Example relating to excluded middle

Define a subset ${\displaystyle A\subset \{0,1\}}$ via

${\displaystyle A:=\{n\in \{0,1\}\mid (P\land n=0)\lor (\neg P\land n=1)\}}$

Clearly ${\displaystyle P\leftrightarrow 0\in A}$ and ${\displaystyle (\neg P)\leftrightarrow 1\in A}$, and from the principle of non-contradiction one concludes ${\displaystyle \neg (0\in A\land 1\in A)}$. Further, ${\displaystyle (P\lor \neg P)\leftrightarrow (0\in A\lor 1\in A)}$ and in turn

${\displaystyle (P\lor \neg P)\to \exists$ !(n\in \{0,1\}).n\in A}

Already minimal logic proves ${\displaystyle \neg \neg (P\lor \neg P)}$, the double-negation for any excluded middle statement, which here is equivalent to ${\displaystyle \neg (0\notin A\land 1\notin A)}$. So by performing two contrapositions on the previous implication, one establishes ${\displaystyle \neg \neg \exists$ !(n\in \{0,1\}).n\in A} . In words: It cannot consistently be ruled out that exactly one of the numbers ${\displaystyle 0}$ and ${\displaystyle 1}$ inhabits ${\displaystyle A}$. In particular, the latter can be weakened to ${\displaystyle \neg \neg \exists n.n\in A}$, saying ${\displaystyle A}$ is proven non-empty.

As example statements for ${\displaystyle P}$, consider the infamous provenly theory-independent statement such as the continuum hypothesis, consistency of the sound theory at hand, or, informally, an unknowable claim about the past or future. By design, these are chosen to be unprovable. A variant of this is to consider mathematiacal propositions that are merely not yet established - see also Brouwerian counterexamples. Knowledge of the validity of either ${\displaystyle 0\in A}$ or ${\displaystyle 1\in A}$ is equivalent to knowledge about ${\displaystyle P}$ as above, and cannot be obtained. Given neither ${\displaystyle P}$ nor ${\displaystyle \neg P}$ can be proven in the theory, it will also not prove ${\displaystyle A}$ to be inhabited by some particular number. Further, a constructive framework with the disjunction property then cannot prove ${\displaystyle P\lor \neg P}$ either. There is no evidence for ${\displaystyle 0\in A}$, nor for ${\displaystyle 1\in A}$, and constructive unprovability of their disjunction reflects this. Nonetheless, since ruling out excluded middle is provenly always inconsistent, it is also established that ${\displaystyle A}$ is not empty. Classical logic adopts ${\displaystyle P\lor \neg P}$ axiomatically, spoiling a constructive reading.

Example relating to choice

There are various easily characterized sets the existence of which is not provable in ${\displaystyle {\mathsf {ZF}}}$, but which are implied to exist by the full axiom of choice ${\displaystyle {\mathrm {AC} }}$. As such, that axiom is itself independent of ${\displaystyle {\mathsf {ZF}}}$. It in fact contradicts other potential axioms for a set theory. Further, it indeed also contradicts constructive principles, in a set theory context. A theory that does not permit excluded middle does also not validate the function existence principle ${\displaystyle {\mathrm {AC} }}$.

In ${\displaystyle {\mathsf {ZF}}}$, the ${\displaystyle {\mathrm {AC} }}$ is equivalent to the statement that for every vector space there exists basis. So more concretely, consider the question of existence of a Hamel bases of the real numbers over the rational numbers. This object is elusive in the sense that are different ${\displaystyle {\mathsf {ZF}}}$ models that either negate and validate its existence. So it is also consistent to just postulate that existence cannot be ruled out here, in the sense that it cannot consistently be negated. Again, that postulate may be expressed as saying that the set of such Hamel bases is non-empty. Over a constructive theory, such a postulate is weaker than the plain existence postulate, but (by design) is still strong enough to then negate all propositions that would imply the non-existence of a Hamel basis.