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Academian comments on Second-Order Logic: The Controversy - Less Wrong
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So after reading that, I don't see how it could be true even in the sense described in the article without violating Well Foundation somehow, but what it literally says at the link is that every model of ZFC has an element which encodes a model of ZFC, not is a model of ZFC, which I suppose must make a difference somehow - in particular it must mean that we don't get A has an element B has an element C has an element D ... although I don't see yet why you couldn't construct that set using the model's model's model and so on. I am confused about this although the poster of the link certainly seems like a legitimate authority.
But yes, it's possible that the original paragraph is just false, and every model of ZFC contains a quoted model of ZFC. Maybe the pair-encoding of quoted models enables there to be an infinite descending sequence of submodels without there being an infinite descending sequence of ranks, the way that the even numbers can encode the numbers which contain the even numbers and so on indefinitely, and the reason why ZFC doesn't prove ZFC has a model is that some models have nonstandard axioms which the set modeling standard-ZFC doesn't entail. Anyone else want to weigh in on this before I edit? (PS upvote parent and great-grandparent.)
Here's why I think you don't get a violation of the axiom of well-foundation from Joel's answer, starting from way-back-when-things-made-sense. If you want to skim and intuit the context, just read the bold parts.
1) Humans are born and see rocks and other objects. In their minds, a language forms for talking about objects, existence, and truth. When they say "rocks" in their head, sensory neurons associated with the presence of rocks fire. When they say "rocks exist", sensory neurons associated with "true" fire.
2) Eventually the humans get really excited and invent a system of rules for making cave drawings like "∃" and "x" and "∈" which they call ZFC, which asserts the existence of infinite sets. In particular, many of the humans interpret the cave drawing "∃" to mean "there exists". That is, many of the same neurons fire when they read "∃" as when they say "exists" to themselves. Some of the humans are careful not to necessarily believe the ZFC cave drawing, and imagine a guy named ZFC who is saying those things... "ZFC says there exists...".
3) Some humans find ways to write a string of ZFC cave drawings which, when interpreted --- when allowed to make human neurons fire --- in the usual way, mean to the humans that ZFC is consistent. Instead of writing out that string, I'll just write <ZFC is consistent> in place of it.
4) Some humans apply the ZFC rules to turn the ZFC axiom-cave-drawings and the cave drawing <ZFC is consistent> into a cave drawing that looks like this:
"∃ a set X and a relation e such that <(X,e) is a model of ZFC>"
where <(X,e) is a model of ZFC> is a string of ZFC cave drawings that means to the humans that (X,e) is a model of ZFC. That is, for each axiom A of ZFC, they produce another ZFC cave drawing A' where "∃y" is always replaced by "∃y∈X", and "∈" is always replaced by "e", and then derive that cave drawing from the cave drawing "<ZFC axioms> and <ZFC is consistent>" according to the ZFC rules.
Some cautious humans try not to believe that X really exists... only that ZFC and the consistency of ZFC imply that X exists. In fact if X did exist and ZFC meant what it usually does, then X would be infinite.
4) The humans derive another cave drawing from ZFC+<ZFC is consistent>:
"∃Y∈X and f∈X such that <(Y,f) is a model of ZFC>",
6) The humans derive yet another cave drawing,
"∃ZeY and geX such that <(Z,g) is a model of ZFC>".
Some of the humans, like me, think for a moment that Z∈Y∈X, and that if ZFC can prove this pattern continues then ZFC will assert the existence of an infinite regress of sets violating the axiom of well-foundation... but actually, we only have "ZeY∈X" ... ZFC only says that Z is related to Y by the extra-artificial e-relation that ZFC said existed on X.
I think that's why you don't get a contradiction of well-foundation.