

>>16805540 (OP)
>Clearly, the diagonal argument made in the paragraph above contained no statement about comparing infinities, so it is logically impossible to deduce a statement comparing infinities from the diagonal argument in the above paragraph.
What do you think "comparing infinities" means? The only way to compare them is to establish a bijection between two infinite sets, or prove the impossibility of doing so. That's the meaning of comparing cardinalities, and that's what the diagonal argument does (it shows that you can't have a bijection between the naturals and reals).
>You may complain "But not being able to find a countable enumeration is by definition the same as being an uncountable infinity". Well, you can make that definition if you want to but then you would be confusing epistemology for ontology. Just because a countable enumeration of all the reals can't feature in your proofs does not mean the reals are "uncountably large" (whatever that term is really supposed to mean) or that a countable enumeration does not "exist".
That's not the definition. The definition is about finding bijections.
>(whatever that term is really supposed to mean)
What it means is very simple, I pointed it out above.

>>16805586
>That's not the definition. The definition is about finding bijections.
Finding a countable enumeration of R is the same as finding a bijection to N. Not sure why you're saying they're different.

>>16805552
>if you think toroid wheels go faster than tetrahedron wheels, give me everty single possible wheel shape between the two

>>16805612
>Can't answer or guess the most basic question arising from the theory of uncountable infinities
>Claims it's intuitive
Embarrassing

>no statement about comparing infinities
it's implicit by virtue of taking things to their limit. there's no assumption here besides the idea that you can in fact form a bijection, yet doing so fails immediately by the intrinsic nature of having the diagonal entry not mapped. it's essentially a proxy for a supremum that will always violate the bijection when taken to the limit.

a finite list can be added onto, to a finite end, eventual, or deterministic in some eventual case.

an infinite list cannot be added onto.

the diagonal argument is incorrect because it assumes a new number can be created onto an infinite list, thereby increasing the size of the established infinite list; and it gets there by assuming the created number is not already part of the established infinite list, which is a bad assumption, because it already is part of the infinite list, it just wasn't part of the finite list representing a viewable perspective within the infinite list already. For example, Z=[1,2,3,4,5,...] (which is basically incorrect notation by assuming the end bracket ever comes), in the very least displays a finite viewable perspective which relates to the conception of the infinite list of integers Z, showing us "1,2,3,4,5".

>>16805540 (OP)
words words words words words
you know someone is a math hack when they make arguments without a single concrete mathematical statement

>>16805643
>Argument from illiteracy

>>16805646
more words

>>16805646
>Argument from muh dik
Preach nigger queen

>>16805540 (OP)
The fundamental problem with real numbers is that they include undefinable numbers, which means that MOST real numbers can not be represented by any finite expression. In other words, almost all “real” numbers are completely useless and imperceptible to us humans. It doesn’t even make sense to say they “exist” as there is no proof of them. It’s not even clear that there should be undefinable numbers at all. It could very well be the case that any string of digits after the decimal actually corresponds to some finite expression, but then that would mean all real numbers could be ordered by their lengths, and therefore countable, which would show that the diagonal argument has some other type of error that has gone unnoticed all this time.

>>16805662
just because almost all real numbers are not computable doesn't mean you can't reason about them or make statements about their properties

>>16805664
Again, it’s not proven that such numbers exist in the first place. And it makes no difference to reason about their properties since it’s ultimately irrelevant. Suppose I have two undefinable numbers a and b. Then does a + b equal b + a? Who cares? It doesn’t affect anything. And why should they be called “real” numbers anyway? Why not given them a different category? Can you prove that they in fact have the properties of the definable reals?

>>16805673
in math, you have to formalize your argument so it can be analyzed
you seem incapable of formalizing your reasoning, meaning you don't have a well formed idea and instead are reasoning based on feelings
by trying to reason with your feely wheels, you will bend like a reed whenever presented with anything that demonstrates you are retarded
often times when one tries to formalize their ideas, they realize the limits of their understanding when they hit an impasse
seriously, go ahead, try to form a mathematical argument demonstrating your idea, we're waiting

>>16805540 (OP)
>Clearly, the diagonal argument made in the paragraph above contained no statement about comparing infinities, so it is logically impossible to deduce a statement comparing infinities from the diagonal argument in the above paragraph.
It's true that the paragraph doesn't use that term. But it does prove that no surjective function from natural numbers to real numbers can exist, where a a surjective function from real numbers to natural numbers does exist. That seems to me like a perfectly reasonable notion of size comparisons between infinite sets (it is a total order, for one thing, and agrees with notions of set size for finite sets). While that doesn't obviously make it *the* notion of size of sets, do you disagree that it is a plausible such candidate? If so, why?

>>16805678
I just asked you to prove that undefinable numbers exist and that they have the properties of the definable reals and you refused to do it, because you can’t.

>>16805687
it's your thread, sweetheart
i'm not going to lift a finger for you until you do some work yourself, starting with making the thread topic more concrete

>>16805636
>bla-bla diagonal argument assumes bla-bla can be created onto an infinite list bla-lba increasing the size of the list bla-bla
no
>assumes number is not already part of the established infinite list
it not assumes, it literally constructs such a number. this number is different from all other numbers by construction.

>>16805540 (OP)
we have established bijection. but we made such a real number that it doesn't have enumeration. thus, it's isn't a bijection. if it was, this number has to be in the list, but we are contstructing such a number so it will be different from all numbers in the list. OP, this is math 101 stuff, come on

>>16805540 (OP)
>1-to-1 correspondence, wut?
Ngmi, Setless.

>>16805697
It purports to construct the number in theory. If you disagree with me I urge you to name any number that satisfies the criterion. Pro tip: you cant.

>>16805540 (OP)
we have this thread every week. OP, you are just a midwit lacking any mathematical talent. go away.

>>16805740
>name any number that satisfies the criterion
give me your suppose bijection to natural numbers then

i think you don't understand two differences:
1. actual/existential infinity
2. potential infinity

in this example with diagonal argument it builds on the foundation of classical mathematics, in which we work with actual infinities (actual doesn't mean "physically real"). if you don't agree with that then you must use constructivist foundation. and i don't know well constructivistic mathematics to say what relationship it has with diagonal argument. anyway, i think your problem in diagonal argument lies in this. in the classical sense diagonal argument is OK

so stop being some sophist and stop the demagoguery

>>16805746
not OP, but I don't think that distinction makes any sense in math, or indeed at all. let's not dirty this thread with philosophizing. we talk about infinite sets, not special quantities or giod forbid numbers called 'infinities'.

>>16805540 (OP)
>Clearly, the diagonal argument made in the paragraph above contained no statement about comparing infinities
We compare the cardinality of infinities using bijections.
The diagonal argument shows bijection is impossible.
Therefore ones must be bigger than the other.

>>16805748
no, it makes sense in math.

if we speak in intuitionistic tradition (bishop, troelstra, markov's constructivism) then there is this distinction in potential and actual infinity. in constructivistic framework we only give an algorithm for constructing any natural number, but we don't have them all, unlike in classical mathematics where we literally have [math]\mathbb{N}[/math].

your computer doesn't have all natural numbers, right?

>>16805540 (OP)
>You may complain "But not being able to find a countable enumeration is by definition the same as being an uncountable infinity". Well, you can make that definition if you want to but then you would be confusing epistemology for ontology

levels of retardation beyond human comprehension.

>>16805682
start with n=1
[eqn]\\ a_n^i[b(f_n)] = b[f_{n1}, f_{n2}, f_{n3},\ddots,f_i]
\\ a_n^{i}[b(\frac{1}{10^n})] = b[0.1,0.01,0.001,\ddots,\frac{1}{10^i}]
\\ a_n^i\big[c(a_z^9[b_n(\frac{z}{10^n})]\big] = c\big[b_1[0.1,0.2,0.3,\ddots,0.9],b_2[0.01,0.02,0.03,\ddots,0.09],
\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \enspace b_3[0.001,0.002,0.003,\ddots,0.009],\ddots,
\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \enspace
b_i[\frac{1}{10^i},\frac{2}{10^i},\frac{3}{10^i},\ddots,\frac{9}{10^i}]\big] =
\\ c\big[_{n1}\overset{b1_1}{[0.1]}, \overset{b2_1}{[0.01]}, \overset{b3_1}{[0.001]}, \overset{b4_1}{[0.0001]},\overset{b5_1}{[0.00001]},\overset{b6_1}{[0.000001]}, \overset{bi_1}{\ddots}
\\ \enspace _{n2}\overset{b1_2}{[0.2]},[0.02],[0.002],[0.0002],[0.00002],[0.000002], \ddots
\\ \enspace _{n3}\overset{b1_3}{[0.3]},[0.03],[0.003],[0.0003],[0.00003],[0.000003], \ddots
\\ \enspace _{n...},{\ddots}
\\ \enspace _{n9}\overset{b1_9}{[0.9]},[0.09],[0.009],[0.0009],[0.00009],[0.000009], \overset{bi_9}{\ddots} \big] =
\\ c\big[ [b1_1 \to b1_9] ,\to [b2_1 \to b2_9] ,\to [b3_1 \to b3_9],\to \ddots \big]
\\ \sum_{n=1}^{\infty}c[n]= 4.999... \approx 5
\\ \text{which is equivalent to}
\\ \sum_{n=1}^\infty \sum_{x=1}^9 \frac{x}{10^n} \approx 5
[/eqn]
set c contains every possible decimal expansive part required to make any real number between 0 to 1. for example, the number [math] (\pi-3)[/math] is 0.14159..., also [eqn] \overset{+}{c}[b1_1, b2_4, b3_1, b4_5, b5_9, ...] [/eqn].
pi is also summable from elements of c, for example:
[eqn] \overset{+}{c}[b1_9, b1_8, b1_7, b1_6, b1_1,b2_4,b3_1,b4_5,b5_9,b6_2,...][/eqn]
which is trivial because of the ability to construct any integer between 0 to 5 from elements of b1.
since it is true that every number following the pattern [eqn]\frac{1\to 9}{10^1 \to 10^{\infty}} [/eqn] appears in this c set, essentially indexable by coordinates [x,y] where x is a number 19 and y is a number > 0

