I don't understand how everything relates. It seems that ZFC is a "first order theory" with axioms described in the language of first order logic, and it can recreate all the same axioms of Peano arithmetic (but not the other way around), so I suppose this makes PA a first order theory as well.

But then I am hearing that Peano's axioms are technically a second order theory? But then there's the first order theory that isn't as strong? Then I am unsure where natural numbers are defined exactly, and if this technically requires us to have set theory first in order to talk about membership? And what about functions? Don't these require set theory as well? Does this mean functions require ZFC? And if not, then what exactly are the "sets" we're using here?

I'm just totally lost as to what's defined where in terms of what and what's required to do this or that, it's all so hazy and vague and unclear and after reading countless answers on this website where everyone recommends the same unclear links that only partially answer the question, I'm losing a bit of hope.

Can anyone just plop the stuff down in a super easy to understand relationship hierarchy that clearly delineates what builds on what?

  • 2
    $\begingroup$ This is a great great good question (+2). $\endgroup$ Sep 29 '19 at 22:36
  • $\begingroup$ Look at Eric Wofsey's answer to a similar question I asked. Well actually, my question was different, but I think his answer will satisfy you. $\endgroup$
    – saulspatz
    Sep 29 '19 at 23:08
  • $\begingroup$ We have numbers and we have theories studying them. Peano's axioms is the "best choice" : it can be perfectly formalized with second order logic. Unfortunately, SOL has some "difficulties" and thus we can use first-order theory of arithmetic: it has limitations, but it is very simple to manage it. In $\mathsf {ZFC}$ (and far less : the theory of Hereditary Finite Sets is enough) we can define a mathematical structure that is a perfect proxy for the natural numbers. $\endgroup$ Oct 1 '19 at 8:17
  • $\begingroup$ @MauroALLEGRANZA Is the version of Peano's Axioms in Tao's Analysis Vol 1 first order or second order? $\endgroup$
    – user709833
    Oct 1 '19 at 13:34
  • $\begingroup$ Tao's exposition is informal; but basically it is FOL (see Remark 2.1.10, page 19) $\endgroup$ Oct 1 '19 at 13:47

Peano's name is attached to two different theories about the natural numbers, which unfortunately don't always have clearly different names. The following convention is fairly common, though:

  • The "Peano axioms" is a second-order theory, which just describes the successor function and a general induction axiom. With some amount of set theory as a background feature of the logic, we can then define addition and multiplication without needing specific axioms for them.

  • "Peano Arithmetic" is a first-order theory, developed long after Peano's time as a "best-effort" first-order approximation of the second-order Peano axioms. It has specific axioms for the successor function and addition and multiplication, and an induction axiom schema that only works for properties that can be expressed in its first order language of successor+addition+multiplication.

Peano Arithmetic is what is usually meant by just the abbreviation PA. (Note capital A and no "the" for PA).

Because the induction axiom in Peano Arithmetic is not as strong as the full second-order induction axiom, the theory is weaker -- it has models that are not isomorphic to the usual $\mathbb N$. (It is hard-to-impossible to describe one of these non-standard models; we just have an existence proof for them. It depends crucially on the fact that first-order logic is complete: every consistent theory has a model. This is not true about the standard semantics for second-order logic, which is why the second-order axioms are stronger).

Despite being weaker, first-order PA has a lot more theoretical interest, because first-order logic is a lot better behaved than second-order.

For "don't functions require ZFC?", see When does the set enter set theory? or perhaps What is the dependency hierarchy in foundational mathematics?.

  • $\begingroup$ I don't think this really answers the question, a lot of this is just restating what I already mentioned/am aware of, in the OP. $\endgroup$
    – user709833
    Sep 30 '19 at 3:18
  • $\begingroup$ @user709833: To me the question looked like you were asking why different sources describe Peano's system as alternately first-order and second-order, because you were not aware those sources speak about two different systems. If you did know that, I'm afraid I don't understand what the question is. $\endgroup$ Sep 30 '19 at 8:35
  • $\begingroup$ Perhaps this will be helpful too. $\endgroup$ Sep 30 '19 at 8:48
  • $\begingroup$ My question is basically the same as the second link you posted and I have read that answer (and most of yours on the subject), I just have a hard time understanding them. When we define natural numbers, what are we using? PA? Peano's axioms? ZFC? I understand that ZFC axiomatically defines how sets work, but then when I see Peano's axioms or PA they both seem to rely on sets already? And why does PA "overshadow" Peano axioms despite being weaker? I also see FOL using functions, which are defined in terms of sets... it's all so confusing and seemingly circular. $\endgroup$
    – user709833
    Sep 30 '19 at 14:21
  • 1
    $\begingroup$ @user709833: No. When we do ordinary mathematics we're not assuming any formal foundational system "powering something underneath". Mathematics works just fine without them, as it did for thousands of years before any of the formal foundational systems were even thought of. The formal systems are models of ordinary mathematical thought, not what mathematical thought "really is". Read this! $\endgroup$ Oct 1 '19 at 23:19

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