We presented the proof that “-trichotomy” implies choice. The following is still open:

Question. () Assume that is non-well-orderable. Is there a countably infinite family of pairwise size-incomparable sets?

We mentioned a few (familiar) statements that fail in the absence of choice, like the existence of bases for any vector space, Tychonoff’s theorem, or the “surjective” version of the Schröder-Bernstein theorem.

We defined addition, multiplication and exponentiation of cardinals, and verified that addition and multiplication are trivial. We stated the continuum hypothesis , and the generalized continuum hypothesis .

We want to prove (in subsequent lectures) a few non-trivial results about the behavior of exponentiation. In order to do this, we need the key notion of cofinality. We proved a few basic facts about cofinality and defined regular cardinals.

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The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

Let $s$ be the supremum of the $\mu$-measures of members of $\mathcal G$. By definition of supremum, for each $n$, there is $G_n\in\mathcal G$ with $\mu(G_n)>s-1/n$. Letting $G=\bigcup_n G_n$, then $G\in \mathcal G$ since $\mathcal G$ is closed under countable unions, and $\mu(G)=s$, since it is at least $\sup_n\mu(G_n)$ but it is at most $s$ (by definiti […]

The result you are trying to prove is false. For example, if $a=\omega+1$ and $b=\omega+\omega$, then $a+b=\omega\cdot 3>b$. Here is what is true: first, the key result you should establish (by induction) is that An ordinal $\alpha>0$ has the property that for all $\beta

Very briefly: Yes, there are several programs being developed that can be understood as pursuing new axioms for set theory. For the question itself of whether pursuing new axioms is a reasonably line of inquiry, see the following (in particular, the paper by John Steel): MR1814122 (2002a:03007). Feferman, Solomon; Friedman, Harvey M.; Maddy, Penelope; Steel, […]

This is a very interesting question and the subject of current research in set theory. There are, however, some caveats. Say that a set of reals is $\aleph_1$-dense if and only if it meets each interval in exactly $\aleph_1$-many points. It is easy to see that such sets exist, have size $\aleph_1$, and in fact, if $A$ is $\aleph_1$-dense, then between any tw […]