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In my set theory course, we introduced the Axiom of Union.

For any set $x$ there exists a set $y$ s.t. all elements of $y$ are elements of some $z\in x$. Notation for this set is $y= \bigcup x$.

$\forall x\ \exists y \ \forall z (z\in y \leftrightarrow \exists w \ (z\in w \land w\in x))$

My professor then said

We define $\{x_1,...,x_{n+1}\}$ recursively as $\{x_1,...,x_n\}\cup\{x_{n+1}\}$.

This makes intuitive sense to me, but the idea of a "recursive definition" doesn't come up until way later in the course! We need the axiom of infinity to define things recursively on $\mathbb{N}$. This definition of the union was given in lecture 1. Showing recursion on $\mathbb{N}$ was in lecture 13!

If this definition doesn't work, we can't guarantee existence of sets of an arbitrary finite size. If I wanted to work with a set of $n$ elements I could easily just apply Pairing and Union and show it existed, but I can't generalise this without the recursiveness definition. Am I missing something obvious here? Or is this just an informal metalinguistic definition that we kinda have to work with?

Asaf Karagila
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Amitai
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  • There is some meta-theoretical use of recursion going on. Basically, you do math in a formal setting, a "petri-dish" sets are like little bacteria- meta-metamathematics is us talking objects in the petri dish. For example, you were probably introduced to your variables in set theory as being enumerated by natural numbers, $x_1$, $x_2$ etc... however natural numbers arent defined till later, using those very same indexed variables. So, what gives? The point is that we use our mathematical notions, to talk about actual objects in the formal theory, using intuition from outside the theory. – Michael Carey Mar 15 '24 at 02:53
  • So, we "talk" about the theory using notions that may not yet be developed in theory. This is important if you care about distnguishing between proofs within a model VS proofs about a model. Which is usually a grad school logic course topic. – Michael Carey Mar 15 '24 at 02:58
  • Your question has a similar spirit as my question https://math.stackexchange.com/q/4290850. It seems that there are "intuitively pre-existing concepts" which are formalized later. – Kritiker der Elche Mar 15 '24 at 09:25

1 Answers1

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There is a subtle point, so subtle and confusing, that it is often better to leave it out and let students ask for clarification than to confuse the entire class so early on.

When we do set theory, or really any sort of mathematics, we still work inside a universe. We simply study set theory as an object. This mathematical universe can be "material" (i.e., a larger universe of some set theory) or "syntactic" (i.e., manipulating strings of characters) but it has some kind of understanding of what is induction and recursion. After all, formulas that we use to write down things are often defined by recursion.

The difficulty arises when one is exposed to the fact that the meta-universe may disagree with the universe on the concept of the natural numbers. In other words, given a model of set theory inside a large universe of set theory, the model and the universe may very well disagree about the natural numbers and some of their properties. One thing, however, that we can say, is that the universe's natural numbers will always be represented inside the model, it just that the model can disagree on whether or not these are all of the natural numbers.

The recursion to define $\{x_0,\dots,x_n\}$ is done "externally". In the outside universe, and it will guarantee that there are finite sets of any size that the universe considers finite. Because of more complicated reasons, however, this guarantees that there are in fact finite sets of any finite size internally to the model.

You are correct that the Axiom of Infinity is needed when defining the natural numbers as an internal object, but at the end of the day, we don't actually need it to define the natural numbers as a concept and define by recursion over it, even internally to the universe. It's just that if the Axiom of Infinity is false in a given model, it happens to be that the natural numbers are exhaustive of the entire class of ordinals. This is fine, we can still use induction and recursion even in arithmetic theories such as Peano Arithmetic, and even weaker theories than that (although one has to start being more selective as to what can or cannot be inducted over). It turns out that $\sf ZF$ without the Axiom of Infinity (or rather, with its negation) is "roughly equivalent" to Peano. So in either case, recursion and induction are fine.

Asaf Karagila
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    Forgive me for how many questions I'm going to ask; don't worry about answering all of them, I can ask my professor after the vacation ends.

    You wrote: "You are correct that the Axiom of Infinity is needed when defining the natural numbers as an internal object, but at the end of the day, we don't actually need it to define the natural numbers as a concept and define by recursion over it, even internally to the universe."

    What's the difference between an internal object and a concept?

     – Amitai Mar 14 '24 at 13:52
  • Also, if we try to use set theory as a "basis" for mathematics to be built up on, taking a foundational approach to derive mathematics from it, does this now pose a problem (in which we can't talk about the meta-universe's recursiveness) or am I misunderstanding - I've only seen this kind of stuff at an undergrad level where it does not get into meta content whatsoever – Amitai Mar 14 '24 at 13:56
  • The concept of a natural number is something that "we understand" but is not a formal understanding. We then formalise it and prove that there is a unique object that satisfies the basic properties of the natural numbers. But this is like saying that there is a unique smallest number in a set. Which set you're taking will dictate if this smallest number is $0$ or $1$ or something else. Objects live inside your model, concepts live our shared understanding of how things "should be". – Asaf Karagila Mar 14 '24 at 16:33
  • Using set theory as a foundation of mathematics is fine. We can do that without a problem, but set theory itself is formalised in a larger universe which may very well be set theoretic, and yes, it is in fact turtles all the way down. See https://math.stackexchange.com/questions/121128/when-does-the-set-enter-set-theory and https://math.stackexchange.com/questions/linked/121128?lq=1 for more. – Asaf Karagila Mar 14 '24 at 16:35
  • If we're going to use techniques (such as recursion) that aren't defined in our object language, what's the point of some axioms? Why not say that you can just pair two elements? Why state it as the Axiom of Pairing? It seems like we're going to use intuitive ideas from the larger universe anyway... – Amitai Mar 14 '24 at 16:47
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    The best advice I can give you is to keep these thoughts in your back pocket. Study set theory, study maths, study more. Get back to them once you have a better grasp of these things. – Asaf Karagila Mar 14 '24 at 17:07
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    @Amitai usually the answer to the question " why do mathematicians bother with formalizing notion x" is the same. Because the defininition allows mathematicians to prove things about the notion, that otherwise wouldn't be provable. In the case of the axioms- one often wants to prove if some Class is a Model of the Axioms, which can be related to the complexity of the formula- of the rigourpusly defined statement. – Michael Carey Mar 15 '24 at 03:07