As discussed during Lecture 13, for the theories one encounters when studying set theory, no absolute consistency results are possible, and we rather look for relative consistency statements. For example, the theories “There is a weakly inaccessible cardinal” and “There is a strongly inaccessible cardinal” are equiconsistent. This means that a weak theory (much less than suffices) can prove . Namely: is a subtheory of , so its inconsistency implies the inconsistency of . Assume is inconsistent and fix a proof of an inconsistency from . Then a proof of an inconsistency from can be found by showing that each is a theorem of , and this argument can be carried out in a theory (such as ) where the syntactic manipulations of formulas that this involves are possible.

It is a remarkable empirical fact that the combinatorial statements studied by set theorists can be measured against a linear scale of consistency, calibrated by the so called large cardinal axioms, of which strongly inaccessible cardinals are perhaps the first natural example. Hypotheses as unrelated as the saturation of the nonstationary ideal or determinacy have been shown equiconsistent with extensions of by large cardinals. One direction (that models with large cardinals generate models of the hypothesis under study) typically involves the method of forcing and won’t be discussed further here. The other direction, just as in the very simple example of weak vs strong inaccessibility, typically requires showing that certain transitive classes (like ) must have large cardinals of the desired sort. We will illustrate these ideas by obtaining large cardinals from determinacy in the last lecture of the course.

We defined the axiom of determinacy . It contradicts choice but it relativizes to the model . This is actually the natural model to study and, in fact, from large cardinals one can prove that .

We illustrated basic consequences of for the theory of the reals by showing that it implies that every set of reals has the perfect set property (and therefore a version of is true under ). Similar arguments give that implies that all sets of reals have the Baire property and are Lebesgue measurable. In the last lecture of the course we will use the perfect set property of sets of reals to show that the consistency of implies the consistency of strongly inaccessible 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 […]

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 […]

Say that the triangle is $ABC$. The vector giving the median from $A$ to $BC$ is $(AC+AB)/2$. Similarly, the one from $B$ to $AC$ is $(BA+BC)/2$, and the one from $C$ to $BA$ is $(CB+CA)/2$. Adding these, we get zero since $CB=-BC$, etc.

The usual definition of a series of nonnegative terms is as the supremum of the sums over finite subsets of the index set, $$\sum_{i\in I} x_i=\sup\biggl\{\sum_{j\in J}x_j:J\subseteq I\mbox{ is finite}\biggr\}.$$ (Note this definition does not quite work in general for series of positive and negative terms.) The point then is that is $a< x