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Suppose that $\kappa$ is $\lambda$-strongly compact and $\lambda$ is regular. Let $j:V\to M$ be the ultrapower by a fine measure $\mu$ on $P_\kappa\lambda$. The object $s=[\text{id}]_\mu$ in $M$ is a …

Update. This answer is not correct, because it is a subtler matter to ensure that the levels are closed under relative constructibility while also maintaining $L_{\alpha+1}\cap\mathfrak{L}_\alpha=L_\a …

The answer is no, you cannot have measurable cardinals consistently with your theory. Your theory includes the axiom "V=L or V=L[c] for an $L$-generic Cohen real $c$". This statement is provable from …

With regard to the title question, I believe that the main argument people would provide would be that it is the actual existence of the large cardinals that explains the consistency assertions that o …

The answer to Q1 is no. The least rank-to-rank cardinal is never supercompact. The reason is that a cardinal $\kappa$ being rank-to-rank is a $\Sigma_2$ property, being witnessed by the existence of a …

Every large cardinal property admits a formalization with the desired property. That is, every large cardinal property $\text{LC}$ admits a ZFC-provably equivalent formulation $A$ for which $\newcomma …

There are numerous examples of such statements. Let me organize some of them into several categories. First, there is the hierarchy of large cardinal axioms that are relatively consistent with V=L. Se …

This is never true in the circumstances you request, quite apart from your uniformity requirement, since some $U$ admit no such $f$ at all. The reason is that if $[\text{id}]_U$ is generated by $\kapp …

I believe that the answer is no, using some methods from my old paper: Hamkins, Joel David, Destruction or preservation as you like it, Ann. Pure Appl. Logic 91, No. 2-3, 191-229 (1998). arXiv:1607.0 …

In general, no, because $\kappa$ might not be $\lambda$-supercompact, even if it is $\lambda$-strongly compact. The two large cardinal notions are not provably equivalent (although it is an open quest …

Indeed, many of the structural features that hold in the constructible universe $L$ are obtainable by forcing in a way that accommodates large cardinals. GCH. The standard forcing of the GCH is an Eas …

It seems to me that if you build the constructible universe using $\mathcal{L}_{\omega_1,\omega}$ logic, you will get the inner model $L(\mathbb{R})$. The reason is that every $\mathcal{L}_{\omega_1,\ …

The explanation is philosophical rather than mathematical. The idea is simply that the inner-model theory provides a rich account of what it would be like for the large cardinal axioms to be true, and …

The assertion that all proper classes are equinumerous is equivalent over GBc class theory to the axiom of global choice, which is the assertion that there is a global well-ordering of the universe of …

A Grothendieck universe is known in set theory as the set Vκ for a (strongly) inaccessible cardinal κ. They are exactly the same thing. Thus, the existence of a Grothendieck universe is exactly equiv …