>the true nature of numbers is... uhhhh... sets with empty sets inside them
who comes up with this? set theory was a mistake

I don't like it either but if you think about it all you need is

1) There is at least *Something*
Therefore logically I can have Nothing.
Let Nothing be 0
Let Nothing(Nothing) be 1
...

Noone is making claims about the supposed true nature of numbers. It's just necessary axiomatic foundations.

The natural numbers are called natural for a reason but vibes are not a good foundation for a logical system.

>>16714675
>>16714883
>[math] 0.999\ldots = \{\emptyset\}[/math]

>>16714920
Ok so come up with your own construction of the natural numbers then.
If you just wanted to make a thread to complain that you find this definition of the natural numbers ugly then I don't see the point. And if you didn't want to complain then you just misinterpreted axiomatic foundations into the subjective notion of "the true nature of numbers".

There's also the Peano axioms construction if that suits your vibes better
https://en.wikipedia.org/wiki/Natural_number#Peano_axioms

Type theory has its own problems but is more natural than set theory.

>>16714670 (OP)
all of these issues are problems with describing things and ideas. language. there's no set of things that don't exist and will never exist in nature. i don't know of any paradoxes in nature either.

One nice thing I can say about the set-theoretic construction over its alternatives is that it makes the connection to cardinality of sets simple. 3 is a set with 3 elements, so a set has 3 elements iff it has a bijection with 3.

Category theory has its own problems but is more natural than type theory

>>16714670 (OP)
isn't the lower part of the pic missing a branch?

Homotopy type theory has its own problems but is more natural than Category theory

>>16715084
Causality is a paradox.
Another paradox is that for contradiction to be an exclusion, the knowledge of the contradiction must exist. There is some sense where A&~A must exist and be knowable; otherwise the contradiction isn't grounded.

Counting numbers with your fingers has its own problems but is more natural than Homotopy type theory

>>16715614
those seem to be language abstractions.

Infinity category theory has its own problems but it’s more natural than homotopy type theory.

>>16714920
This is actually incorrect. On the left you have a cauchy sequence approaching (the real number) 1. On the right, you have (the natural number) 1. In set theory, these are two different things, as if types are implicit to the theory. This shows that type theorists never really understood set theory in the first place
>B-b-but |N| =S6
You used the wrong encoding.

>>16716065
> 1 =/= 1
i hate tranny set theorists

>>16716097
>There is some fundamental "1" that is agnostic of the model that it exists in
That's what Frege thought 100 years ago before he got clowned on by Russel. There is no defining the unique, true 1. There's only recognizing the 1 you already know in an unfamiliar situation. Even in your precious type theory, you have to "coerce the natural type to real type" or whatever CS inspired nonsense in order to do your math.

>>16716065
wrong, the left is clearly the rational number 9/9

>>16716119
>Still not equal to (one)

>>16716101
>got clowned on by Russel
Phrases which inherently have no truth to them

>>16716195
He found a flaw in his magnum opus the day before it was scheduled to be released. Knowing Russel, I'd say he waited to tell Frege on purpose.

We write informally that N is a subset of R, but what we really mean is that N canonically injects into R in a way that we can treat as an inclusion without loss of meaning. This happens absolutely everywhere in math, for example we often say that continuous function spaces are subsets of appropriate Lp spaces even though that is never true in the strict sense.

>>16716101
>>16716216

>>16716216
>Without any loss of meaning
Proof? Well, we lose the encoding, but I'm sure nobody cares.

>>16716222
The encoding is canonical. It's threefold. There is the canonical inclusion of N into Z when both are interpreted as monoids, then the canonical inclusion Z to Q, where Q is interpreted as the field of fractions over Z, then the canonical inclusion of Q into R defined by its Dedekind completion. All of these are universal properties in the sense that everything I described is unique up to isomorphism.
t. categorypilled anon

>>16714883
>not accepting that it’s all made up

>>16716271
Essentially unique != unique, though.

t. strictnesspilled anon

>>16716823
strict equivalence is whatever
like do you really care that what symbol we use for number 6? What matters is how elements in a structure interact with one another.
t. equivalenceprinciplepilled Yonedamaxxing anon

>>16716826
Even with univalence, not everything is stable by propositional equality. Definitional equality is the real thing.

t. conversionmaxxing turbotypetheorist

>>16716832
Just work in the skeleton category and everything becomes strictly equivalent. Or don’t. It’s your choice.

P.S. That was an axiom of choice pun.

>>16716847
I'm afraid you've been utralfiltered.

>>16716889
They're something anal about logicists in the Freudian sense.