>>16805540 (OP)
rent free

>>16805552
>can you use your intuition to answer whether there are any infinities of size strictly between the reals and the naturals?
but tell us how you really feel about non-euclidean geometry

>>16805636
>an infinite list cannot be added onto.
do the naturals include 0?

>>16805697
if you refer to the op image, it is using binary (0 and 1) to construct a binary number.

s1 [0 0 0 0 0 0 0 0] = all 0's (equivalent to the minimum value)
s2 [1 1 1 1 1 1 1 1] = all 1's (equivalent to the maximum value)
s3 [0 1 0 1 0 1 0 1] = swapping from 0 to 1 to 0
s4 [1 0 1 0 1 0 1 0] = swapping from 1 to 0 to 1
s5 [1 1 0 1 0 1 1 0] = 11010 over and over (arbitrary)
s6 [0 0 1 1 0 1 1 0] = 011 over and over, after an initial 0 (very arbitrary)
s7 [1 0 0 0 1 0 0 0] = 1000 over and over (arbitrary)
s8 [0 0 1 1 0 0 1 1] = 0011 over and over (arbitrary)
s9 [1 1 0 0 1 1 0 0] = 1100 over and over (arbitrary)
10 [1 1 0 1 1 1 0 0] = no determinable pattern given (exceedingly arbitrary)
11 [1 1 0 1 0 1 0 0] = 11, 01,01,001,001 (arbitrary)

the point is, the list is not meaningful. the additional lists are seemingly arbitrary, which means an arbitrary list could ever exist containing
s = [1 0 1 1 1 0 1 0]

s1 [0 0 0 0 0 0 0 0 0 0] = all zeros
s2 [1 1 1 1 1 1 1 1 1 1] = all ones
s3 [0 1 0 1 0 1 0 1 0 1] = 01 alternating
s4 [1 0 1 0 1 0 1 0 1 0] = 10 alternating
s5 [1 1 0 1 1 0 1 1 0 1] = 110 alternating
s6 [0 0 1 0 0 1 0 0 1 0] = 001 alternating
s7 [1 1 1 0 1 1 1 0 1 1] = 1110 alternating
s8 [0 0 0 1 0 0 0 1 0 0] = 0001 alternating
s9 [1 1 1 1 0 0 0 0 1 1] = 11110000 alternating
10 [0 0 0 0 1 1 1 1 0 0] = 00001111 alternating

s= [1 0 1 1 0 0 0 0 0 1] = could be another number added onto the list, or
11 [1 1 1 1 1 0 0 0 0 0] = 1111100000 alternating, could also be added onto the list, or
12 [0 0 0 0 0 1 1 1 1 1] = 0000011111 alternating, could also be added onto the list

you can add things onto the ends of lists, and then the list increases in size.
a list of numbers, for example the integers, which affirms to try to be 'all' the integers, would be an ever-increasing list, and it's size infinite, as [for every n, n+1].

there is no meaning to the op image for determining a new number that did not exist.

>>16805767
oh wow, latex

>>16805777
>you can add things onto the ends of lists
you do realize that the diagonal argument produces a number NOT ON THE LIST, right?, please do point at any mathematicians explanation of the diagonal argument right at the place in why they say "add things onto the ends of lists" or whatever the hell you niggerbabble could be understood as, thank you

>>16805780
you have been diseasing this website for a long time. commit sudoku, retard-chan.

>>16805767
what's the point of this post even? a lot of letters but no coherent conclusion

>>16805682
>But it does prove that no surjective function from natural numbers to real numbers can exist
Does it really? I think it only proves that there is no surjective function from N to R that you can write down. To give an analogy, Godel proves using a diagonal argument that there are statements of arithmetic which can be proved neither true nor false in some given proof system, but this is not the same as saying that truth values for those statements do not exist. I suspect something similar is happening here - what the diagonal argument is is a statement on the limitation on the sorts of functions that can appear in your proof system, rather than a statement about the size of infinities.

Actually, if you go by this >>16805662 anon's suggestion, then the real numbers are, in some sense, countable since there will only be countably many finite expressions. So a bijection might exist, but you won't be able to write it down.

>and agrees with notions of set size for finite sets
There are also places where it disagrees with finite sets. You can have a sets which have the same size as proper subsets according to this definition. Is the set of even numbers really the same size as the set of all naturals? It's obvious that the former is smaller than the latter.

>>16805794
>It's obvious that the former is smaller than the latter
prove to us, that the set of even numbers is smaller than the set of natural numbers. you know what will happen? you will not prove this. why? because there is a bijection between even numbers and natural ones: [math] f(n) = 2n, f^{-1}(m) = \frac{m}{2}[/math].

>the limitation on the sorts of functions that can appear in your proof system
you are telling us that some "functions" can't appear in our "proof system" because of this? do you even understand what do you mean by a "proof system", big brain?

my fucking god it feels like anon learned about naive set theory and doesn't understand it's paradoxes and inconsistencies

>>16805811
>prove to us, that the set of even numbers is smaller than the set of natural numbers
Is the set {1, 2} smaller than the set {1, 2, 3}? Obviously yes because the former is a subset of the latter. Is the set {2, 4, 6, ...} smaller than {1, 2, 3, 4, 5, 6, ....}? Obviously yes because the former is a subset of the latter.

>you are telling us that some "functions" can't appear in our "proof system" because of this?
Yes, I am particularly talking about functions which exhibit a bijection from N to R. Those functions might exist, but they just won't appear in your proof system. This seems to be a much more reasonable conclusion to draw from the diagonal argument than the claim that, suddenly, this finite proof somehow shows the existence of uncountably many real numbers.

>>16805818
>bla-bla thos functions might exist bla-bla
you didn't answer the question about "proof system". i think you just have some kind of inner "inuition" about this "proof system", and this "intuition" is just wrong

>subset means smaller
when the sets are finite, then yes (proper subsets will be "smaller"). when the sets are infinite - no.

just for you dumb-dumb, let's speak about even numbers. can i just use sign "1" to denote two? yes, it's just a fucking letter. can i denote four as "2"? the same logic. in the end the set of even numbers will look like [math]\{ 1, 2, 3, 4, 5, ... \}[/math]. and what it looks like? like fucking set of naturals.
>b-b-but it's just letters!
yes, it's fucking letters. the same logic can be used for naturals: we can denote them anyway we fucking want. for example, one as 2, two as 4, three as 6, and so on.

>>16805818
>Those functions might exist, but they just won't appear in your proof system
The mathematics we use is built on top of a set of axioms.
We use those axioms to prove theorems.
The real and natural numbers, as defined in our axiomatic system, cannot have a bijection, which is what the diagonal argument proves.
This means that in our mathematical system, such a function cannot exist, which is the whole point of the theorem.

You are free to create your own mathematical system and define the concept of natural numbers, real numbers, and bijections in a way that allows for them to exist.
Point is: your system will be useless

>>16805830
It looks like you're taking a purely formalist approach to mathematics. That's possible, but then you're not the target audience for my post. My post is intended for people who think there really are real numbers out there and they really are uncountably larger than the natural numbers. I think the real numbers exist too, but I just don't believe it's been shown that there are uncountably many of them.

>>16805829
>when the sets are finite, then yes (proper subsets will be "smaller"). when the sets are infinite - no.
Under your definition of "smaller", sure. But you haven't convinced me that I should use your definition, since I already provided a very reasonable sense of "smaller" for which the even numbers are smaller than natural numbers.

