The Universe of Discourse

My mistake about errors in the presentation of axiomatic set theory

[ Content warning: highly technical mathematics ]

A couple of weeks ago I claimed:

Many presentations of axiomatic set theory contain an error

I realized recently that there's a small but significant error in many presentations of the Zermelo-Frankel set theory: Many authors omit the axiom of the empty set, claiming that it is omittable. But it is not.

Well, it sort of is and isn't at the same time. But the omission that bothered me is not really an error. The experts were right and I was mistaken.

(Maybe I should repeat my disclaimer that I never thought there was a substantive error, just an error of presentation. Only a crackpot would reject the substance of ZF set theory, and I am not prepared to do that this week.)

My argument was something like this:

• You want to prevent the axioms from being vacuous, so you need to be able to prove that at least one set exists.

• One way to do this is with an explicit “axiom of the empty set”: $$\exists Z. \forall y. y\notin Z$$

• But many presentations omit this, remarking that the axiom of infinity (“!!A_\infty!!”) also asserts the existence of a set, and the empty set can be obtained from that one via specification.

• The axiom of infinity is usually stated in this form: $$\exists S (\varnothing \in S\land (\forall x\in S) x\cup\{x\}\in S).$$ But, until you prove that the empty set actually exists, it is not meaningful to include the symbol !!\varnothing!! in your axiom, since it does not actually refer to anything, and the formula above is formally meaningless.

I ended by saying:

You really do need an explicit axiom [of the empty set]. As far as I can tell, you cannot get away without it.

Several people tried to explain my error, pointing out that !!\varnothing!! is not part of the language of set theory, so the actual formal statement of !!A_\infty!! doesn't include the !!\varnothing!! symbol anyway. But I didn't understand the point until I read Eike Schulte's explanation. M. Schulte delved into the syntactic details of what we really mean by abbreviations like !!\varnothing!!, and why they are meaningful even before we prove that the abbreviation refers to something. Instead of explicitly mentioning !!\varnothing!!, which had bothered me, M. Schulte suggested this version of !!A_\infty!!:

$$\exists S (\color{darkblue}{(\exists Z.(\forall y. y\notin Z)\land (Z \in S))} \\ \land (\forall x\in S) x\cup\{x\}\in S).$$

We don't have to say that !!S!! (the infinite set) includes !!\varnothing!!, which is subject to my quibble about !!\varnothing!! not being meaningful. Instead we can just say that !!S!! includes some element !!Z!! that has the property !!\forall y.y\notin Z!!; that is, it includes an element !!Z!! that happens to be empty.

A couple of people had suggested something like this before M. Schulte did, but I either didn't understand or I felt this didn't contradict my original point. I thought:

I claimed that you can't get rid of the empty set axiom. And it hasn't been gotten rid of; it is still there, entire, just embedded in the statement of !!A_\infty!!.

In a conversation elsewhere, I said:

You could embed the axiom of pairing inside the axiom of infinity using the same trick, but I doubt anyone would be happy with your claim that the axiom of pairing was thereby unnecessary.

I found Schulte's explanation convincing though. The !!A_\infty!! that Schulte suggested is not a mere conjunction of axioms. The usual form of !!A_\infty!! states that the infinite set !!S!! must include !!\varnothing!!, whatever that means. The rewritten form has the same content, but more explicit: !!S!! must include some element !!Z!! that has the emptiness property (!!\forall y. y\notin Z!!) that we want !!\varnothing!! to have.

I am satisfied. I hereby recant the mistaken conclusion of that article.

Thanks to everyone who helped me out with this: Ben Zinberg, Asaf Karagila, Nick Drozd, and especially to Eike Schulte. There are now only 14,823,901,417,522 things remaining that I don't know. Onward to zero!

[ Addendum 20220224: A bit more about this. ]

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