Question

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.

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??

Answer 1

Constructing ZFC* by simply inverting the membership relation in ZFC is unintuitive as it risks consistency, and challenges common set theory notions, since the axioms of ZFC are designed to avoid paradoxes that could arise with concepts like a 'universal set'.

Step-by-step explanation:

The notion of forming a dual theory like ZFC* by inverting the membership relation in ZFC (Model I), which leads to a universe with a universal set in ZFC* (Model II), indeed challenges common intuition about sets. In standard ZFC, the axioms forbid the existence of a universal set due to the risk of paradoxes such as Russell's Paradox. When constructing a new model like ZFC*, the relations and axioms of the theory must be notably distinct from those of ZFC to maintain a consistent framework, as simply inverting the membership relation does not yield a direct translation of concepts like 'empty set' to 'universal set.'

Intuitively comprehending these notions necessitates a shift in how we think about set membership and the nature of sets themselves. While in some logical systems such inversions might maintain consistency, in ZFC the axioms are meticulously constructed to avoid certain well-known paradoxes and problems.