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Wed, 23 Feb 2022
My mistake about errors in the presentation of axiomatic set theory
[ Content warning: highly technical mathematics ] A couple of weeks ago I claimed:
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:
I ended by saying:
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:
In a conversation elsewhere, I said:
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. ] [Other articles in category /math] permanent link |