fin

>non-Platonic point of view
>problematic
What did she mean by these?

Multiplication makes it undecidable.

>>8377352
infinite sets exist

>>8377462
Prove it.

>>8377462
Although a logical domain is sort of like a set, let's not get into sets and look at the axioms.

The standard axiomatization is what I get at btw., not the longer more algebraic one in the pic

>>8377472
Where does [math]\mathbb{N}[/math] come from if not the axioms? This "set" is what I consider problematic, as asked by the OP.

>>8377352
The last one.
0 is not a natural number

>>8377478
You don't need any set theoretic notions in these axioms. You can formulate them the same way in types, categories, toposes, etc.

>>8377462
>>8377465
>>8377472
>>8377478
>>8377484
>>8377497
How many subsets are there within an infinite set?

>>8377508
exactly two

>>8377508
For the simplest notion of "exists", the theory says 2^|X|, where |X| is the quantity of elements of X (the cardinality of X)

>>8377352
Can you philosophers please learn some model theory before you go asking stupid questions?

Point is, as far as we can tell the Peano axioms are consistent, so there is nothing "problematic" about them. You pretty much have to assume something like them to do any math. And if they are inconsistent by some miracle, the proof of inconsistency will probably involve nearly all of the axioms.

So the only meaningful question is, removing which axioms will give you an interesting/useful system. One obvious thing to do is remove the order axioms (which aren't really part of the Peano axioms to begin with). Removing things like commutativity will give you rigs (rings without negation), groups, etc. so associativity and commutativity are probably the "least problematic" ones.

>>8377774
And, if you remove #15 and change #12 slightly, you get a theory that is also true of the integers. So in that sense 15 is the "most problematic" or the most specific.

>problematic
It's not social science m8

>>8377774
I didn't mean for the thread to be about consistency at all. Rather for positions like finitism, for example.

Besides, the question about consistency of the Peano axioms of course always also casts Gödel into play, and if you're a formalist, then "consistent as far as we know" is pretty odd statement.

>>8377783
PA is equiconsistent with PRA+quantifier-free induction up to ε0

>>8377795
Yeah bravo, this involves some imaginary numbers

>>8377783
>"consistent as far as we know" is pretty odd statement.

Why? It's the only statement we can make.

Finitism (a la Wildburger and his ilk) again is also based on an inability to understand model theory.

I'm not a formalist either because mathematical concepts do mean something. But we are finite beings, so we can never verify things about the infinite without assuming some axioms.

>>8377798
No, it doesn't, it's all natural. The consistency of PA basically boils down to one "believing" in the well-foundedness of ε0.

>>8377802
>Why? It's the only statement we can make.
Okay, but this doesn't mean it's also odd.

Why would understanding model theory necessarily make one into a non-finitist?

>>8377834
>Why would understanding model theory necessarily make one into a non-finitist?

It wouldn't necessarily, but at least you wouldn't be an edgelord claiming that the Peano axioms are "not true" like most finitists do. If you understand model theory then there is no point in being a partisan towards a particular theory, any more than using ZFC makes you a "setist", or studying graph theory makes you a "graphist" etc.

File: ctr-tinfoil.png (14KB, 224x280px) Image search: [Google]

14KB, 224x280px

>>8377842
Have you ever actually discussed this with actual people instead of fantasy strawman? Because I can't see evidence of that from your shitposting. Wildberger isn't making videos for you. He doesn't care about you at all.

>>8377872
I'm not sure what your point is - I don't care about Wildberger either except for the fact that he might be convincing people of shitty ideas.

Why do we even need axioms? Aren't definitions of objects/interactions enough?

>>8377352
>non-Platonic

get >>>/out/