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The procedure you suggest really cannot get too far. Here is an abstract result explaining what I mean (see Aspects of incompleteness by Per Lindström): Given an r.e. sequence of r.e. theories that in …

Ricky: I assume you mean to ask your question in $L({\mathbb R})$. In general, without choice, the ordering of cardinals tends to be rather pathological, although we do not yet know by how much. He …

The answer to the first question is no: You can add a club subset of a stationary subset of $\omega_1$ by forcing. The closure of any uncountable subset of the generic club is a club contained in the …

Hi Yu, No, your statement is equiconsistent with $\mathsf{ZFC}$. In Leo Harrington. Long projective wellorderings, Annals of Mathematical Logic 12 (1977) 1-21, MR0465866 (57 #5752). it is show …

If $\mathcal M$ is $\omega$-small, then many $\mathcal J^{\mathcal M}_\beta$ may think that (a large fragment of $\mathsf{ZF}$ holds and) there are plenty of Woodin cardinals. What matters is that for …

(This was a reply to John that grew beyond the alloted limit.) large cardinals have consequences that were initially surprising to set theorists. But, John, this is simply because our intuitions …

There is a fantastic (and not too well-known) result of Shelah stating that $L({\mathcal P}(\lambda))$ is a model of choice whenever $\lambda$ is a singular strong limit of uncountable cofinality. T …

If $X$ is a set, then $L[X]$ does not have strong cardinals. In fact, $X^\sharp$ does not belong to $L[X]$, so any assumption that implies that $V$ is closed under sharps fails in these models. Now, …

As you say, Hugh's precise result is unpublished. I have not seen any written reports of it, so I do not know the precise hypotheses it uses. For purely historical reasons, I would be interested if so …

What I would call the standard source of examples is the series of very nice "finitary" combinatorial statements that Harvey Friedman has been working on. You can see plenty of such statements in his …

Rupert, I will explain the argument for $Y=\omega$ (this makes no difference, but you may find it easier to visualize) when the continuous function is particularly nice (in a way I will make precise. …

If rather than large cardinals we concentrate on their consistency strength, a source of examples comes from applications of forcing axioms. A notorious case is Justin Moore's result that there is a …

This is just a comment, but too long to be one: $\mathsf{PFA}(\mathfrak c)$ settles the size of the continuum, this was proved by Velickovic in Forcing axioms and stationary sets. Adv. Math. 94 (2) (1 …

A famous result of Jensen is the coding theorem showing that the universe can be extended by class forcing to a model of the form $L[a]$, with $a$ a real, in a way that the original ground model can b …

Just some comments to complement Joel's answer: That forcing can destroy and then recreate measurability is due to Kunen: Kenneth Kunen. Saturated ideals, The Journal of Symbolic Logic 43 (1) ( …