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I don't know if it's lesser known, but it is certainly not on par with some of the other results on this page. Theorem. (Doyle–Conway) Assume $\sf ZF$. If there is a bijection between $3\times A$ …

It is known for a while now that $\frak p=t$, as a result of Malliaris-Shelah. The original paper draws from model theoretic methods. I've heard rumors that there was a proof which was purely set the …

A friend of mine and myself (both grad students with a relatively decent set theoretic background) want to venture into the universe of inner models. [pun intended :-)] I would very much like to get …

Finally, a good use for the newly purchased copy of "Zermelo's Axiom of Choice". Moore writes that Cantor formulated the following problem in 1878: Every infinite subset of $\Bbb R$ is either den …

If by the Hilbert cube you mean only $[0,1]^\mathbb N$ then the answer is yes. There is such proof, you can find it in Herrlich's The Axiom of Choice as Theorem 3.13. If you mean the general case of …

The argument, given to me by Hugh Woodin over a drink in Barcelona in September 2016, is a nice retort to "AC is obviously false since there cannot be a well-ordering of the real numbers". It is argua …

In the early 20th century there was a lot of fuss over the axiom of choice implying that there are Lebesgue non-measurable sets of reals. In his book about The Axiom of Choice, Gregory Moore points to …

You can find it, amongst other places in my write up: Zornian Functional Analysis or: How I Learned to Stop Worrying and Love the Axiom of Choice. If you need a source to cite, my money is on Handbo …

In the vast majority of papers forcing is always developed over ZFC. Not surprisingly too, since infintary combinatorial principles are often used to prove results based on properties such as chain c …

First of all, the axiom of countable choice says that given a countable family of non-empty sets, you can choose from each set simultaneously. If you want to choose from one, then from another, then f …

No. Most of the work of Pincus that involved these sort of iterated constructions, where one "pushes the counterexamples out" has never been reworked into modern terms. In my Ph.D. one of the reasons …

Alex Lubotzky was a parliament member in the late 1990s in Israel. Menachem Magidor was, while being the president of the Hebrew university (a political position in itself), the head of the "preside …

It seems that your question has a positive answer, as shown by Galvin and Komjáth in their paper Galvin, F.; Komjáth, P., Graph colorings and the axiom of choice, Period. Math. Hung. 22, No.1, 71- …

In set theory, in particular the context of forcing, if $M$ is a model of $\sf ZFC$ and $P\in M$ is a partial order, we say that $G\subseteq P$ is a generic filter (or $M$-generic or generic over $M$) …

This is difficult to prove using forcing, for one simple reason. If $M\models\sf ZFC$, and $G$ is an $M$-generic filter (for some forcing notion), then $M[G]\models\sf ZFC$. In other words, the only …