Anonymous
10/4/2025, 2:31:28 AM
No.16805586
>>16805540
>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.
>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.