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Are there any examples of a set which is uncountable in a given axiomatization, which is denumerable in another?

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    Of course, let $X$ be defined by "$\omega_1$ if CH is true and $\omega$ if CH is false". Depending on your axioms this might define a countable or an uncountable set. Not sure if this is what you are looking for... your question is a bit unclear. – 2017-02-28
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    But maybe you are interested in [absoluteness](https://en.wikipedia.org/wiki/Absoluteness#In_set_theory). The property of being uncountable is not absolute for models of ZFC. – 2017-02-28

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Here is one natural example, $\Bbb R^L$ is the set of all constructible reals. Assuming $V=L$, this is the set of all real numbers and therefore uncountable. On the other hand, assuming $\sf PFA$ the set is very much countable.

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    +1, but PFA's a bit overkill . . . – 2017-02-28
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    Sure, but it's also a rather natural axiom. Compared to sharps, which are technically natural... But technically natural. – 2017-02-28
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    That's fair - I find it less natural than sharps. The [inner model hypothesis](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.109.7482&rep=rep1&type=pdf) is another fun one. (I'm sure you're aware of it, I'm listing it for the OP's sake.) – 2017-02-28
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    Well, there is also "every uncountable set of reals has a perfect subset", which is also somewhat natural and implies that $\omega_1$ is a limit cardinal in $L$... :) – 2017-02-28