>. the same logic can be used for naturals: we can denote them anyway we fucking want
Okay, but I'm not talking about how you "denote" them. I'm talking about the actual sets themselves. It's like if I said 1 + 1 = 2 and you said "umm ackshually you can denote 1 by 2 and 2 by 3 and + by -, so you should ackshally say 2 - 2 = 3"

>>16805794
>I think it only proves that there is no surjective function from N to R that you can write down.
>the real numbers are, in some sense, countable since there will only be countably many finite expressions.
You got that the wrong way around. The real numbers *that you can write down* are indeed countable. The set of all real numbers is not. The diagonal argument proves this, as it makes no assumption about being able to write down a real number; it only assumes the existence of a surjection.

(Technically, the version you depict in OP shows the uncountability of the sequences of bits, not the real numbers. Applying that to the real numbers is an extra step. But not a difficult one, and the argument can indeed be adapted that way.)

>There are also places where it disagrees with finite sets.
Oh? Can you give an example?

>You can have a sets which have the same size as proper subsets according to this definition.
Not finite ones.

>Is the set of even numbers really the same size as the set of all naturals?
You will note that those sets are not in fact finite.

>>16805836
dumb-dumb, let's use the evens that i denoted that way. what will be two plus two? four, right. 1 + 1 = 2, and just to remind you: '1' is two, '2' is four. coincidence? no, it works for ANY number. six plus eight = fourteen. 3 + 4 = 7, what is 7? fourteen. even numbers are isomorphic as a semigroup to naturals. so addition works literally like on naturals.

>i'm not talking about how you "denote" them
bijections are like denoting elements of a set differently (using a set bijective to it). this bijections doesn't mean isomorphism (conservation of operations)

>>16805839
>it only assumes the existence of a surjection.
Not quite. It assumes the existence of a surjection and it also assumes that you can treat that surjection like any other function and manipulate it as needed by the diagonal argument. It is only the conjunction of these premises that let you derive a contradiction and the diagonal argument doesn't itself tell you which of those premises you must discard.

>Not finite ones.
Well, you said it was reasonable since it agrees with corresponding statements for finite sets sometimes and I pointed out that it also disagrees with statements for finite sets sometimes. If P is the statement "If A is a proper subset of B, then there is no bijection from A to B", then it is true for finite sets but false for infinite sets.

>>16805836
>My post is intended for people who think there really are real numbers out there and they really are uncountably larger than the natural numbers
dude, nobody is the fucking target of your post then

>>16805847
I don't think so. There are plenty of people who think the real numbers really are uncountably large.

>>16805852
That's not what you just wrote

>>16805857
? Explain.

>>16805858
>My post is intended for people who think there really are real numbers out there AND they really are uncountably larger than the natural numbers

>>16805860
The second statement already presumes the first anyway. How do you believe that the real numbers are really uncountably infinite and that they don't really exist?

>start with an undefinable infinite list containing undefinable infinite decimal strings and you will get an undefinable infinite decimal that’s different from all of them!
woah..

>>16805864
Correct. You're apparently one of the rare alpha retards who can formulate how the argument works into a valid sentence while still being so retarded that you have to add a sarcastic element proclaiming your own retardation.

>>16805861
Because math is made up? Are you dense?

>>16805868
If you think the real numbers are made up, this thread is not for you like I already said.

>>16805871
Nobody thinks real numbers exist "in the wild" as physical things. Your thread is about nothing

>>16805874
Nobody mentioned anything about "physical things" other than you. If you believe the real numbers really exist, then you believe that they are really uncountably large or you don't. If you don't believe in real numbers, you are free to ignore the thread.

>>16805876
>Nobody mentioned anything about "physical things"
>>My post is intended for people who think there really are real numbers OUT THERE
out there where? in the real physical world. you don't even know what you wrote 5 minutes ago

>>16805878
>in the real physical world
Nice addition you just made which wasn't in my post at all. Now can you stop derailing the thread, please?

>>16805880
what does out there mean then, come on make up some excuse about how it means "out there in the magical realm of mathematics"

>>16805883
Like I'm saying for the third and final time, if you don't believe in the real numbers, you are free to ignore the thread.

>>16805884
You have proven you are incapable of clearly expressing yourself even in simple english. You got exposed and now you're deflecting.

If real numbers don't exist in the real world, then the diagonal argument proves that the reals are bigger than the naturals within the axioms we all use for our mathematics.

If real numbers exist in the real world then you first have to tell me to which physical objects they correspond before any discussion could be had

>>16805886
>the diagonal argument proves that the reals are bigger than the naturals within the axioms we all use for our mathematics.
No, read the OP.

>>16805888
>no rebuttal
LMAO
>read the OP
nothing in the OP rebukes what I wrote
>Clearly, the diagonal argument made in the paragraph above contained no statement about comparing infinities
this statement is already wrong

>>16805888
>contained no statement about comparing infinities
NTA, the diagonal argument literally says that all the indexed s are complete and infinite and that the index is complete and infinite. You're apparently one of the beta retards.

>>16805889
Apart from addressing it in the OP, I've already addressed it in several posts. You think being able or unable to find bijections is the same as comparing sizes and I am saying that facts about sizes of sets are or should be independent of being able to find bijections in your axiomatic systems.

>>16805890
Reread the OP or see >>16805892.

>>16805892
>facts about sizes of sets are or should be independent of being able to find bijections in your axiomatic systems
1) Why?
2) What alternative that works better or makes more sense do you propose?

>>16805893
>contained no statement about comparing infinities
This is from your own OP. You can either try to rewrite it or be wrong forever. As it stands, you're simply wrong.

>>16805894
Because a fact is true or false regardless of the axioms you choose.
I'm inclined to think all sets are countable but I have not yet fully developed a system around this guess and I wouldn't post it here if I did.

>>16805896
>contained no statement about comparing infinities
And this "fact" is false.

>>16805895
It's factually correct. The first paragraph just doesn't contain the words like "comparing infinities" or anything like that.

>>16805896
>Because a fact is true or false regardless of the axioms you choose.
"Parallel lines never intersect" is a true fact in the axioms of Euclidean geometry and a false fact in the axioms of spherical geometry, so what you said is demonstrably wrong

>>16805906
If they never intersect then they were never parallel to begin with. Chew on that one, nerd.

>>16805899
>The first paragraph
Of what? Are you like a gamma retard or something? This is the diagonal argument paper.
https://www.digizeitschriften.de/id/37721857X_0001%7Clog29?tify=%7B%22pages%22%3A%5B83%5D%2C%22view%22%3A%22info%22%7D#navi

>>16805906
The axioms are about different things here (Euclidean vs spherical), so your example is not relevant.

>>16805909
That's not what parallel means

>>16805910
Seriously, why bother posting if you didn't even read the OP? I mean the first paragraph of the OP.

>>16805912
>the axioms are about different things
if you change the axioms you get a different thing, exactly. which is why a fact can be true under certain axioms and false under others, thus your previous statement us false

>>16805913
You might say that they seem parallel but at no section on the sphere are they truly parallel. They only approximate being parallel starting at a certain point, but points can’t be parallel. As soon as you add another point onto the lines, they cease to be parallel.

>>16805915
You can't point out a fallacy in someone else's paper by pointing out a fallact your own retarded cliff notes about it. Fucking lol.

>>16805920
No, then those would just be facts about different things. 1 + 1 = 2 is true regardless of what axioms you use.

>>16805923
I tried to engage with you in good faith but you seem to be genuinely mentally challenged, sorry.

>>16805922
What is the definition of parallel, then?

>>16805926
You said your own "first paragraph just doesn't contain the words like "comparing infinities" or anything like that." So? Cantor is dead. You're not Cantor, you retard.

>>16805925
I can easily create an axiomatic system where 1 + 1 = 1, so again you are wrong

>>16805931
No, you fucking cannot, not unless you think all mathematics is nonsense.

>>16805927
If I select any point on line A and find the shortest distance to a point on line B, the distance of this line should be the same regardless of which point I start with. Which is to say that the distance between the lines never changes, and that they never intersect.

>>16805933
Why would an axiomatic system where 1 + 1 = 1 be nonsense?

>>16805937
Because 1 + 1 = 2 and 1 is not equal 2.

such a retard thread at this point

>>16805933
in trivial group this is true, braindead

>>16805939
Why is 1 + 1 = 2 better than 1 + 1 = 1?

>>16805942
>>16805943
If you cannot agree to such basic things like 1 + 1 = 2, this thread is not for you. You should take up your issues with other mathematicians. Sorry, I won't discuss this any further. You are just derailing my thread.

>>16805940
Of course it's a retard thread. The only legit criticism of the diagonal argument is to agree with it sarcastically, like this alpha retard did >>16805864

>>16805949
>your axioms are nonsense because they don't agree with my axioms
ok? something more sustantial? I could say your axioms are nonsense just as well

>>16805949
Are you implying that 1 + 1 = 2 is true because some mathematicians say so?

