Since you have already linked to some of the contemporary
primary sources, where of course the full accounts of those
views can be found, let me interpret your question as a
request for summary accounts of the various views on CH.
I'll just describe in a few sentences each of what I find
to be the main issues surrounding CH, beginning with some
historical views. Please forgive the necessary simplifications.
Cantor. Cantor introduced the Continuum Hypothesis
when he discovered the transfinite numbers and proved that
the reals are uncountable. It was quite natural to inquire
whether the continuum was the same as the first uncountable
cardinal. He became obsessed with this question, working on
it from various angles and sometimes switching opinion as
to the likely outcome. Giving birth to the field of
descriptive set theory, he settled the CH question for
closed sets of reals, by proving (the Cantor-Bendixon
theorem) that every closed set is the union of a countable
set and a perfect set. Sets with this perfect set property
cannot be counterexamples to CH, and Cantor hoped to extend
this method to additional larger classes of sets.
Hilbert. Hilbert thought the CH question so
important that he listed it as the first on his famous list
of problems at the opening of the 20th century.
Goedel. Goedel proved that CH holds in the
constructible universe $L$, and so is relatively consistent
with ZFC. Goedel viewed $L$ as a device for establishing
consistency, rather than as a description of our (Platonic)
mathematical world, and so he did not take this result to
settle CH. He hoped that the emerging large cardinal
concepts, such as measurable cardinals, would settle the CH
question, and as you mentioned, favored a solution of the form $2^\omega=\aleph_2$.
Cohen. Cohen introduced the method of forcing and
used it to prove that $\neg$CH is relatively consistent
with ZFC. Every model of ZFC has a forcing extension with
$\neg$CH. Thus, the CH question is independent of ZFC,
neither provable nor refutable. Solovay observed that CH
also is forceable over any model of ZFC.
Large cardinals. Goedel's expectation that large
cardinals might settle CH was decisively refuted by the
Levy-Solovay theorem, which showed that one can force
either CH or $\neg$CH while preserving all known large
cardinals. Thus, there can be no direct implication from
large cardinals to either CH or $\neg$CH. At the same
time, Solovay extended Cantor's original strategy by
proving that if there are large cardinals, then increasing
levels of the projective hierarchy have the perfect set
property, and therefore do not admit counterexamples to CH.
All of the strongest large cardinal axioms considered today
imply that there are no projective counterexamples to CH. This can be seen as a complete affirmation of Cantor's original strategy.
Basic Platonic position. This is the realist view
that there is Platonic universe of sets that our axioms are
attempting to describe, in which every set-theoretic
question such as CH has a truth value. In my experience,
this is the most common or orthodox view in the
set-theoretic community. Several of the later more subtle
views rest solidly upon the idea that there is a fact of
the matter to be determined.
Old-school dream solution of CH. The hope was that
we might settle CH by finding a new set-theoretic principle
that we all agreed was obviously true for the intended
interpretation of sets (in the way that many find AC to be
obviously true, for example) and which also settled the CH
question. Then, we would extend ZFC to include this new
principle and thereby have an answer to CH. Unfortunately,
no such conclusive principles were found, although there
have been some proposals in this vein, such as Freilings
axiom of symmetry.
Formalist view. Rarely held by mathematicians,
although occasionally held by philosophers, this is the
anti-realist view that there is no truth of the matter of
CH, and that mathematics consists of (perhaps meaningless)
manipulations of strings of symbols in a formal system. The
formalist view can be taken to hold that the independence
result itself settles CH, since CH is neither provable nor
refutable in ZFC. One can have either CH or $\neg$CH as
axioms and form the new formal systems ZFC+CH or
ZFC+$\neg$CH. This view is often mocked in straw-man form,
suggesting that the formalist can have no preference for CH
or $\neg$CH, but philosophers defend more subtle versions,
where there can be reason to prefer one formal system to
another.
Pragmatic view. This is the view one finds in
practice, where mathematicians do not take a position on
CH, but feel free to use CH or $\neg$CH if it helps their
argument, keeping careful track of where it is used.
Usually, when either CH or $\neg$CH is used, then one
naturally inquires about the situation under the
alternative hypothesis, and this leads to numerous consistency or independence results.
Cardinal invariants. Exemplifying the pragmatic view, this is a very rich subject
studying various cardinal characteristics of the continuum,
such as the size of the smallest unbounded family of
functions $f:\omega\to\omega$, the additivity of the ideal
of measure-zero sets, or the smallest size family of
functions $f:\omega\to\omega$ that dominate all other such
functions. Since these characteristics are all uncountable
and at most the continuum, the entire theory trivializes
under CH, but under $\neg$CH is a rich, fascinating
subject.
