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Is adding abstractions (in the below mentioned manner) over $ZFC$ inconsistent? if not then does it result in increment of consistency strength?

The idea is to add a new primitive binary relation "is abstracted from", denoted as: $``\curvearrowleft"$, to the language of set theory (first order logic with equality and membership) and we add the following extensionality and comprehension axioms for abstractions.

  1. Extensional abstractions: $\forall \zeta \eta \ [\forall x (\zeta \curvearrowleft x \leftrightarrow \eta \curvearrowleft x) \to \zeta=\eta]$

In English: Abstractions abstracted from the same objects are identical.

  1. Abstracting scheme: If $R$ is a binary relation symbol defined after a formula in the language of set theory that doesn't use the symbol $``\zeta"$, then:

$ R \text{ is equivalence relation } \to \forall x \ \exists \zeta \ \forall y (\zeta \curvearrowleft y \leftrightarrow y \ R \ x)$

is an axiom.

So to re-iterate my questions:

  1. Is the above addition on top of $ZFC$ consistent?

  2. If Yes, would it increase the consistency strenght or not?

  3. if we unleash the abstracting schema for abstractions to come from any equivalence relation, i.e. just remove the restriction to language of set theory; would that result in an inconsistent theory?

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  • $\begingroup$ My gut feeling is it would do nothing to consistency strength because there's not really any interaction with with $\in$. I don't see how you'd be able to prove any $\in$-sentences you couldn't prove before. $\endgroup$ – Malice Vidrine May 5 at 20:06
  • $\begingroup$ @MaliceVidrine, I'm particularly suspicious of 3. I think it might lead to an inconsistency? $\endgroup$ – Zuhair May 5 at 21:14
  • $\begingroup$ All the weakened version of 2 says, in conjunction with 1, is that a thing exists for every one-parameter definable class, right? And $\curvearrowleft$ can't appear in $R$? A model of this should be trivial to cook up from a model of the set theory. $\endgroup$ – Malice Vidrine May 5 at 22:57

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