TLDR:
>OP believes mathematics should follow how the real world works and human intuition, but he cannot admit it ITT because then everyone would leave laughing

>>16805969
Infinity doesn’t exist within human intuition, it’s more of a divine concept.
It’s more defined by what it’s not, and types of infinities are also defined similarly.
What op is missing is that we’ve already done something unintuitive which is begin in a space we dont fully comprehend or understand, and we applied a test that failed meaning our assumption the test works are wrong which is why it implies the existence of uncountable sets
If we started with a more intuitive space like a countable set we wouldn’t really progress our knowledge
That’s a big deal, and frankly I think it demonstrates the fragility of intuitionism. Of you lack understanding your set of truth is smaller than the real set of truths
People’s personal gnosis will always be limited by genetics and education
And they will always push against you, but formalism also kills them because it demonstrates truth without understandingSo even a machine could be convinced about a truth. But machines have no souls so there is no personal development in formalism it’s mostly just for sanity
So in general intuition developed from a divine level like the pedagogical figure of platonism works better since it begins sight from above the ground rather than beneath it, but you’ll need both to transverse towards knowledge andunderstanding.it’s like this: I seriously doubt a god would understand what it’s like to be a worm, a man strives to know of the worm and god equally, the god and the worm seek to know nothing equally
Thus man is not a slave like gods or worms, he is born free
This is why I’m highly against excessive abstraction that claims to unify all issues, as well as excessive practice which claims to increment knowledgeBoth are slaves to the knowledge of nothingwhen I already know nothing, so that I might seek to know something
Not slave nor god slave, I am the king of my garden
TL;DR: You’re burning in hell for being retard, meanwhile I am in heaven for being based

>>16806121
T. 30IQ btw, so even a retard like me can be free and not enslaved, what’s your excuse OP?
Here’s how I think about it. Sophia is Athena, and she has many virgin daughters in touchable by men, and they have many daughters of lower rank, some touchable. The daughters of math are my concubines, meanwhile you are a concubine-less faggot.

>>16806121
>divine concept
stop reading after that

>>16805931
Would your proposed axiomatic system be able to prove "if x + y = x + z then y = z"?

>>16806132
in trivial group this works, dumbass. starting reading your fucking books, anon

>>16806128
My iq is sub 30 and I read the Bible. Problem with that?
Sorry you can’t into figure and will remain a sup 100 iq science bitch made faggot.
https://youtu.be/X_jmEOGvriw
Here’s the proof using the Bible, imagine god had an infinite name. Take the set of all names of god and enumerate it, there is a name of god distinct that exists via construction not in the enumeration.
SOIENCE BITCH BTFO’D BY A SUB30 IQ NEOPLATONIST CHRISTIAN. Common science bitch L

>>16806124
>>16806131
>>16806140
No one cares about your shitty IQ foid

>>16806143
So uhhh are you just going to keep being off topic or actually try to refute me?
Maybe >>>/lit/ is more your speed

>>16806159
>Maybe >>>/lit/ is more your speed
says the anon that does all his "math" in paragraphs

>>16806132
>Would your proposed axiomatic system be able to prove "if x + y = x + z then y = z"?
It depends, but how is it relevant?

Who the fuck are these retards and why are they derailing my thread?

>>16806163
Because if you're trying to axiomatically treat addition then it should, but then you can easily get 0 = 1 if 1 + 1 = 1.
Of course if your point is that you can make *some* axiomatic system where 1 + 1 = 1 as long as + (and 1 and =) are allowed to mean whatever you want, then yeah but that's not an interesting observation

>>16806171
because your thread is stupid

>>16806175
Fuck this place. I'm never posting here again.

>>16805540 (OP)
its impossible to actually finish this proof fyi

>>16806180
small dick energy

>>16806171
>Indian reply syntax

>The famous diagonal argument goes like this.
>Suppose you had a countable enumeration of some real numbers
wrong and facetious to make the argument seem weak
>The fallacy occurs when you then claim that this somehow shows that the set of reals is a bigger infinite than the set of naturals.
also wrong. mathematical induction is not a claim it's a logical extension of the proof on the base case.
>it is logically impossible to deduce a statement comparing infinities from the diagonal argument in the above paragraph.
see above. you are struggling with this because you don't understand the laws of truth.
>You may complain
fabrication to discredit
>Well, you can make that definition
fabrication to discredit
>Just because a countable enumeration of all the reals can't feature in your proofs
fabrication to discredit
>I'm posting this because I too once believed
fabrication to appeal
>I think the entire field is built on a fallacy.
irrelevant
>This might help liberate others out there
fabrication to discredit

low quality reasoning levels coupled with malicious layering of false pretenses to frame yourself as dealing with a weaker version of the argument. a juvenile toolkit that's neither intriguing nor valuable.

>>16805794
By completeness theorem any statement that is true in every model has to have a proof. If your "proof system" leaves behind all this biyective functions out there, aka there is no proof of any of their properties, then that means there has to be one model in which their existence is true and also one in which its false since your "proof system" doesn't speak about them and you could add them to your axioms. Build a model in which cardinality of natural numbers is the cardinality of R (there exists a biyective function f:N->R) and then prove that given T consistent then T + [N]=[R] is also consistent

>>16805794
>I think it only proves that there is no surjective function from N to R that you can write down
Where in the diagonal argument is it assumed that the function is "written down"?

>>16805552
>can you use your intuition to answer whether there are any infinities of size strictly between the reals and the naturals
Yes, there are.

>>16806121
>the fragility of intuitionism
i don't like the attitude of a lot of intuitionists, but i don't think that what they are about applies to what you are writing, it is a really piss poor name that they chose, so that's where the confusion might have arisen, other than that yeah you are right

>>16806275
my dude dropped gold in a pile of shit. good job

>>16805540 (OP)
The diagonal argument is much more informal than Cantor's Theorem. Any questions of its rigor are irrelevant.

>>16806593
Search for lawvere theorem. Even though you may label diagonal arguments as "informal" it allows you to see many theorems as tentacles of the same lovecraftian being that was out there all this time just in the same way that everything was hodge star operator all this time. The core cause of godel is the same as turing, tarski etc: a problem that arises when a system tries to talk about itself, manifested as a fixed point theorem and caused by a difference in size between the set and all the possible truth values

>>16805751
talk to me when lim {n->actual infinity} a_n starts differing from lim {n->potential infinity} a_n. until then it's philosophical masturbation.
>your computer doesn't have all natural numbers, right?
Inductive nat : Set :=
O : nat
| S: nat -> nat.

don't they kind of get less important though?

why do all these threads turn out to be troll threads?

>>16806870
oh yes, it's literally storing every natural number, great job, dumbass

you do understand that limits in intuitionistic/constructivistic analysis are different? i, not as fucking arrogant like you, don't know how they define limits with their notion of infinity

>>16805540 (OP)
>so it is logically impossible to deduce a statement comparing infinities from the diagonal argument in the above paragraph
Not really, the diagonals + the linears is always going to be a larger set than the linear values themselves when you have established that there are in fact unique values that appear in the diagonals that cannot appear in the originals.

op just wanted to say there are not multiple infinities or varying sizes of multiple infinities.
there is just the one.

>>16805689
>'m not going to lift a finger
Clearly a liar since you had to lift your fingers dozens of times to make that retarded post.

>>16807569
We all know what retard OP said. The small problem is that he just claimed it's true without proof, meanwhile a proof of the contrary exists in the maths we all use

>>16805740
>I urge you to name any number that satisfies the criterion.
It can not only be named through enumeration, but there is an algorithm to generate infinite numbers that satisfy the criterion.

>>16806121
>infinity ooga booga
shut the fuck up, inf has a definition

Infinity is an unbounded quantity greater than every real number.

>>16806953
can't people be retarded without being trolls?

>>16807649
well then, he ain't going to curl a dumbell, is that better?

>>16807671
mind explaining your pic?

>>16807805
OC is a number
once A & B touch you are at infinity

>>16807651
claimed what was true without proof?
the diagonal argument is a bad argument, not a proof of more infinities. If you believe it is a proof of more infinities, you're logically solving it incorrectly. We can explain the logic well if you're having problems.

>>16807802
Yes, but usually in these threads they are not actually interested in the topic and just are trying to rile people up.

>>16808403
>We can explain the logic
>We
>logic
lmao

>>16808469
the diagonal argument is a bad argument.
Are you contradicting for the sake of contradiction?

>>16808403
you claimed the diagonal argument is bad without proof

>>16808403
>the diagonal argument is a bad argument, not a proof of more infinities
Why can't it be both? Why don't you explain how it is "bad" and why it doesn't actually show how to construct infinite numbers that can't be directly rationally constructed from the integers alone?

>>16806593
That's a slick proof and all but there's nothing inherently informal about the diagonal argument. Some presentations of it are informal, skipping over messy details with decimal expansions (it's much clearer to demonstrate the diagonal argument first with the uncountability of natural number sequences).