Canonical Inner models. The paradigmatic canonical
inner model is Goedel's constructible universe $L$, which
satisfies CH and indeed, the Generalized Continuum
Hypothesis, as well as many other regularity properties.
Larger but still canonical inner models have been built by
Silver, Jensen, Mitchell, Steel and others that share the
GCH and these regularity properties, while also satisfying
larger large cardinal axioms than are possible in $L$. Most
set-theorists do not view these inner models as likely to
be the "real" universe, for similar reasons that they
reject $V=L$, but as the models accommodate larger and
larger large cardinals, it becomes increasingly difficult
to make this case. Even $V=L$ is compatible with the
existence of transitive set models of the very largest
large cardinals (since the assertion that such sets exist
is $\Sigma^1_2$ and hence absolute to $L$). In this sense,
the canonical inner models are fundamentally compatible
with whatever kind of set theory we are imagining.
Woodin. In contrast to the Old-School Dream
Solution, Woodin has advanced a more technical argument in
favor of $\neg$CH. The main concepts include $\Omega$-logic
and the $\Omega$-conjecture, concerning the limits of
forcing-invariant assertions, particularly those
expressible in the structure $H_{\omega_2}$, where CH is
expressible. Woodin's is a decidedly Platonist position,
but from what I have seen, he has remained guarded in his
presentations, describing the argument as a proposal or
possible solution, despite the fact that others sometimes
characterize his position as more definitive.
Foreman. Foreman, who also comes from a strong
Platonist position, argues against Woodin's view. He writes
supremely well, and I recommend following the links to his
articles.
Multiverse view. This is the view, offered in
opposition to the Basic Platonist Position above, that we
do not have just one concept of set leading to a unique
set-theoretic universe, but rather a complex variety of set
concepts leading to many different set-theoretic worlds.
Indeed, the view is that much of set-theoretic research in
the past half-century has been about constructing these
various alternative worlds. Many of the alternative set
concepts, such as those arising by forcing or by large
cardinal embeddings are closely enough related to each
other that they can be compared from the perspective of
each other. The multiverse view of CH is that the CH
question is largely settled by the fact that we know
precisely how to build CH or $\neg$CH worlds close to any
given set-theoretic universe---the CH and $\neg$CH worlds
are in a sense dense among the set-theoretic universes. The
multiverse view is realist as opposed to formalist, since
it affirms the real nature of the set-theoretic worlds to
which the various set concepts give rise. On the Multiverse
view, the Old-School Dream Solution is impossible, since
our experience in the CH and $\neg$CH worlds will prevent
us from accepting any principle $\Phi$ that settles CH as
"obviously true". Rather, on the multiverse view we are to study all the possible set-theoretic worlds and especially how they relate to each other.
I should stop now, and I apologize for the length of this answer.
Best Answer
The basic contention here is that Wiles' work uses cohomology of sheaves on certain Grothendieck topologies, the general theory of which was first developed in Grothendieck's SGAIV and which requires the existence of an uncountable Grothendieck universe. It has since been clarified that the existence of such a thing is equivalent to the existence of an inaccessible cardinal, and that this existence -- or even the consistency of the existence of an inaccessible cardinal! -- cannot be proved from ZFC alone, assuming that ZFC is consistent.
Many working arithmetic and algebraic geometers however take it as an article of faith that in any use of Grothendieck cohomology theories to solve a "reasonable problem", the appeal to the universe axiom can be bypassed. Doubtless this faith is
predicated onabetted by the fact that most arithmetic and algebraic geometers (present company included) are not really conversant or willing to wade into the intricacies of set theory. I have not really thought about such things myself so have no independent opinion, but I have heard from one or two mathematicians that I respect that removing this set-theoretic edifice is not as straightforward as one might think. (Added: here I meant removing it from general constructions, not just from applications to some particular number-theoretic result. And I wasn't thinking solely about the small etale site -- see e.g. the comments on crystalline stuff below.)Here is an article which gives more details on the matter:
Note that I do not necessarily endorse the claims in this article, although I think there is something to the idea that written work by number theorists and algebraic geometers usually does not discuss what set-theoretic assumptions are necessary for the results to hold, so that when a generic mathematician tries to trace this back through standard references, there may seem to be at least an apparent dependency on Grothendieck universes.
P.S.: If a mathematician of the caliber of Y.I. Manin made a point of asking in public whether the proof of the Weil conjectures depends in some essential way on inaccessible cardinals, is this not a sign that "Of course not; don't be stupid" may not be the most helpful reply?