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Mr. Chris Freiling
Department of Mathematics
California State University at San Bernardino
San Bernardino, CA

Dear Mr. Freiling:

	I enjoyed your recent paper in JSL and was convinced by its
evidence that the continuum hypothesis is false.  I think
yours is the kind of axiom G\"odel was looking for.  (See
Feferman's introduction to the first volume of his collected works).

	I wasn't convinced by the evidence against AC, because
your Theorem 1 ($\Rightarrow$ part) is a proof of $¬A↓{<\aleph↓1}$ in ZF.
Why should one then credit $A↓{<2↑{\aleph↓0}}$?

	There is a little bit more to be gotten from the
$\Leftarrow$ part of Theorem 1.  In the first place, any
three infinite cardinals $\alpha$, $\beta$ and $\gamma$
with $\alpha < \beta < \gamma$ can
play the role played by $\aleph↓0$, $\aleph↓1$ and $2↑{\aleph↓0}$.
proving a version $A↓{\alpha\beta\gamma}$ of your axiom.  Secondly,
it gives a finite version of the axiom as a theorem, namely
if $f$ has range of cardinality $k$, then the axiom holds if
the cardinality of the domain is larger than $k(k+1)$.
Perhaps the finite version can be regarded as evidence for
the infinite forms of the axiom.

Sincerely,
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