>>16808480
>>16808485
Z:[1,2,3,4,5,6,...]
Zlol:[13,54658,2455,755651136875,122,...]
if i combine a number from Z with Zlol, do i get a number that is not present in Z?

>>16808618
>>16808480
>>16808485
op's pic relies on binary however it is a retarded interpretation of binary.
>s1 = 00000000000...
>s2 = 11111111111...
if it is assumed the trailing ... part of the sequence is meant to imply that the sequence never ends, then how did we get to s2? how does s2 get to s3?

s1, assuming it is all zeros, is 0
s2, assuming it is all ones, is the number infinity.
>the number infinity
it's the equivalent of actually trying to write out, in decimal
>s1 = 0
>s2 = 99999999999999999999999999999...
which is not a normal way to use trailing [...] (on integers), but is also the unreasonable existential claim that there would ever be such a number "999..." where {the least significant whole common integer "1"} added onto it wouldn't just turn it into "10000000000000000000000000000....", also a poor claim against the value of 1, but also indicative that the number "0.999..." does in fact, after-all, not equal 1; that rather than trying to imagine an unreasonable infinitesimal of non-point existentialism that any value "0.000...1" would somehow stop mattering and simply disappear (lol wtf no it wouldn't), it would instead make more sense to understand a smallest part must exist and every smallest part does matter, lest 1 never add up to infinity; and that by the multiplicative sense of allowing the construction of an infinitely-long binary number "11111111111111...", it would always be a lesser number finite number for a countable amount of 1's than it's decimal expansion, given 11111111 = 256 (8 binary digits equal 3 decimal digits), even attempting to divine that an infinite amount of work in binary would have ultimately been suboptimal.

allowing the construction of a binary number as "111...", then it attempting to exist as the maximum value, in the very least infinity-1, which regardless of the retarded intent, placing it back in the context of the op pic, s1-s11 are 11 numbers, lesser than decimal, and not a complete list of all numbers.

heres a logical conundrum question:
do you believe there have, in history thus far, been fewer counted and seen sequential integers than seen irrational numbers, rational numbers, and all the digits of those irrational and rational numbers combined?

how would one determine that there would be 50 trillion digits of pi, without having sequentially counted from 1 to 50 trillion?
it's an imagination game to sieve the supposed necessary existence of large numbers.

if your mind cannot even capture the quality that 2000 characters, the 4chan post restriction, is not enough to encapsulate even a whole idea translated as best as can be to english text, requiring re-reading the post and removing a few choice words (desu i start with deleting the egregious amounts of "nigger" "retarded" "fucking", etc. to free up some room so i can post within the 2000 character limit); then how is your mind expected to capture the quality that 50 trillion is only 50,000,000,000,000?
what about the number 2,500,000,000,000,000,000,000,000,000?
if you used every character in a 4chan post to write out 9's, 2000 nines, that is a large number and still nowhere near infinity.

infinity is bigger than a nigger dick. open your eyes please.

>>16808671
>op's pic relies on binary however it is a retarded interpretation of binary.
That's just a retarded interpretation of OP's pic. No infinite string of 0s and 1s with a 1 anywhere in it translates to a binary number unless you put a radix point somewhere.

>>16808475
dunno, are you retarded for the sake of retardation?

>>16805540 (OP)
>Just because a countable enumeration of all the reals can't feature in your proofs does not mean the reals are "uncountably large"
That's quite literally the definition of uncountably infinite you absolute moron.
>the diagonal argument made in the paragraph above contained no statement about comparing infinities
It contained a statement about enumerating them, and you proved there is at least one number not listed anywhere in your emueration.

>>16809126
broooooooooo
s1 = yadayada
s2 = badabing
s3 = donkeydoo
...
...
...
"s" = yaddayadabaddabingdonkeydoodoo

look at this part right here my broski
>...
>...
>...

it's the equivalent of saying
>s1 = [1]2345
>s2 = 2[3]451
>s3 = 34[5]12
>s4 = 451[2]3
>s5 = 5123[4]
>,,,
>"s"=13524
>"LOOK S=13524 THAT IS A NUMBER THAT WASN'T ON THE LIST THERFORE INFINITY IS SMALLER THAN INFINITY"
while in all reasonable technicality, it's also equivalent to saying
>c1 = 1
>c2 = 2
>c3 = 3
>c4 = 4
>c5 = 5
>...
>c = 10

the situational utility of composing a graphic or setting of dots enumerating like this is to imply to a finite end, not to imply to an infinite end.
the dots to the right of s1, s2, s3, imply the numbers continue infinitely (which is stupid because it is binary)
the dots to the bottom, between s11 and s, imply the continuity of s values end finitely and deterministically, and for the sake of belief of suggesting that s is actually a new value to the list, s must not have already been a member of that finite list; otherwise the assumption holds via the above c's that it is obvious that 10 can be enumerated from 1,2,3,4,5, and would also be obvious to be an enumeration within a count towards infinity, and is not a novel element to the list.

>>16809136
>equivalent to saying c = 10
Only if you're writing 1,2,3,4,5... in "base infinity," in which case, yes, the double digit number 10 will never appear in your list.

>>16805933
That's literally how it works in the trivial group you absolute retard

>>16809126
>>16805745
>>16805548
>>16805771

>>16805546
>>16805586
It only shows there can't be a bijective correspondence between each element of a set S and the set of all subsets of S. To put it simple n < 2^n.
The problem when mapping natural numbers to reals is that reals can grow left and right infinitely. However that is only a problem of notation. If you write the real number in binary and arrange the bits in such a way they grow infinitely to the left, like integers, then that problem vanishes.

>>16809191
Reals can't be infinite to the left. In binary, that's called 2-adic and also proves the diagonal argument.

>>16809194
Skill issue.
The diagonal argument is a fallacy.

>>16809197
A conditional can't be false unless the antecedent is true. The antecedent here is P (you can complete an infinite list of complete infinite strings). The consequent is Q (there's another string not on the list). Even if P is false, the diagonal argument P -> Q is still vacuously true.

>>16809210
That is a red herring.

>>16809219
You can only prove it's false if you accept P as true and show that Q is false based on P being completely true. Sorry, them's the rules.

>>16809210
>you can complete an infinite list of complete infinite strings
does one need to complete something that already is?, are you incomplete?

>>16807802
>can't people be retarded without being trolls?
they can, but I was kinda just blogging here about my approach: when having to choose an alternative, I assume OP is a troll rather than a retard. I might be wrong, but I won't betrolled.

>>16805540 (OP)
Advanced set theory has managed to build a theory which is equiconsistent with ordinary set theory ZF (if one is proven inconsistent the other automatically is and for your information this is far from being done, some higly skilled people like Edward Nelson attempted to do this but haven't found a contradiction) and in which we have:
1°) every single axiom of ZFC is satisfied
2°) there is a countable (!!!) transitive set M in which every axiom of ZFC is satisfied too.
Being transitive, M has exactly the same integers as M (but pretty much anything else is altered, for instance the R of M is countable, but the bijection f between it and N is simply outside N, hence there is no contradiction a priori with the diagonal argument).
(for the construction above see e.g. Jech set theory, at the chapter about forcing)

The diagonal argument is the single and trivial fact that, picking any doubly indexed family (u(p,n))_{(p,n) in E^2} of elements of {0,1}, where E is **any** set, you build a sequence n in E -> v(n) which is not of the form (n -> u(p,n) for any p, by defining v_n as 1 - u(n,n) for every n in E.

Where is the lie here? Nowhere, OP is just showing a huge strawman.

>>16809353
>for instance the R of M is countable
So you admit that the diagonal argument doesn't show that R is uncountable. Tell that to everyone in the thread who thinks that R has to be uncountable because of the diagonal agreement..

Why did all the infinitists suddenly shut up?

>>16805540 (OP)
>Given any real number, rational or irrational, reverse its digits and you get a corresponding natural number. You can do the same to convert natural to real number
How can you refute this bijection?

>>16805543
>>16805544
>>16805545
You guys just wrote a dr. seuss book

>>16810186
Explain what you mean with reverse the digits using 1/9 as an example.

>>16810198
use 1/7
1/7 = 0.14285714285...
it natural mapping will be ....58241758241
mapping of 1/9 will be ...11111

>>16810169
Bump. The infinitists should respond to >>16809401 if they have any shame left.

>>16809401
>>16810169
>>16810333
Read the post you were replying to.
>but the bijection f between it and N is simply outside N, hence there is no contradiction a priori with the diagonal argument

>>16810345
So is the R in M countable or is it not? Answer the question, coward.

*crickets*

*crickets*

>>16809353
>1°) every single axiom of ZFC is satisfied
>2°) there is a countable (!!!) transitive set M in which every axiom of ZFC is satisfied too.
doesn't ZFC + Con(ZFC) already prove this by Lowenheim-Skolem?

