I don’t see how it could be true even in the sense described in the article without violating Well Foundation somehow
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 in place of it.
4) Some humans apply the ZFC rules to turn the ZFC axiom-cave-drawings and the cave drawing 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 ” and ” 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+:
“∃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.
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 in place of it.
4) Some humans apply the ZFC rules to turn the ZFC axiom-cave-drawings and the cave drawing 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 ” and ” 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+:
“∃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.