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ZFC is the most well known set theory which is considered by many as the foundation of mathematics but I am confused to understand it intuitively. Most of us have a clear understating of empty set and universal set (set of all sets or an entity which contains all sets). Suppose that we have a model of ZFC (Model I) with the usual interpretation of all symbols of the theory (such as membership ∈) then we can construct another theory which I call it ZFC*. Consider the model of ZFC :

Model I : we interpret x∈y just as usual x in y.

In this model, the existence of empty set axiom ∃x∀y~(y∈x) have a usual intuitive meaning.

The axioms of ZFC* are just the axioms of ZFC by replacing the relation ∈ with ∉. It is clear that ZFC* is consistent if and only if ZFC is consistent, so we have a model of ZFC* where I call it (Model II)

ZFC* : The axiom of the existence of empty set is transformed to the existence of the universal set. We have to change all other axioms appropriately.

Model II : In this model we have the universal set.

I think the Model II is also a model of ZFC if we interpret x∈y as x not in y.

My question : Empty set in one model is just the universal set in another model(please look at the whole story intuitively), I think it is against our common sense, isn't it??

I may have made a mistake in explaining the problem and I welcome any explanation.

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    Is it true that "ZFC* is consistent if and only if ZFC is consistent"?
    – Frank
    Mar 31, 2023 at 1:19
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    By the axioms of regularity and pairing, no set is an element of itself. So you do not get a model of ZFC by interpreting ∈ as ∉.
    – Conifold
    Mar 31, 2023 at 5:48
  • Perhaps it might be easier for everyone to see the theory in full if you were to sketch out your proposed ZFC* axiom schema explicitly? Intuitively, just changing what syntactic symbol you use shouldn’t change the interpreting model, but perhaps that’s not quite what you’re getting at.
    – Paul Ross
    Mar 31, 2023 at 6:39
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    I don't understand the question. Empty set in one model is not the universal set in another model. I'm not convinced the model you describe would work, but if it did, all this means is that the symbols such as set membership and the empty set symbol have different denotations with different models. Which is what you would expect. Mar 31, 2023 at 9:24
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    Well, I don't think anyone claims that ZFC is the most intuitive set theory. The most intuitive set theory is inconsistent. Mar 31, 2023 at 10:05

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"I think Model II is a model of ZFC". call Mod II's underlying structure K.

Suppose K satifies ZFC*, in particular empty*. So there is a universal set as you have said. Now, use separation to generate Russell's paradox.

So we had better hope that the consistency of ZFC doesn't hinge on the consistency of Model II.

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  • No, Russell paradox does not generate in ZFC. It is a theorem in ZFC (that there is no universal set~∃y∀x(x∈y)) which is true in all models of ZFC (please note that model II does not leads to Russell paradox, universal set in that model is a model depended notion and not a syntactic one ). In ZFC* Russell paradox implies that there is no empty set (instead of universal set).
    – Arian
    Mar 30, 2023 at 23:06
  • you've said it yourself: "model two has a universal set. also, it is a theorem in ZFC that there is no universal set." This is exactly tantamount to the fact that all models of ZFc do not have a universal set. So model II is not a model of ZFC.
    – emesupap
    Mar 30, 2023 at 23:29
  • there is no universal set in ZFC means that ∃y∀x(x∈y) is not a theorem. when I said that we have a universal set in model II, I used universal set as an intuitive notion.
    – Arian
    Mar 30, 2023 at 23:40

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