>>16810188
Incredibly rare 100+ IQ /lit/ post.
>>16809401
Common sub-100 IQ /lit/ post.

>>16810769
Incredibly common sub 50IQ /sci/ post.

>>16810773
Cite three examples or you lose.

>>16810776
Three examples of sub 50 iq /sci/ posts? See >>16810769, >>16805548, and >>16805643

>>16810779
Three examples that correlate. Not one example of you putting your finger in your ass and pulling out the first three things you run into.

>>16810786
Begone, sub 50iq poster. This thread is not the place for you.

>>16810790
Beg on.

>>16805540 (OP)
>But not being able to find a countable enumeration is
nonsense. pure non-sequitur. the universe is the hard limit. whether you can observe it or not is irrelevant, it is encoded within the universe.

>>16809163
explain the belief of what base binary infinity is.
how many bits do the binary numbers have? infinity? not real.
111111.... is as meaningful of a number as 99999...., and neither are. it's stupid notation for dumb.

>>16810811
For christ sakes the OP pic is demonstrating that the set of all {0,1}-sequences is uncountable (you would know this if you read the wiki page you nabbed the image from). It is not about numbers. The next step is establishing a bijection between [0,1] and these {0,1}-sequences via binary expansions of numbers in [0,1].

>>16810811
Where did you get "base binary infinity" from?
The anon wrote 1, 2, 3, 4, 5... and said 10 isn't on the list. If you're writing in base ten, it's tenth on the list. If you're writing in base twelve, it's twelfth on the list. The only way it's not on the list is if you have an infinite alphabet of single digits that doesn't include "10" as a single digit.

>>16810808
>DoctorNeger
What was your MCAT score?

>>16810813
>>16810821
idk what you niggers are on about but i hope you're not seriously defending a belief of multiple infinities or quantitative infinities of greater quantity than other infinities; because that would be objectively retarded.

>>16805540 (OP)
Objection! You change from arguing the proof can't show the reals are bigger than the naturals to arguing the proof doesn't show the reals are uncountably infinite. Those are different claims and the proof demonstrably shows the former by literally enumerating all the naturals with reals and showing there are uncounted reals.

>>16810960
Nigger be quiet and listen to me right now: if 10 isn't on an infinite list of single digits 1, 2, 3, 4, 5..., it can only be because it's an infinite list of single digit bases.

>>16806990
so, an example where the two limits differ? handwaving like 'they differ in schizoid analysis' aren't examples, give formulae and numeric values.

>You HAVE to make a table, where any real number could be found in it
>... done
>Heh, now using Diagonal Argument, I can make numbers that are guaranteed to not exist in it. Therefore R is larger th-
>.. done. I just add your new numbers to the top of my table. In fact, anytime you make a diagonal number, I will make a new table with that number on top. Because you give yourself the freedom to walk through infinite digits and infinite entries, I also give myself the right to make infinite tables.
>NOOOOOOOO YOU MUST USE ONLY ONE TABLE, AND YOU MUST NOT MODIFY IT NEVER EVER OR ELSE IT'S NOT FAIR NOOOOOOOOO

>>16805548
>intuitive
HOLY FUCKING SHIT MATH IS LARP

>>16811103
>still in his rigor autist stage
cute

>>16805540 (OP)
Notice that cantor's argument doesn't work in all counting systems. Just use a base-1 tally system and its fucked.

>>16811610
oh is the diagonal thing a cantor thing? lmao
i can prove cantor's interpretation of infinity unintentionally didn't mean what he thought it meant, inclusive of a history of computer engineers that adapted the same or similar belief system. IEEE computer standards had casually identified the largest value of a float32 to be represented as "infinity", which in a binary register was: 1111 1111 1111 1111 1111 1111 1111 1111

32 binary 1's might be a big number, but it surely isn't infinity.
the context comes from
>Z:[1,2,3,4,5...]
>the set of all integers has infinite size
>C:[1,2,3,4]
>the subset C has size 4
>a set can be enumerated upon to have the size of it's integered last-countable integer belonging to the set
>C has size 4, the 4th element of C is also 4
>Z has size infinity, the infiniteth element of Z is also infinity
>assumes to self contain infinity as a number within the set of Z, interpreting Z as a maximum possible value for a given finite construct

cantor's theories are taught common in some certain sense, but there are logical issues even there that his axiom of infinity could be interpreted to apply differently. Didn't have a perfect mind, didn't have a perfect communication, leading to interpretations and misinterpretations.

>>16811610
You're retarded. There's only one infinite string that uses one digit. Where are you getting a "diagonal" from?

>>16811620
Well, every number in such systems will produce n entries in the list. In general, there is no number and radix that actually satisfy his conditions. There is no subset of filled-rank numbers where diagonalization allows you to create a number that is not on the list. If I take base-10 0-999, the diagonal only covers 3 of them and only 10 in a binary encoded strings of 0-1023.
He invokes dark math when appealing to infinities and other absurdities.
This is why we choose a base-1 tally system. His pseud rambling is revealed.

Our tally system involved two elements: marks and piles.

Each pile is given a designated size w_n such that if the number of marks in a pile > w_n, then subtract w_n from the pile and add 1 mark to the pile above it. Just a standard carryover system.
It becomes very clear that cantor is trying to pretend shifting 1 mark makes a new number, but it doesn't. It makes some number that is further down the list.

>>16811610 >>16811629
tally can't even into decimals

>>16811656
Pick a demarcation between piles. Done.

>>16810589
Less is needed, the countable set is built in steps in the following fashion (see e.g. Jech or Krivine set theory books):
Pick a constant symbol M; let's assume you can provide a formal contradiction in a theory T whose axioms contain at least every axiom of ZF + the dependent choice; + the assumption that "M is a countable transitive set" + all axioms of T, relativized to M.

0°) This proof with only require a (explicitely) finite set of axioms A_1,...,A_d, B_1^M, ... B_e^M (where B_i is an axiom of T and B_i^M ist relativitzation to M).

Then in plain T you do the following:

1°) Let B_0 be the extensionality axiom; there is (Montague reflexion principle) a set m such that B_0 /\ B_1 /\ ...B_e <=> (B_0^m /\ B_1^m /\ ... /\ B_e^m) is provable in plain ZF (B_i^m is of course B_i relativized to m for any i)
2°) there is (countable Loweinheim Skolem adapted for set theory, which actually only requires the dependent axiom of choice) a subset m' of m such that m' is countable and m is an elementary extension of m' (it means that for every sentence F without free variables, T proves that F^m <=> F^m')
3°) Using the Mostowski collapse theorem, there is m'' transitive and isomorphic to m with the belonging relationship.

That being said, if we reproduce the proof in 0°) by replacing M by m'', we immediately obtain a contradiction in T alone (which can be only ZF with dep choice). Hence the result (and the consistency of a theory featuring such a strange structure M: this turns out to be a common tool in independence proofs).

>>16810589
>>16811885
thus the extra assumption cons(ZFC) or even cons (ZF + dep choice) was unnecessary

>>16811885
>>16811886
What does this tell you about R (or the set of all binary sequences)? Is it countable or is it not?

>>16811894
the notion of countability is RELATIVE.
It is important to notice this. In the above we deal with a situation where there is not one but **two different** structures S_1, S_2 satisfying the axioms of common set theory (thus in which virtually all math can be made meaningful), and such that
1°) S_1 is contained in S_2
2°) the R defined in S_1 is, in S_2, a countable set
3°) the bijection f from the R of S_1, with the N of S_2 (which turns out in fact to be the N is S_1 as well but this is secondary here) is not an element of S_1 (and couldn't be, unless there is a contradiction in common set theory ZFC)
4°) if the points 1,2,3 above lead to a contradiction, that contradiction can mechanically and reliably be translated in a contradiction in common set theory (ZFC), however this hasn't been found so far.

################

Working on a single structure S satisfying the axioms of common set theory, it is not possible to build a bijection (belonging thus to S), from the R (of S) into the N (of S); which doens't contradict what has been said above.

So is the infinite "broken" (a.k.a. weird), in a sense YES; however it doesn't implies that it is contradictory.

>>16811900
I agree it's not contradictory, but do you think there is a fact of the matter about the set of all binary sequences being countable or not? Do you think your conclusion that countability is relative is a side-effect of your model theory, i.e. your epistemology for reasoning about set theory, not being strong enough to capture everything about R, or do you think it is the final word on the matter?

the proof is literally on wikipedia, idk why OP brought up "ontology" or "epistemology", it's just math lol

>>16811906
>the proof is literally on wikipedia
The proof of what? Not the uncountability, as my OP shows.

>>16811908
>shows that set of reals is a bigger infinite than the set of naturals
It does show this, retard.

