Can Sets be Defined from Peano Axioms Alone?

[Dmobb Jr.]

Is it possible to define sets from just the peano axioms?

Usually when people use the peano axioms as the basis of their math they just assume the existence of sets but without axioms regarding sets we technically shouldn't just say they exist.

Oh, also there are two versions of the induction axiom. Obviously you can't use the one that mentions sets.

 

[Fredrik]

Is it possible to define sets from just the peano axioms?

I don't see a way to do that.

Usually when people use the peano axioms as the basis of their math they just assume the existence of sets

That's because the modern approach is to show that ZFC set theory admits the existence of a set that satisfies Peano's axioms.

I think that originally, the axioms were meant to define a branch of mathematics that deals with integers, without mentioning sets at all. It was an approach to integers that had nothing to do with sets.

 

[Dmobb Jr.]

I agree that it was not Peano's intent to ask these sort of questions. I still think, however, that it would be interesting.

I would like to modify the question to "Define sets from the Peano axioms or prove that it can't be done."

 

[Fredrik]

I would like to modify the question to "Define sets from the Peano axioms or prove that it can't be done."

I'm having difficulties making sense of this. To prove that "it" can be done, we need some criterion by which we measure success. So are you talking about using Peano's axioms to prove that we can define $\in$ as a relation on the integers in a way that ensures that there are integers that satisfy the ZFC axioms?

 

[Dmobb Jr.]

I had not really thought about it thoroughly but I suppose yes, that is what I am asking.

Oh except they don't have to be integers as the peano axioms define natural numbers.

 

[Dmobb Jr.]

Now that I think about it, I do not even know how to define integers without sets.

Integers are traditionally defined as equivilence classes of ordered pairs of natural numbers.
Equivelence classes and ordered pairs are both sets.

 

[yossell]

Usually when people use the peano axioms as the basis of their math they just assume the existence of sets

How so? What do you have in mind here? The second order induction axiom?

 

[Dmobb Jr.]

How so? What do you have in mind here? The second order induction axiom?

For example integers and rationals are usually defined by equivalence classes. I don't see how you could do that without sets.

 

[yossell]

If the project is to start with the natural numbers, and then to define other mathematical objects in terms of equivalence classes in terms of them, then classes will of course be needed. But just as the Peano axioms can be used to characterise the natural numbers (arguably without sets - depends what one thinks of first vs second order induction scheme), so there are axiom systems which can characterise the integers and rationals, again without sets.

I don't think the Peano axioms were ever meant to be a foundation for all of maths.

 

[Fredrik]

I don't think the Peano axioms were ever meant to be a foundation for all of maths.

It's still interesting to consider if they can be a foundation for more than just the natural numbers. I think Dmobb Jr is right that we need some kind of set theory just to go from natural numbers to integers, but it doesn't have to be as sophisticated as ZFC. We really only need to be able to define ordered pairs.

Peano + a simple set theory seems to give us a lot. But without any kind of set theory (or something similar, like a theory of functions or categories), I think the axioms for a type of "numbers" can only define the branch of mathematics that deals with those numbers.

Of course, there are lots of things in mathematical logic that I don't understand. For example, I have never understood the claim that there exist countable models of ZFC. I don't even know what it means. Maybe it means something like what Dmobb Jr suggested about the natural numbers?

 

[HallsofIvy]

You need set theory, at a minimum, to instantiate the Peano axioms but it is easy to show that there is a one- to- one function from one instantiation to any other that preserves the sucessor function.

 

[Fredrik]

You need set theory, at a minimum, to instantiate the Peano axioms

Doesn't that just mean that we need a set theory if we want to show that there's a set that satisfies the Peano axioms?

It has to be possible to view the axioms as defining a theory of numbers (that doesn't involve sets).

 

[yossell]

For example, I have never understood the claim that there exist countable models of ZFC. I don't even know what it means.

I know you know the meaning of `countable model of T' -- so you must have something sophisticated in mind here. What is it?

Every first order theory has a countable model. So you could take the elements of the domain to be natural numbers or integers or rationals. But I'm not sure how that would be considered reducing T to another theory.

It's still interesting to consider if they can be a foundation for more than just the natural numbers. I think Dmobb Jr is right that we need some kind of set theory just to go from natural numbers to integers, but it doesn't have to be as sophisticated as ZFC. We really only need to be able to define ordered pairs.

I agree it's interesting, but I suppose it depends what you what from a foundation. If you want an explicit identification of integers and rationals etc; with something other than the integers, then you will have to have axioms that are stronger than PA, because you will need to introduce new entities. But if you just wanted to show that the theory of integers or rationals could be reproduced within Peano Arithmetic, then I suppose that, using coding, you will be able to PA as a foundation for quite a bit.

 

[Fredrik]

I know you know the meaning of `countable model of T' -- so you must have something sophisticated in mind here. What is it?

I think you're just overestimating my abilities. I'm quite a noob when it comes to mathematical logic. I have read a little here and there, but I never made it very far into any topic, and I have forgotten some of what I read. Right now I'm even struggling to remember the definition of "model". I think I can figure that out, but I have never studied proofs of statements like "every first order theory has a countable model".

 

[lugita15]

IFor example, I have never understood the claim that there exist countable models of ZFC. I don't even know what it means.

As far as what it means, you described it pretty well when you said "So are you talking about using Peano's axioms to prove that we can define ∈ as a relation on the integers in a way that ensures that there are integers that satisfy the ZFC axioms?", except possibly for the "using Peano's axioms to prove" part. You may need to use a bit more machinery to prove it; I haven't really thought it through.

As far as how such a far-fetched statement can possibly be true, read this blog post by Steve Landsburg for an intuitive explanation. (He explains how there can be a countable model of the first-order theory of real numbers, but the reasoning is equally applicable to ZFC, and besides, a countable model of ZFC would have a countable set of real numbers.) By the way, if you find his blog post interesting, you may also find my comment to it interesting.