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What's the relationship between ZFC and Peano axioms? Are they overlapping, complementary or separate?

I'm particularly wondering whether natural number stuff like order can expressed in ZFC at all. Perhaps there's some way to associate "smaller" and "larger" numbers as cardinals or something.

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  • $\begingroup$ It seems that PA could be expressable in ZFC. math.stackexchange.com/a/459050/248602 $\endgroup$
    – mavavilj
    Aug 19, 2018 at 17:04
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    $\begingroup$ In layman's terms they can both express arithmetic - if it's arithmetic, you can write it with them. Everything PA can express, ZFC can too. But ZFC goes further. For example, ZFC can prove that every Goodstein sequence terminates, while PA can't. $\endgroup$
    – it's a hire car baby
    Aug 19, 2018 at 17:12
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    $\begingroup$ @RobertFrost superb comment (+1) Even more, PA cannot prove that any function growing at least at level $\large f_{\epsilon_0}$ , is complete. ZFC goes far beyond the $\large f_{\Gamma_0}$-level, so is much much more powerful. $\endgroup$
    – Peter
    Aug 19, 2018 at 17:14
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    $\begingroup$ Another very interesting link : Every Goodstein-sequence terminates if and only if PA is consistent showing because of Goedel's second incompleteness-theorem, that PA is unable to prove the Goodstein-theorem. $\endgroup$
    – Peter
    Aug 19, 2018 at 17:21

1 Answer 1

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The two system are not related in a direct way. Peano axioms come to axiomatize arithmetic, whereas Zermelo–Fraenkel (with or without choice) come to axiomatize the properties of the set theoretic universe.

Neither pretends to "express what is a natural number". For arithmetic, it is us who said that we have a concept of a natural number, and it seems to satisfy these properties. And the properties were later distilled into Peano axioms, second-order and then the first-order version.

Set theory can be used to encode the natural numbers, usually this is done via the von Neumann ordinals: the finite ordinals satisfy all the axioms of Peano. But since ZFC provides us with an infinite set, which might as well be the set of all finite ordinals (and if not, then it provides us with the means of proving that the collection of all finite ordinals is in fact a set), it lets us do more. For example, since there is a model of Peano axioms, the theory is proved to be consistent, something you cannot do from Peano itself.

Even more than just that. ZFC lets us interpret the second-order Peano axioms. And in that sense, it is much stronger in defining what is a natural number. Of course, encoding "objects" into "sets" is not by any means unique. There are many ways we can think about the natural numbers as being sets, some are more natural and others less, but they are all valid methods.

What ZFC does prove, however, is that any two models of second-order PA are isomorphic in a unique way. And in that sense, ZFC can actually say what is a natural number.

One last thing, perhaps, is the fact that replacing the Axiom of Infinity with its negation, one transforms ZFC into a theory which is equivalent to first-order Peano axioms in a very strong sense.

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  • $\begingroup$ Thanks for the answer! I was wondering why a set theory can be used for studying a number theory or generally a mathematical theory? "Set theory can be used to encode the natural numbers, usually this is done via the von Neumann ordinals: the finite ordinals satisfy all the axioms of Peano. " Does the encoding convert formulas in the language of the number theory to formulas in the language of the set theory, in some (what?) way so that we can convert back results about the set theory to results about the number theory? What kind of theory can we study indirectly by a set theory in such a way? $\endgroup$
    – Tim
    Mar 12, 2021 at 10:44
  • $\begingroup$ Yes, set theory interprets number theory, and indeed, most of mathematics. And this interpretation is "simple enough" that we can almost pretend it isn't there. This is one of the reason that set theory is a very successful foundation to mathematics. Moreover, if you think about number theory as studying the set of the natural numbers and its first/second-order properties, then you already accept the idea it is a set, so it lies in some universe of set theory to begin with. $\endgroup$
    – Asaf Karagila ♦
    Mar 12, 2021 at 11:04
  • $\begingroup$ Can you point me to some introductory readings about the way of using a set theory to study a mathematical theory (a number theory)? $\endgroup$
    – Tim
    Mar 12, 2021 at 11:25
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    $\begingroup$ There's no book about how to study number theory using set theory. Just like there's no book explaining how to study Java programming using machine code. Instead you just need to understand how the machinery works, and everything falls into place on its own. For that matter, understanding what is a language, theory, structure, and model are usually enough. Once you understand that these are all taken as sets, and how set theory internalises induction and recursion, you see that that's it. $\endgroup$
    – Asaf Karagila ♦
    Mar 12, 2021 at 11:52
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    $\begingroup$ You don't use set theory to study number theory. You study number theory, it just takes place inside set theory, very much in the same way that you can write a Java program and run it on your computer, but it doesn't mean that you're using machine code to write, or even understand, your code. $\endgroup$
    – Asaf Karagila ♦
    Mar 12, 2021 at 11:53

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