# Contradiction to axiom of foundation

Consider the usual language and axioms of ZF. Now add constants $x_1, x_2, \dots$ to the language together with the axioms $x_2\in x_1, x_3\in x_2, \dots$ to form a new theory. Then by the compactness theorem, since every finite subset of the axioms has a model, the new theory has a model. But doesn't the set {$x_1, x_2, \dots$} have no $\in$-minimal element, contradicting the axiom of foundation?

I'm thinking that maybe {$x_1, x_2, \dots$} is not necessarily a set in the model, but isn't it by replacement? Maybe not, since we don't necessarily have a copy of $\mathbb{N}$ in our model... Could someone clarify this please?

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You have already answered your question yourself: the set $\{x_1, x_2, \ldots\}$ cannot exist in the model, since this would violate the Foundation Axiom. You cannot get this set from Replacement, since there is no definable function having this set as its range, required to invoke the Replacement Axiom.
I went in and fixed some TeX. At issue is that Markdown, which processes the page before jsMath gets to it, thinks that { is a special character, and that \{ is what you'd type to escape it. So Markdown dutifully strips off the  but does no further processing to the { before passing the text to the webbrowser. Then jsMath (which runs in the browser) comes along and sees just an open brace, so ignores it when rendering the code as LaTeX. There are two fixes: either use \\{, which looks bad in TeX, or protect any string of code by back-ticks, so that Markdown knows that it's code. –  Theo Johnson-Freyd Apr 6 '10 at 20:23
Theo, thanks for fixing that. Actually, jsmath doesn't work for me at all on most of the machines I use regularly, and my usual MO experience is to be looking at a lot of $'s and \'s, which I don't really like. I suspect that a significant but silent fraction of the MO user population is in my situation, and that MO should discourage excessive TeX usage for this reason. – Joel David Hamkins Apr 6 '10 at 20:31 You can also write \lbrace and \rbrace instead of \\{, for example$\lbrace x \mid x \not\in x\rbrace\$. –  Andrej Bauer Apr 6 '10 at 22:43