>>16811916
*sigh* I really don't like dealing with toddlers who start throwing temper tantrums when they can't argue. Look at this anon's proof that countability is relative >>16811900

>>16811916
the only thing your OP shows is that you deserve your mentally illness

>>16811908
the only thing your OP shows is that you deserve your mentally illness

>>16811922
That doesn't change the fact that the
>set of reals is a bigger infinite than the set of naturals
which is what your OP says is a fallacy. Retard.

>>16811930
The fact that you can have a model of real numbers which is countable doesn't change the fact that you think the reals are bigger than the naturals?

>>16811934
Call it what you want, the bijection is outside of N either way.

>>16811935
*outside of M

countability is relative but R is relative too btw

>>16811935
So you admit that the diagonal argument doesn't show that R is uncountable, it just shows that you can't write down the bijection in your particular proof system, like I already said in my OP several days ago and which you only realized now.

>>16811939
You're a babbling idiot. Nothing like what you said happened or is happening.

>>16811941
No argument? Just ad hominem? I think you should thank me for educating you, but you've already shown that you're an immature manchild who's prone to temper tantrums.

>>16811943
People like you who lie, invent a story about it in their head, then use nonsensical language to make a nonsensical point about it are a plague on humanity. You sick vermin deserve nothing but ad hominen.

>>16811944
I accept your concession. You should get your mental illness treated at some point.

>>16811945
>I accept your concession
And there it is, the Indian self-QED. Go back lol.

>>16811939
>you can't write down the bijection in your particular proof system
And in MY particular proof system (which is the one used by the entirety of the math world), showing the impossibility of a bijection between sets proves that one is bigger than the other.

You are free to reject the proof system the entire world is using and that has worked wonders for us in the last centuries, but at least have the humility of either proposing a better alternative or admitting you just believe some fringe shit and your entire thread is for basically nobody

>>16805540 (OP)
>you want to but then you would be confusing epistemology for ontology.
100% certain this retard also makes posts about the ontological nuances of Monty Hall.

>>16812034
>showing the impossibility of a bijection between sets proves that one is bigger than the other.
Except your own system also shows that R can be countable too.
>proposing a better alternative
Obviously I won't do that on 4chan of all places. This thread is just meant to make strong believers in the uncountable infinite question their faith.

>>16812057
>Except your own system also shows that R can be countable too.
And where is this proof?

>>16812060
See this post >>16811900

>>16812063
That post is obvious LLM slop. Either you're a bot or you're getting gaslighted by ChatGPT like countless other midwits into thinking you're a genius.

>>16812066
That post was written by an anon who's proficient at model theory, something they don't teach in undergrad real analysis.

>>16812068
Obvious AI slop. I'm gonna have to conclude you're a bot, otherwise you would have removed all the markup to obscure the source of this vomit.

>>16812071
Keep digging that hole
https://en.wikipedia.org/wiki/Skolem%27s_paradox

>>16812068
>>16812073
>>16811900
Obvious AI slop. I'm gonna have to conclude you're a bot, otherwise you would have removed all the markup to obscure the source of this vomit.

Be sure to keep replying and digging that hole.

>>16812074
I accept your concession.

>>16812063
>the notion of countability is RELATIVE.
False, it's perfectly well defined and never relative. A set is countable if there exists a bijection between it and the set of natural numbers.
>the axioms of common set theory (thus in which virtually all math can be made meaningful)
False, the proof system we are using is not based on common set theory. You are using vague language and vague concepts because you cannot fornally prove anything.

Nowhere in set theory you have "structures". You have sets, you have classes, elements, and so on. You are using that word because you actually don't know any set theory, so you have to resort to your informal, vague colloquialisms

>>16812076
1. I am not the author of that post
2. You are free to publish your revolutionary complaints against model theory in any mathematical journal of your choice.

>>16812085
>quotes the post as proof
>well I didn't write it why are you replying to me
I accept your spineless concession
>publish your revolutionary complaints against model theory in any mathematical journal of your choice.
I opened my book about axiomatic set theory (the system which actually underpins all mathematics, not some vague "common set theory" crap) and there is no mention of any "structure" before reaching the point where bijections, natural numbers, and size are defined.

>>16811900
>Working on a single structure S satisfying the axioms of common set theory, it is not possible to build a bijection (belonging thus to S), from the R (of S) into the N (of S); which doens't contradict what has been said above.
No ones seems to notice that part in the aforementioned post which is kinda sad but remember we are on /sci/, not math overflow

the more detailed content is here:
>>16811885

>>16812091
>I opened my book about axiomatic set theory and there is no mention of any "structure"
Lol, keep embarrassing yourself, undergrad.
https://en.wikipedia.org/wiki/Structure_(mathematical_logic)

>>16812075
>Indian bot loop

>>16812095
Again, since you seem unable to read, I can safely reach the point where bijection, natural numbers, and countabiliy are defined without a single mention of any structure. You'll have to try harder if you want to disprove proven things

>>16812103
>waaahhhh its not reeeal bcuz it's not in my book 4 babbiess!!1 :((((
I don't know how to respond to this.

>>16812103
See also Kunen: set theory p.233 for the countable transitive model of set theory discussed in the so-called bot slop

>>16812106
The difference in size between the natural numbers and the real numbers is a proven thing in ZFC set theory.

So either:
1. You reject ZFC and thus your thread is for the 10 persons in the world who do shit with other axioms
2. You accept ZFC and thus are trying to disprove a proven thing

Choose your own flavour of being a loser

>>16812109
ZFC also proves that there exist models of ZFC which are countable, so R can be countable.

>>16812113
When you do model theory you are operating at a higher level than ZFC. You are doing meta mathematics. Are there models of ZFC that behave in whatever wacky retarded way I want? Sure. Are they what we base our mathematics upon?

No.

>>16812107
NTA but, again, you're just wrong. R is never the same size as N. The surjection from N* to R* is outside of N*. Who cares if it's "countable" in N? You're doing the /lit/ thing where you latch on to a word, like "Planck length" -> universe pixel, then use the word to try to prove some shit that doesn't follow. See also this anon's post >>16812092

>>16812120
>You're doing the /lit/ thing where you latch on to a word
That's because you are arguing with poorly trained bots who post markup on 4chan unironically.

>>16812117
>When you do model theory you are operating at a higher level than ZFC.
Wrong. All of model theory can be done in a much weaker system than ZFC.

structures are legit mathematical concepts btw ; see e.g. https://en.wikipedia.org/wiki/Structure_(mathematical_logic)

>>16805540 (OP)
>>16805543
The diagonal argument is a proof by contradiction.
Which means one of the assumptions is false.
The false assumption is that infinite sets exist.

>>16815808
It doesn't need an infinite set, but it does extend to an infinite set since it is sequential and algorithmic to any arbitrary size. It would apply to a system of any size, though, its just that we already know for a fact that a finite set can not represent all numbers, so its trivial to apply to a finite set.

>>16815823
A finite set of binary decimals of n digits requires size n^2.
That's it. That's all the diagonal argument is.
The "uncountability" only happens when you let n go to infinity.
Hence the problem is the infinity.
Not the set construction.

>>16815866
>A finite set of binary decimals of n digits requires size n^2.
Yes which is not all the digits.

>That's all the diagonal argument is.
No, the diagonal argument is the argument that even with an infinite set of countable digits, you can not represent all the possible combinations of digits because some digit combinations are not in the set of countable numbers, even if you let that set go to infinity, there are still limitations on the set that prevent it from representing all possible combinations despite being infinite.

>>16805540 (OP)
What about the diagonal going the other way?

>>16816699
Count from 0 to 1 in infinitesimal increments and when you finish, I will explain all about the diagonal that goes the other way.

>>16805540 (OP)
While we're on the topic
Is there any reason you can't do diagonalization with integers other than
>NOOO YOU CAN'T DO THAT
?

>>16817065
What does "do diagonalization with integers" even mean?

>>16817065
Integers all have a finite number of integer digits so the list of infinite integers would be empty so there wouldn't be a diagonal.

>>16816948
Sure I will do it if you can name one infinitesimal number by its decimal expansion.

>>16805540 (OP)
OP you fail to realize that these proofs in religion specifically on diagonal statement's only seem to come out when you see it again in another set therefore the infinite amount of possibilities and each set you can make does not need to correlate to any behavior in reality, that defines you to be in a universe where sets are the standard or proof that finiteness of universe can determine your conclusion that people limit themselves to what they cannot see between all the patterns in humanity, life, the universe, or simply the scape or realm of studies and academics they are in. but yet sets still proves that there certain things made for certain things! LOL OP you’re becoming a nepotist OP don’t get mad at humanity just stay WHERE you are and live life you’re GOOD thats all that matter…..

>>16817335
If integers can't have infinite digits how many digits does the largest integer have?

>>16817686
No such thing

>>16817697
if there's no largest integer then integers must have up to infinite digits

>>16817839
Why? Makes no sense.

