the set of all sets cannot exist.
An interesting thing about set theory in the philosophy of mathematics that involves e.g., the Axiom of Choice is what, exactly, is necessary in order for a set to exist (i.e., define a set). In most formulations of set theory, the Axiom of Choice can't define sets and is largely irrelevant to set theory. In fact, it is logically equivalent with everything from Zorn's Lemma (not actually a lemma in the technical sense) to the well-ordering axiom.
It is not quantum logic that is responsible for the success but quantum mathematics.
There really isn't such a thing as "quantum mathematics". The "consensus" interpretation of QM (despite itself having multiple interpretations) is that it is irreducibly statistical. The entire structure of quantum mechancis is mathematical, but this makes it less mathematical physics than statistical physics and no more mathematical than classical mechanics (and less, actually, than quantum field theory).
Until a measurement (a much misunderstood term) is made, a quantum does not have a value classifiable as classical.
QM, despite the nomenclature, isn't really a theory of quanta. That's more a historical byproduct. More importantly, quantum systems absolutely do have values before measurement. Even more importantly, the entirety of physics is concerned with the dynamics of physical systems in abstract mathematical spaces. Classical physics possessed varying mathematical structures depending upon the field and the mathematical spaces used to model systems in it. For example, systems in special relativity exist in Minkowski space, not Euclidean, "observables" in classical physics are represented by values rather than operators in Hilbert space as in QM, etc.
Mathematical structures, from abstract algebras and mathematical logics to topology and non-Euclidean geometries, exist in many forms in applied mathematics, physics, cosmology, theoretical computer science, data analysis, and basically the sciences.
Both classical and non-classical logics have mathematical structures, just as QM, QFT, QCD, and the phase space of classical dynamical systems do. The point of quantum logic is to distill quantum mechanics down to its mathematical structure independent of any experimental outcomes. A bit more precisely (yet still simplistically), quantum logics are representations of the mathematical apparatus used by quantum physicists. Thus quantum logics have a closer tie to physical reality than does classical logic, which is largely a formalization of Aristotle's ~2,300 year old works and hasn't become more closely linked with theories of physical systems since. Quantum logics, however, post-date and are rooted in physical theory.
It is wave spread over a (possibly abstract) space.
It isn't, and the idea that either waves or particles exist is compatible with logic only if one regards both notions as contradicting quantum physics or as incoherently defined. Waves are never measured as discrete, yet the wave-like behavior of quantum "particles" display wave-like behavior even when they are detected discretely. Likewise, particles follow definite trajectories, can't be represented accurately as existing in multiple places, and don't 'interfere" with themselves as do photons, electrons, etc.
The measurement connects with some small portion of the wave and a quasi-classical value is assumed.
The values aren't assumed. The space in which quantum systems "exist" is called Hilbert space not due to any mathematical formulation of a theory of physics the way that Minkowski developed the geometry of Einstein's algebraic formulation of spacetime.
It is not that the quantum has multiple values before measurement. It is that the wave may be intercepted at various places.
That would make it not a wave, as waves are defined as fundamentally lacking any discrete units or location (actually, they are mostly the distributed or "spread-out" nature of force propagating through a medium; however, quantum "waves" don't propagate through media, aren't "spread out", and aren't "waves").
Why the wave function collapses upon measurement and how it always avoids collapsing at more than one place
It never "collapses" at more than one place. That's part of the measurement problem.
Complicating and confusing - the situation is that in many cases the possible detectable values are quantized, often binary.
1) All values are "quantized" in quantum theory. That's the reason for the nomenclature.
2) "Binary units" in information theory have been replaced by "qubits" in quantum physics for a reason.
But most experimental physicists and engineers working in the field do not worry about quantum logic.
That's true. On the other hand, they don't generally worry about set theory or even really mathematics (which, for them, is frequently only a tool; a frequent exception is theoretical physicists, cosmologists, theoretical computer scientists, computational physicists, computational biophysicists, and those like myself concerned with complex systems across disciplines).
They take Bohrs advice and just do the math.
The "shut up and calculate" quote, often attributed to Feynman, was actually from Mermin who himself describes it as follows:
"I'm not proud of having said it. It's not a beautiful phrase. It's not very clever. It's snide and mindlessly dismissive" (
source)
Theoretical physicists are usually so deep into other things that they are beyond worrying about quantum logic.
The development of quantum logic was the work of von Neumann, among the most applied founders of physics.
What I meant was that logic has been found to be limited in unexpected ways, in support of which I gave several examples.
The ways you've expressed, however, have either depended upon the use of logic in the non-formal sense (i.e., the way that proofs in analysis depend upon logic but don't depend upon any logical system), or are irrelevant to logic (i.e., don't involve any formal logic and so far haven't even included non-classical logics).
But Hilbert expected some things that turned out to be impossible, although in the fin de siècle era it seemed intuitively obvious that they were attainable.
It was never "intuitively" obvious that
mathematics (Hilbert
expressly formulated the most important unsolved problems as being those in
mathematics, not logic, and set theory is a field of mathematics while logic even today is a discipline of philosophy) implied what Hilbert thought. That mathematics didn't, however, turned out to be even less intuitive. The use of logic to make discoveries like those made by Cantor, Gödel,, Cohen, etc. are incredibly non-intuitive. Dedekind (the mathematician instrumental into providing a basis for the reals) was a contemporary Cantor and, upon seeing his proof that infinities have different "sizes" (cardinalities) said: "Je le vois, mais je ne le crois pas" ("I see it but I don't believe it").
AC and GCH are intimately intertwined.
To the extent "the AC and GCH are intimately intertwined" we know this because of logic. It seems hard to argue that the use of logic can show the limitations of logic using the ways in which logic has allowed us to conclude, even prove, that which it has. That the AC is, as its name indicates, an axiom, means that it lacks even the proofs behind the continuum hypothesis' undecidability but depends upon logic no less than they.
AC was invented to address GCH. GCH does not merely entail AC, GCH requires AC.
The GCH predates the AC, so can't require it. Moreover, the AC
is an axiom,
NOT a theorem. It is assumed to be true. The GCH has been proven to be undecidable, which means it CANNOT require any axiom of ZF or ZFC or classical logic (or any other formal system).
The two cannot be separated.
They have been. In fact, that's what Cohen did: proved that the axioms of set theory (including the ZF formulation) are consistent if the GHC is false.
It is undecidable PM, ZFC, non-classical logics, and every single set theory, formal logic, etc., there is.
proving one would automatically prove the other. They go hand in hand.
That's not what decidable means. If it is possible to prove that the GCH is consistent with the axioms of set theory and that it is inconsistent with the axioms of set theory (ZF or other formulations) than it is impossible to prove that GCH is "true" (more technically, if it is true, then we require a more powerful form of mathematical proof and logic in order to show that it is true, and part of what Gödel did was show that proof and truth are distinct). More importantly, it is obvious to prove that the AC is false within certain mathematical systems/set theories. Russell showed this with shows, although socks are more intuitive here: it is impossible for the AC to be true in ZF if we define sets as is allowed in ZF, because infinite subsets of infinitely pairs of socks can't be chosen s.t. the result is a set the way that boots (which have a left and right and thus don't require a further axiom) can. Hence ZFC, which incorporates an additional axiom that isn't proven but assumed. The GCH, however, can be shown to consistent is assumed true and also if assumed false. It is compatible with ZF if assumed to be so yet is incompatible if assumed to be so. This isn't true of the AC.