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The question is this: When, in the history of mathematics, have problems "like P vs. NP" arisen and then been solved? In those cases, what were the resolutions? … Until Cohen, there were many proven statements about transfinite sets, and then a whole class of other statements---V=L, GCH, CH, AC, Zorn's Lemma, well-orderability... …

What reasons are there to believe in their truth, rather than merely in their consistency, or at most their truth in some transitive model? … One might be $V = L$, or even the axiom of restriction, but most set theorists dismiss them, as they rule out large large cardinals and inaccessible cardinals, respectively, and instead seek core models …

Without that existence, we wouldn't have any reason to believe in even the much weaker consistency assertions. … According to Steel, the large cardinal set-theorist can still consider the nature of V=L set theory, without the stronger large cardinals, simply by relativizing set theory to L. …

=v, t''= \frac{t'v'}{l'}, t''' = \frac{t''v''}{l''}, etc.$$ He also uses the notation $Div$, which I believe is an abbreviation for "Divisor", since the arithmetical procedure he describes involves the … Secondly, the power of $z$ written there is actually 4; I believe this is just a typo of Gauss. Thirdly, this cubic form has $v=|2^{\frac{1}{3}}|$, and I believe $\epsilon$ is a third root of unity. …

retains its large cardinal property in L, so we get consistency with V=L. … These are all independent of ZFC+V=L, since they are independent of ZFC, and their truth is the same in V as in L. …

National Acad., 1938, he remarks that V=L added as a new axiom seems to give a natural completion of the axioms of set theory, in so far as it determines the vague notion of an arbitrary infinite set in … So Davis (and apparently also Kreisel) believed that Gödel accepted V=L in 1938 as a new axiom. …

Now, consider the incidence correspondence $X = \lbrace (v,L) \in V \times \mathrm{Gr}(k+1,n+1) \mid v \in L \rbrace \subset V \times \mathrm{Gr}(k+1,n+1)$. This is a projective variety. … \cap L) = 0$ for all $L \in U$. …

Since it is a model of $\text{ZFC}+V=L$, one gets on board all of the consequences of that theory in the minimal model. … But there are many further properties true in the minimal model, when it exists, that are not provable from $\text{ZFC}+V=L$. …

I believe it is essentially this perspective that is expressed by the quotations you have made in the original question. … For example, if ZFC is consistent, then we know that it is consistent that ZFC + V=L + ¬Con(ZFC), with the point being that in any world fulfilling this theory, we would have the full fine-structural account …

way of building up the model $L[A]$), which have to have well-founded ultrapowers (in which case they are called mice), and one needs a comparison lemma for establishing which mice are initial segments … I read some of Steel’s “The Comparison Lemma” in my quest for an answer, and it mentioned that for $L$ the mice are just well-founded structures satisfying a fragment of $\mathrm{ZFC}$ and $V = L$, which …

I don't believe "ultimate L" will ever gain popular traction, for reasons that should be apparent. … I think the question of whether $V$ equals $L$ is far more important to settle. My opinion about $V=L$ is in the minority, but it has had some important supporters. …

There cannot be a "true" $V$. There cannot even be a completed collection of all ordinals. … This rules out wanting $V=L$ as an axiom, as that limits the types of collections available. …

I will assume $V \in L^\infty(\Omega)$ is smooth, $V>0$ and $g \in C^1(\partial \Omega)$. … {L^\infty}}{\lambda_1(\Omega)} \int_\Omega |\nabla u|^2 + 2\|V\|_{L^\infty} \int_\Omega w^2 \,, $$ hence $$ J(u) \ge \left(1-\frac{2\|V\|_{L^\infty}}{\lambda_1(\Omega)} \right) \int_\Omega |\nabla u|^2 …

In particular, to prove that a particular $\Sigma^1_2$ statement is true in ZFC, it suffices to prove it under the assumption also that V=L, where one also has all kinds of additional structure available … To prove that there is a proof, is a proof, so I believe ultimately there will be no way to avoid the quibbling over whether it is easy or hard to translate the high-level proof into a low level proof, …

It is interesting to note that Sacks' proof of Theorem 3.1 from his paper gives an indication of what must happen if $V\neq L$ (most set theorists do not believe that $V$=$L$ is an 'acceptable' axiom): … Suppose $\forall x(x\in L\leftrightarrow\{e\}(x)\downarrow)$ and $V\neq L$. Then for some $b\notin L$, $\{e\}(b)\uparrow$. …