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Is it possible to squeeze the hereditarily Cantorian world "$\sf H_{Cant}$" of $\sf NFU$ be the well founded world $\sf WF$ of $\sf NFU$. Moreover, can we have $\sf WF$ of $\sf NFU$ to satisfy all rules of $\sf ZFC$.

In symbols:

Is "$\sf NFU + (WF \models ZFC) + H_{Cant} = WF$" consistent?

Where a well founded set is a set belonging to the cumulative hierarchy of pure sets. Alternatively one can work in $\sf NFU^V$ where all Ur-elements are co-extensional with the universe $V$, and use the usual definition of well founded sets.

If this is consistent, then what would be the consistency strength of such extension of $\sf NFU$?

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    $\begingroup$ What is the precise definition of $\mathsf{WF}$ you are using? $\endgroup$
    – James E Hanson
    Commented Mar 27, 2022 at 23:28
  • $\begingroup$ What is the definition of cumulative hierarchy you are using? What is the definition of well-founded set? This is important for your question because different definitions of well-foundedness fail to be equivalent in $\mathsf{NFU}$. $\endgroup$
    – James E Hanson
    Commented Apr 24, 2022 at 22:56

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