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The "injective continuum function hypothesis" (ICF) is the following statement.

ICF (Version 0). For all cardinal numbers $\kappa$ and $\nu$, we have $2^\kappa = 2^\nu \rightarrow \kappa = \nu.$

I really like this axiom, because its equivalent to:

ICF (Version 1). For all sets $X$ and $Y$, we have:

$$|\mathcal{P}(X)| = |\mathcal{P}(Y)| \rightarrow \mathcal{P}(X) \cong \mathcal{P}(Y)$$

(On the left we have equality of cardinal numbers; on the right, isomorphicness of posets.)

This means that, under ICF, cardinality is "more useful" than it might otherwise be, because all we need to know about $\mathcal{P}(X)$ and $\mathcal{P}(Y)$ to tell whether or not they're isomorphic as posets is their cardinality.

Question. What other potential axioms for set theory can be written in the form: "If mathematical structures $X$ and $Y$ are equipotent, then they're isomorphic"?

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  • $\begingroup$ It isn't an axiom of set theory, but among the saturated models of a fixed first-order theory, any two that are equipotent are isomorphic. $\endgroup$
    – Joel David Hamkins
    Commented Aug 18, 2015 at 18:12
  • $\begingroup$ @JoelDavidHamkins, perhaps the axiom "every model has a saturated elementary extension" can be recast into the desired form. $\endgroup$
    – goblin GONE
    Commented Aug 18, 2015 at 21:52
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    $\begingroup$ Over a finite field, if two vector spaces are equipotent then they are isomorphic. $\endgroup$
    – Asaf Karagila ♦
    Commented Aug 19, 2015 at 15:44

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