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I am assuming the standard Peano-Axioms, which can be found here https://en.wikipedia.org/wiki/Peano_axioms under "Formulation". Does one assume that a set of objects called $\mathbf{N}$ exists? A set is a collection of objects according to naive set theory, therefore, if not assuming that such objects exists, where are they coming from? I see it often that these axioms are used to define a set of natural numbers and that one then shows that any two sets that satisfy those axioms are isomorphic, which allows to speak of the natural numbers. But again, if not assuming that a set with objects exists, how can I construct a set?

I could write down things like $\{0,I,II,III,...\}$ which would satisfy the PA, but is this a set? What are its objects? I think one would have to assume something like: There exists a set $A$ that satisfies the PA and then define $I:=s(0)$, $II:=s(1)$ and so on. This would fix the problem, since in this case I would have assumed objects. However, this would only give me one set of objects. Meaning, I could also model $\{0,1,2,3,...\}$ by defining it the same way as above. This however would lead to $\{0,1,2,3,...\}=\{0,I,II,III,...\}$. Is it possible to have two models that are "only" isomorphic, meaning not equal but isomorphic? Or more strictly, is it possible to have two models with completely different objects?

As far as I see it, I need to assume the existence of some kind of objects, and then give those objects names such as $1,2,3$ or $0,I,II,III$, however, can I reverse this by giving names such as $0,I,II,III,...$ and find objects that model this set (in this case this should be possible when assuming the existence of some sort of natural numbers)? Is this possible for every set? Doesn't this also lead to ambiguity, since one does not know which model is meant?

MaxH
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I think the canonical way to construct the natural numbers today is to use the so called "Von Neumann encoding". We start with the empty set and define this to be $0$. The empty set exists according to the axioms of ZF-set theory (the explanation can be found on wikipedia for example). Now define $$1 := \{0\} = \{\{\}\}, 2 := \{0,1\} = \{\{\}, \{\{\}\}\}, 3 := \{0,1,2\},...$$ This way we end up with a set-theoretic model of the natural numbers (and 0) such that the subset relation on the sets directly gives us the order relation on the naturals.

To answer your other questions: yes it is possible to have multiple models of the same thing with entirely different objects. For the natural numbers we could for example also define $0$ to be the empty set, $1:=\{0\}, 2:=\{1\}$ and so on which would be different but also give us a series of sets that we could then put the structure of the natural numbers on in some way. So one approach is to just model (nearly) everything as sets all the way down, this is (afaik) the basic idea behind set theory as a foundation for mathematics.

If you have some countable set, then you can always model it as the natural numbers since being countable explicitly means being bijective to the naturals. For bigger sets I'm not quite sure, but I'd assume that we could always work with the power-set in this case (using that the power-set is always strictly larger than some set). So by saying "for any set $S$ there exists some ordinal number $\omega$ such that there's a bijection between $\omega$ and $S$" you can model every set as some ordinal which is (or can be modeled as) ultimately a set.

On a set-theoretic level this certainly leads to ambiguity, but the higher up you go the less you actually rely on this set-theoretic footing of mathematics, since you don't want to prove that a theorem about some thing $A$ is true for some specific model of $A$, but rather the concept of $A$. So we basically work with all those isomorphic models at once.

You may find very elaborated answers to your questions in Hamkins' Lectures on the philosophy of mathematics.

SV-97
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  • Thanks! So as far as I see it, the problem that occurs regarding the existence of objects is solved by using the emptyset which is assumed to exist and then combinations of sets containing this as "all forms of objects", right? Furthermore, how can I see that the model does not make a difference the higher up I go, as you say it? – MaxH Sep 27 '21 at 09:51
  • Yes, to my understanding (I'm not a set theorist or logician) that's how it works.

    Oh because we simply don't use the underlying set properties anymore. For example we can define functions, relations etc. as special sets and show that they have the properties we expect from them. And then we simply ignore that we defined them as sets and rely solely on those properties that to our understanding define what it means to be a function, relation etc.. Or we can define the real numbers via different constructions (e.g. degekind cuts or cauchy sequences) but we wouldn't use these models in proofs

     – SV-97 Sep 27 '21 at 10:05
  • about real numbers. Rather we use defining properties like completeness etc. – SV-97 Sep 27 '21 at 10:06
  • Incidentally, your post and comments touch nicely on the same points in the posts I linked in a comment on the question. That is, whenever we prove a theorem about naturals, we are usually in fact proving a theorem about every model of PA. And whenever we prove a theorem about reals, we are actually proving a theorem about every Dedekind-complete ordered field. – user21820 Nov 20 '21 at 14:00