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This is a question from Jech's Set Theory (Ex. 17.12) which I'm reading at the moment and pretty much stuck on. If $D$ is a normal measure on $\kappa$ and $\{ \aleph_\alpha \colon > 2^{\aleph_\a …

Suppose $j\colon V\to M$ is an elementary embedding and $\kappa$ is the critical point of $j$, then $\kappa$ is measurable, and we can define the ultrafilter $U$ over $\kappa$ as: $$A\in U\iff \kappa\ …

It is a theorem of A. Levy, if $\kappa$ is an inaccessible cardinal, then $V_\kappa\prec_{\Sigma_1} V$ namely $V_\kappa$ is an elementary submodel when considering only $\Sigma_1$ sentences. One migh …

It is known that $L\models 2^\kappa=\kappa^+$, and that for a set of ordinals $A$ we know that $L[A]\models \exists\lambda\forall\kappa>\lambda(2^\kappa=\kappa^+)$. In this sense, there is some simil …

Suppose that $\kappa$ is a strong limit cardinal. The singular cardinal hypothesis states $2^\kappa=\kappa^+$. We know that the failure of SCH requires large cardinals, and in fact is equiconsistent w …

If $\kappa$ is an inaccessible cardinal then $V_\kappa$ is a model of $\sf ZFC_2$ ($\sf ZFC$ with a second-order replacement axiom). If there are many inaccessible cardinals then there are many model …

Suppose that $\kappa$ is the critical point of $j\colon V\to M$, and suppose that $\mathcal F=(F_\alpha\mid\alpha\leq\kappa)$ is a sequence such that for every limit ordinal $\alpha$, $F_\alpha$ is a …

Assuming the axiom of choice there is a neat way defining inaccessible cardinals as uncountable, regular, strong limit cardinals. Without the axiom of choice we have several notions of inaccessibilit …

Recall that $\kappa$ is a worldly cardinal if $V_\kappa$ is a model of $\sf ZFC$. While every worldly cardinal is a strong limit cardinal, it is not necessarily regular. The point being that the short …

(This question was posted on math.SE over two weeks ago, but received no answer. I am therefore posting it here as well.) It is well-known that there are difficulties in developing basic category the …

One of the magnificent theorems of $\sf ZFC$ is that there exists an Aronszajn tree on $\omega_1$. Namely, a tree of height $\omega_1$ in which every level is countable, but no branch is cofinal. On …

In short, what can we say about the consistency strength of "$\kappa$ is a singular worldly and inaccessible in an inner model"? Clearly, $0^\#$ exists since we have a singular cardinal which is regul …

It is a well-known theorem that if $\kappa$ is measurable, then there is a generic extension in which $\kappa$ is no longer weakly compact, but we can force its weak compactness back and recover the f …

In a recent question on Math.SE it was asked whether or not For every infinite cardinal $\mathfrak m$ there is no $\aleph$ number, $\kappa$, such that $\mathfrak m<\kappa<2^{\mathfrak m}$. By requiri …

In the last part of Kanamori's excellent "The Higher Infinite" there is a small diagram about the strength and consistency strength of some major large cardinal axioms. Below supercompact cardinals t …