Does there exist a statement P in the language of first-order arithmetic, such that P is is a theorem of ZFC, and not P is a theorem of ZF + not C? In other words, can the question of whether the axiom of choice is true or not have an effect on what the truths of first-order arithmetic are? What about if we replace first-order arithmetic with higher-order arithmetic?
If there exists no such statement, then does there exist any statement X independent of ZF, for which there is a statement P in first-order arithmetic such that P is a theorem of ZF + X and not P is a theorem of ZF + not X? (Or replace ZF with ZFC.) Of course there are examples like Con(ZF) and not Con(ZF), but I'm looking for statements X such that neither ZF + X nor ZF + not X is obviously unsound. Or to put it another way, I want a statement which we're not obliged to believe or disbelieve if we accept the soundness of ZF (or ZFC).
Now the continuum hypothesis can be written as a statement of third-order arithmetic, and it's independent of ZFC. Does it and its negation have contradictory consequences for first-order arithmetic? Do they have contradictory consequences for second-order arithmetic?
Any help would be greatly appreciated.
Thank You in Advance.
If $P$ is a statement in first-order arithmetic then $P$ is absolute between $V$ and $L$. This means that $P$ is true in $V$ if and only if it is true in $L$. Since the axiom of choice is true in $L$, this means that the answer is negative. Similarly for the continuum hypothesis, which is always true in $L$.
The key theorem here is Shoenfield's absoluteness theorem. This theorem also holds for "simple enough" second-order statements.
As for the statement $X$ which might exist, note that $V=L$ decides most "natural looking" statements of set theory (at least those which don't require a greater consistency strength, like large cardinals). So the same logic applies here as well. If we do want to talk about large cardinals, then note that you can always consider $V_\kappa$ and $L_\kappa$ for the least $\kappa$ that these are models of $\sf ZF$. The natural numbers are there, and it is easy to see that the same statements are true in these smaller universes, which do not contain large cardinals, as in the full universe.
On the other hand, there are number theoretical statements which can be taken as a set theoretical axioms. For example $\newcommand{\con}{\operatorname{Con}}\con(\sf ZFC)$ and $\lnot\con(\sf ZFC)$ are such statements. If there is a model of $\sf ZFC$ in the universe, then $\con(\sf ZFC)$ is true, but otherwise it is false. Moreover if $\sf ZFC+\con(ZFC)$ is consistent then we can find a model of $\sf ZFC+\lnot\con(ZFC)$. Note, however, that in a model of $\sf ZFC+\lnot\con(ZFC)$ there are non-standard integers (standard ones cannot encode a proof of the contradiction from $\sf ZFC$).
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