# How does equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ in $\sf{ZF}$ relate to the axiom of choice?

Usually, the equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ for infinite cardinal $\mathfrak{m}$ is proved like that: $$2^\mathfrak{m} \leqslant \mathfrak{m}^\mathfrak{m} \leqslant (2^\mathfrak{m})^\mathfrak{m} = 2^{\mathfrak{m} \cdot \mathfrak{m}} = 2^\mathfrak{m}$$ The last step relies much on the axiom of choice, because the fact $\mathfrak{m}\cdot\mathfrak{m} = \mathfrak{m}$ does. Moreover, Tarski's theorem states that in $\sf{ZF}$ this fact is equivalent to $\sf{AC}$.

Note, by a cardinal I mean an equipollence class of an arbitrary set, not necessarily admitting a well-ordering.

That has motivated my question: what can we say about relationship of the initial equality and the axiom of choice in pure $\sf{ZF}$?

To avoid question about definition of cardinals without $\sf{AC}$, we may consider the following statement instead:

For any infinite set $X$ exists a bijection $\; F \, \colon X^X \rightarrow 2^X$

where $X^X$ is the set of all functions from $X$ to $X$, $\; 2^X = \mathcal P(X)$ is the power set, and an infinite set is such that cannot be bijectively mapped onto a finite ordinal.

• Tarski's theorem states that $X \times X \approx X$ for every infinite set $X$ implies AC, but $\kappa.\kappa = \kappa$ for any infinite cardinal $\kappa$ can be proven in ZF. Feb 17 '16 at 21:03
• @nombre, you probably call cardinal thing I call aleph, that is cardinality of an ordinal. In that sense you are right, but I don't want to debate on the definition of a cardinal. Feb 17 '16 at 21:55
• Ok. I really don't know whether the second statement implies AC but it is an interesting question. Feb 17 '16 at 22:20

The relation of the subject statement to the axiom of choice seems poor. So, the subject statement is consistent with $\sf ZF + \neg\sf AC$, as in final remark in this paper. More, its negation also may hold in models of $\sf ZF$ without choice, as in this paper!
For reference, permutation models of $\sf ZF$ with atoms (or urelements) used in the above papers are described in book "The Axiom of Choice" by Thomas J. Jech.