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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?

Xander Henderson
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user709833
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    This is a great great good question (+2). – hamam_Abdallah Sep 29 '19 at 22:36
  • 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. – saulspatz Sep 29 '19 at 23:08
  • 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. – Mauro ALLEGRANZA Oct 01 '19 at 08:17
  • @MauroALLEGRANZA Is the version of Peano's Axioms in Tao's Analysis Vol 1 first order or second order? – user709833 Oct 01 '19 at 13:34
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    Tao's exposition is informal; but basically it is FOL (see Remark 2.1.10, page 19) – Mauro ALLEGRANZA Oct 01 '19 at 13:47

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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?.

hmakholm left over Monica
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  • 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. – user709833 Sep 30 '19 at 03:18
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    @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. – hmakholm left over Monica Sep 30 '19 at 08:35
  • Perhaps this will be helpful too. – hmakholm left over Monica Sep 30 '19 at 08:48
  • 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. – user709833 Sep 30 '19 at 14:21
  • @user709833: Ultimately the natural numbers are defined as "the counting numbers you learned about in primary school". We have nothing better than that in mathematics, though we certainly have things that sound more obfuscatively learned. PA and/or the Peano axioms are best thought of as "some claims we're pretty sure are true about the natural numbers we already know and love, and which seem to be a good basis for using our nice systems for formal proofs to reason about those natural numbers". That is a worthy task for them, but they do not work well as an authoritative definition. – hmakholm left over Monica Sep 30 '19 at 15:46
  • So when we are using real numbers, doing algebra, calculus, etc are we usually assuming Peano's axioms or PA "powering" everything underneath? – user709833 Oct 01 '19 at 22:30
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    @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! – hmakholm left over Monica Oct 01 '19 at 23:19
  • Sure but if I do something in calculus and someone wants a proof that it all works I will need to resort to those frameworks to show that it's all true, and depending on how far down I go (e.g. real numbers, etc) I may need to resort to some kind of foundational system would I not? – user709833 Oct 02 '19 at 00:48
  • @user709833: No you won't. Most proofs in mathematics are not carried out in formal systems. The only time anyone even contemplates writing down proofs in formal systems is when they're convincing themselves that the formal systems actually can work as models of ordinary mathematical reasoning. But the model is not the thing itself, and formal proofs in any foundational system is NOT "what ordinary mathematics really is". – hmakholm left over Monica Oct 02 '19 at 00:50
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    Well I mean for example, raising a real number to a real power or something similar, doesn't proving such results require us to agree on what a real number is and how it works, how it's defined, etc? – user709833 Oct 02 '19 at 00:58
  • @user709833: I think most real analysis texts work from a set of (informally stated) axioms of the real numbers, which tells how the real numbers behave, but not "what they really are". Some do present a construction of the reals based on the naturals and some set theory, but as a rule they never even attempt to define "what the natural numbers really are", nor even to present axioms for them. You're supposed to know how the counting numbers work, from primary school! – hmakholm left over Monica Oct 02 '19 at 01:04
  • Maybe I am being imprecise with my language but I don't mean some metaphysical claim like "what they are," more like "whatever they are, here's some descriptors of how they behave," yes. So for a real number I am referring to the formalization of how they work and the rules for exponentiating them etc. – user709833 Oct 02 '19 at 01:10
  • @user709833: You're wrong when you expect that there's a foundational formalization in play at all when we're doing ordinary mathematics. Read this already, dammit! – hmakholm left over Monica Oct 02 '19 at 01:12
  • I have read that answer many times, even before this week! Maybe an easier example to ask: Let's say you have written some proof for something. I look at your work and say, "Hey right here you use this thing called the 'distributive law' $a(b + c) = ab + ac$. That's kind of peculiar, how do you know for sure you can do that?" – user709833 Oct 02 '19 at 01:14
  • @user709833: Then I tell you that my work assumes we're working in a context where the distributive law holds. If you're in a different situation where the distributive law doesn't hold, then my work may not apply to it -- caveat emptor. – hmakholm left over Monica Oct 02 '19 at 01:15