I wrote about how existential axioms are related to non-constructive arguments, non-constructive sets, and incompleteness. This might be a bit repetative, but you might prefer one explanation over the other. I did not give examples before because that would have been a distraction. Here are some examples.

	If

1. The Continuum hypothesis (CH) cannot be proven or disproven from Zermelo-Frankel set theory with the axiom of choice (ZFC).
2. The contention / assumption that the rational field is a subfield of the real field requires CH. (Does not have to be true just consistent!)
3. The Birch-Swinnerton-Dyer conjecture (BSD) includes this assumption.
4. The Riemann hypothesis (RH) includes this assumption.

	Then

5. BSD cannot be proven or disproven from ZFC.
6. RH cannot be proven or disproven from ZFC.

7. Are real numbers, the continuum, and ranking the continuum vs discrete infinity each distinct concepts according to the axioms (ZF, AC, CH)?

We take for granted that logic and set theory work together. Logical completeness guarantees that the syntax and semantics of logic always correspond. Set theory is the description of the values of the variables of the logical statements (ie an object theory for the semantics of logic). When (and where) we choose to break completeness, we loose the justification to switch back and forth between logic and set theory. 

To break completeness, we use an existential axiom that adds something new. Those new things exist logically (ie are logically constructable so long as we choose a consistent system), but cannot be constructed via set theory. In this situation, we need to understand the philosophy of mathematics to know the difference between formal / deductive and informal / non-deductive arguments. An incomplete system will include invalid but satisfiable statements, yet conceptually satisfiable statements will not always actually be satisfiable. Due to incompleteness, proof by contradiction / the law of excluded middle is not valid for non-constructive sets for the given axioms. In other words, we cannot rely on counter examples to show us when we are making a mistake. Not keeping to / acknowledging axioms goes unnoticed because implicitly adding / generating additional axioms by using completeness with non-constructive objects is guaranteed to be consistent. Ergo argument number two. 

I have long thought that it would be good for students to know that the continuum is different than real numbers. That inevitably comes to uncountability. Uncountability is defined non-constructively. It uses completeness with an existential axiom. (A limit process itself does not need infinity.) Let us take that as implying another axiom. Thus, uncountability and continuum seem to need AC, rather than justifying each other.

8. Do (non-redundant) existential axioms form non-constructive arguments and non-constructive sets and thus cause incompleteness?