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When we say $M$ is a model of ZFC, we tacitly assume $M$ to be a set model. But wait, $M$ can't be a set in the very ZFC by the axiom of foundation. Then where is $M$ a set?

To say/verify something is a set, we need a set theory. So, does this means that when we say $M$ is a model of some theory $\Gamma$, we are actually saying we have a set theory $\Gamma_{set}$, then we assume $\Gamma_{set}$ to be consistent, and then we can manage to construct within $\Gamma_{set}$ a set which satisfies $\Gamma$?

If that is the case, it looks kind of weird, because that seems to imply that when we are talking about models, consistency or the like, we are actually working inside a certain set theory. Besides, it also seems to allow the possibility that some theory is consistent in a working set theory (the set theory in which we are working) $\Gamma_1$ but inconsistent in another working set theory $\Gamma_2$.

I'm not a student majoring in math or something, and therefore has no professor or someone to turn to. Any help is welcomed! Thanks!

Michael
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    This confusion should be discussed at the beginning of any textbook on model theory. – GEdgar Sep 28 '22 at 12:22
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    First paragraph: no, the axiom of foundation has nothing to do with this. Second paragraph: no, we are not constructing a model, we are assuming we are given one (by an ineffable god, if you like). Third paragraph: yes, it can happen that a consistent set theory believes itself to be inconsistent. // I think it would be good for you to takes things slowly and think through what the mathematics is telling you carefully, rather than taking natural language explanations literally. – Zhen Lin Sep 28 '22 at 12:47
  • @ZhenLin About first paragraph: If $M$ is a set in ZFC, then since it is the Von Neumann universe, it belongs to itself, contradicting the axiom of foundation. About second paragraph: If we are given one set model, then what does the set mean here? – Michael Sep 28 '22 at 13:03
  • @GEdgar What model theory book did you read? I have just skimmed through the first chapters of the model theory books by David Marker and Chang C.C. separately. I read some introduction notes on model theory earlier. They all give terse definitions of a model and the like only at least in the beginning. They seems to not concern about this part of the matter and only care about the techniques. – Michael Sep 28 '22 at 13:10
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    @Michael why must M be the Von Neumann universe? It’s just some set that satisfies some properties. – spaceisdarkgreen Sep 28 '22 at 14:03
  • In the search box up there, try "model set theory" and look at what you get. Several previous questions are asking essentially the same thing as you ask. – GEdgar Sep 28 '22 at 17:38
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    @Michael There is no "the" von Neumann universe. At least, what $M$ believes to be "the" von Neumann universe is different from what you believe "the" von Neumann universe to be, if you see $M$ as a set-sized model. This is one of the things I mean when I say not to take natural language explanations literally. – Zhen Lin Sep 28 '22 at 22:07
  • I have attempted an answer here. – C7X Oct 02 '22 at 03:13

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