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If we start with axioms of $\sf ZF$ replace the axiom of Infinity with an axiom stating the existence of a set that doesn't have an injection to a von Neumann natural, and replace the axiom of Power sets by the axiom of unbounded cardinality, that is for each set $x$ there is a set $y$ such that $x$ is injective to $y$ but $y$ not injective to $x$.

Is it consistent with this theory to have all its sets being Dedekind finite?

Can this theory interpret $\sf ZFC$, through interpreting $L$ within it?

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    $\begingroup$ The answer to the question in the title, which is quite different from the question in the question, is yes: e.g., take $\mathrm{ZF_{fin}+Con_{ZFC}}$. $\endgroup$
    – Emil Jeřábek
    Commented Sep 13 at 11:50
  • $\begingroup$ @EmilJeřábek, OK, I've changed the title to better suit the question. $\endgroup$
    – Zuhair Al-Johar
    Commented Sep 13 at 12:26
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    $\begingroup$ What do you mean by constructing $L$ if you don't have $\omega$? $\endgroup$
    – Joel David Hamkins
    Commented Sep 13 at 12:32
  • $\begingroup$ @JoelDavidHamkins, I thought $\omega$ could be interpreted, even though not provable in the system. $\endgroup$
    – Zuhair Al-Johar
    Commented Sep 13 at 12:49
  • $\begingroup$ But then you wouldn't be constructing $L$, but interpreting a model. $\endgroup$
    – Joel David Hamkins
    Commented Sep 13 at 14:10

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