>>16817839
Up to, but not including

>>16817845
take the one that's up to and multiply by 10

>>16817845
>>16817849
No such thing as one less than infinity.

>>16817065
eh, can't do that in the integers, but you can in the p-adics

>>16817840
Then what is the other option that makes sense to you?
You claimed that integers all have a finite number of digits instead of being infinite, so you should be able to identify that finite limit if you have confirmed it definitely can't be infinite.
Why can't you, how can there be "no such thing" as the upper finite limit if there definitely is some upper finite limit to the number of digits in an integer?

>>16818027
None of what you said makes sense. Cant help you.

>>16818206
So if you can't make sense of the difference between finite and infinite on your own an nothing anyone else presents clicks either, you think maybe the problem is (You)?

>>16818280
Don't know what to tell you. You either don't know what words mean or you don't know how to put words together. Not worth trying to guess which one it is.

>>16818294
Except you are the one who can't explain how a finite integer can have infinite digits and can't understand any of the explanations that have been provided, so you are the one who has failed to communicate your understanding of math jargon and you are essentially admitting you are basically guessing as to the actual meanings of finite vs infinite, so you should just come out and admit science/math isn't for you and go back to some other place more aligned with your mathematical illiteracy.

>>16818303
k

>>16818305
Cool, I really respect you for owning up to your own shortcomings and accept your admission with no strings attached, best of luck to you wherever you end up now that you know /sci/ isn't right for you.

>>16818322
k bye

>>16818324
Thanks and sincerely good luck in your future endeavors wherever else that leads you.

>>16818327
Polite of you to say bye so many times.

>>16818335
A fond farewell to you as well and have a forever goodbye for the rest of your journey elsewhere.

>>16818347
Well

>>16818303
for the 25 000 000 times the diagonal argument doesn't mention any infinity notion anywhere, it just says that for every set E and every map f: E -> (E -> {0,1}) there is a map g outside the range of f, obtained by setting g(x):= 1 - f(x,x) for every x in E.

Where did you see that E was even assumed to be "infinite" ?

>>16818525
Are you blocking OP's image or something?

>every set
So how many finite sets are there if every set is not infinite?

>>16818549
>how many
you are very confused

>>16818592
You could clear up any confusion if it weren't clearly you who were confused and didn't even notice/understand all the ellipses in OP's image and doesn't understand the boundless assertion embedded in the invocation of every or all sets instead of a specific finite number of sets.

If you still think there is some confusion that can be cleared up, I will give you one more chance to try to identify the finite number of sets E applies to if infinity is not implied in the setup to your understanding of the argument.

>>16818610
>specific number
you are very, very confused.

I'll give you one more chance to ask grok about arbitrarily many vs infinitely many.

>>16818633
No, you still can't explain yourself and are lazily outsourcing your thinking to some other chatbot, but "every x in E" definitely implies x is from some standard number set with no upper limit. Your formula doesn't just apply to some arbitrary finite x, it applies to an infinite set which is why OP's image uses the ellipses that are standard indication of tendency to infinity.

>>16818641
you are a dumb nigger:

https://en.wikipedia.org/wiki/Arbitrarily_large#:~:text=Furthermore%2C%20%22arbitrarily%20large%22%20also%20does%20not%20mean%20%22infinitely%20large%22.%20For%20example%2C%20although%20prime%20numbers%20can%20be%20arbitrarily%20large%2C%20an%20infinitely%20large%20prime%20number%20does%20not%20exist—since%20all%20prime%20numbers%20(as%20well%20as%20all%20other%20integers)%20are%20finite.

>>16818648
No, you are, the diagonal argument applies to arbitrarily more than arbitrarily many and arbitrarily more than that (and so on) since you can replace x with x+y and y with y+z (and so on) and it still holds, so it holds for infinitely many rather than just just arbitrarily many values.

>>16818648
https://en.wikipedia.org/wiki/Infinitely_large
>something which is boundless, limitless, endless, or larger than any natural number.
So what is the known bound or limit to E and x in your setup and how much smaller is it than any natural number?

>>16818653
you are really, really confused and really, really stupid. No one cares about your nigger headcanon math you made up and your fake definitions lol.

also see >>16809353

>>16818656
>known bound or limit
confused as fuck you are.
>how much smaller is it than any natural number
gibberish. I do not give a rat's ass about your nigger headcanon math.

>>16818663
Its literally from the "infinitely large" link on the wiki page you posted.
>Furthermore, "arbitrarily large" also does not mean "infinitely large". For example, although prime numbers can be arbitrarily large, an infinitely large prime number does not exist—since all prime numbers (as well as all other integers) are finite.
>>https://en.wikipedia.org/wiki/Infinitely_large
>something which is boundless, limitless, endless, or larger than any natural number.
The x in the diagonal argument is boundless, limitless, and larger than any natural number, so by the definition in the source you provided, it is infinitely large.

Also, your arbitrarily large page specifically says
>arbitrarily large, arbitrarily small and arbitrarily long are used in statements to make clear the fact that an object is large, small, or long with little limitation or restraint, respectively.
>little limitation or restraint
so the onus is on you to prove there is some small amount of limit or restraint to E and x in the diagonal argument since it is never specified which is why the illustrations such as OP's image specifically use the ellipses to indicate infinitely large.

If you don't care so much, then just walk away and accept the L to stop embarrassing yourself with this self imposed humiliation ritual.

>>16818669
>so the onus is on you
and the anus is on you, dumb faggot. Go back to /lit/ or whatever retard incubator you crawled out of.

>>16818672
Concession accepted, you can show no little limit or restraint to x because it is infinitely large by the standards of your own sources.

>>16818676
>Concession accepted
Indian debate syntax.
congrats, you played yourself

>>16818658
No, your confusion is such that you can't even accept your own sources.

>>16818679
Sure that namecalling projection totally proves your original point, so you are now the smartest jeet in jeetville.

>>16818681
or, much more likely, your confusion is that my sources aren't in Hindi

>>16818687
No, my confusion is that you posted a source, then refused to accept the exact words written on the page you linked when you realized they didn't actually support your original claims.

>>16818696
you don't even understand the words, you literally think
>how much smaller is it than any natural number
is a valid question. Lol. Retard. I enjoy mocking sea lion faggots like you, tho, so please go on.

>>16818702
Nope, you don't understand words and that the diagonal argument you presented specifically said every x in E instead of arbitrarily many x in E.

>how much smaller is it than any natural number... is a valid question
No because I don't think there being little limitation or restraint to x is a valid premise in the first place (that would only follow from your severely confused logic, but you can't provide an answer because your premise is severely flawed) since it says every x in E rather than all but some little amount of x in E.

>>16818705
I'm not this anon >>16818525
I literally just butted into the conversation to mock you for your nigger tier follow up question
>>16818549
>So how many finite sets are there if every set is not infinite?
What a fucking stupid question. Displays the exact same ignorance/ stupidity/ sealioning/ jeetery as this one lol
>how much smaller is it than any natural number

>>16818713
The question were not meant to make sense given the premise that lead to them isn't sensible due to the fact that the anon you totally are not doesn't understand that there either has to be infinite digits or a largest finite number that is arbitrarily smaller than infinite.

>>16818731
>there either has to be infinite digits or a largest finite number
oh, is that how it works in your nigger headcanon math? Fucking lmao

>>16818740
No, that what the source that was provided by anon >>16818648 says is the difference between arbitrarily large >>16818669 and infinitely large >>16818656.

>>16818746
>source [...] says is the difference
Lol what? you retarded ESL gas bag. maybe try google translating into your native language?

>>16818751
Its a straightforward english sentence, if you can't make sense of it, you are the one lacking an intuitive grasp of english likely due to problems translating it to your native tongue.

>>16818756
not gonna play your faggot game with you, just here to mock you and piss down your throat.

>>16818760
Alrighty, since that sounds like the best accomplishment you will ever possibly have in life, enjoy yourself while you can, I would feel bad taking away what little joy you could ever conceivably muster for yourself.

>>16818765
don't overburden your feelings, you have enough to feel bad about already. Like this howler
>there either has to be infinite digits or a largest finite number

>>16818769
Take it up with wikipedia >>16818648 >>16818656 >>16818669 and tell the anon that totally isn't you that they should read their sources before they link them to avoid having to refute their own sources.

>>16818772
this anon IS me dumb fuck >>16818648
Not my fault you can't read

>>16818779
Its your fault that you are retarded enough to think writing something to someone who can't read is sensible while yourself completely ignoring the exact text in your own source while desperately trying to change the subject because the source you posted contradicts your initial premise.

>>16818781
good, step 1 is to admit you can't read lol. Let's see if we can get you to step 2.

>>16818790
Is step 2 being retarded enough to think that writing something to someone who can't read makes any sense?

>>16819057
>2. Came to believe that a Power greater than ourselves could restore us to sanity.
step 2 in your case is to renounce your retard act and bow to me.