The following theory is another way of dealing with naive comprehension. It uses the double extension principle, broadly speaking similar to what's used in Double Extension Set Theory of Andrzej Kisielewicz (1998), but possibly simpler? I'm deliberately presenting a rather possibly strong version of this theory, which might even turn to be inconsistent. I also had presented a version of this approach to tag alternative set theories of Math.StackExchange, which seems to have a simply written axioms also. However the version presented here is bolder!

Language: first order logic

Primitives: Equality $=$, and Two set membership relations $\in_1; \in_2$

Extensionality: $i=1,2: \forall x,y [\forall z (z \in_i x \leftrightarrow z \in_i y ) \to x=y]$

Define: $uniform(x) \iff \forall y (y \in_1 x \leftrightarrow y \in_2 x)$

Comprehension: If $\phi^1$ is a formula not using $\in_2$, in which $x$ doesn't occur free, then all closures of:$$ \exists x \forall y (y \in_2 x \leftrightarrow \phi^1)$$; are axioms.

Uniformity: $\forall x [uniform(x) \leftrightarrow \forall y \in_2 x (uniform(y))]$

Induction: over naturals for any formula $\phi$.

This theory would interpret and prove the consistency of Zermelo set theory, over the realm of uniform sets. However, it does go far beyond Zermelo.

Question: Is this theory subject to some forms of the known paradoxes of set theory?

The second question is rather not very specific, but it presents itself. The matter is that it appears, generally speaking, that such approach, i.e. using double extensions, can be effective in evading paradoxes and one seemingly can develop strong theories with rather simple axiomatization that springs directly from naive contemplation of this method. Why then this approach was not favored generally? If one see the cited article of Randall Holmes on alternative set theories, he presents it at the end and only as a curiosity. He mentions that it is hard to contemplate. But is this the only deterrent? or there are certain technical drawbacks to it? What are those?


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