Search Results

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... …

[Reinhardt suggested an ultimate axiom of the form "there is a non-trivial elementary embedding $j:V\to V$". … For some of them, set theory is about all possible (natural) extensions of $\mathsf{ZFC}$, and there are certainly many interesting such extensions (such as $V=L$) that rule out large cardinals. …

V=L isn't alone in this ...). … V=L, by contrast, is genuinely complicated. …

The special case when d=1 was done in 1870 by Georg Cantor... V. L. … operator is bounded on $L^2$. …

In the second place, even a Platonist might believe the consistency of some axioms for reasons other than a belief in their truth. … On the other hand, I've never been able to persuade myself of the actual existence of even subtle cardinals (which are so small that they're consistent with $V=L$). …

(For example, $V=L$ abbreviates "all sets are constructible.") … In particular, when I talk about statements being true in $V$, I mean simply that the quantified variables are to be interpreted as ranging over arbitrary sets. …

=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. …

A person who believes only in the constructible universe might reject the same ordinal collapsing functions if cardinals incompatible with $V=L$ are used. … So in a way, this "real world" value changes depending on our philosophical stance. …

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, …

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. …

$\vdots$ These are a part of arguments which could be used against large cardinal axioms, but many set theorists not only believe in the existence of large cardinals, but also refute every statement like … $V=L$ which is contradictory to their existence. …

The doubts, however, are usually of the form, "I don't feel that there's enough evidence to make me believe in them," rather than any concrete arguments against their existence. … Once in a while you may run into someone who believes $V=L$ strongly enough to use that as a reason to disbelieve in measurable cardinals, but that's pretty rare. …

It is known that if the theory $\mathrm{ZF + V = L(0 ^{\#}})$ has transitive model of height $\alpha < \omega_1$, then $\alpha \in U$; coupled with Friedman's above theorem, this implies that $\mathrm{ … ZF + V = L(0 ^{\#}})$ has more than one transitive model of height $\alpha$. …

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. …

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 …