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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Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run of the game for $A$ is coded by a real, […]

Yes. This is obvious if there are no such cardinals. (I assume that the natural numbers of the universe of sets are the true natural numbers. Otherwise, the answer is no, and there is not much else to do.) Assume now that there are such cardinals, and that "large cardinal axiom" is something reasonable (so, provably in $\mathsf{ZFC}$, the relevant […]

Please send an email to mathrev@ams.org, explaining the issue. (This is our all-purpose email address; any mistakes you discover, not just regarding references, you can let us know there.) Give us some time, I promise we'll get to it. However, if it seems as if the request somehow fell through the cracks, you can always contact one of your friendly edit […]

The characterization mentioned by Mohammad in his answer really dates back to Lev Bukovský in the early 70s, and, as Ralf and Fabiana recognize in their note, has nothing to do with $L$ or with reals (in their note, they indicate that after proving their result, they realized they had essentially rediscovered Bukovský's theorem). See MR0332477 (48 #1080 […]

For $\lambda$ a scalar, let $[\lambda]$ denote the $1\times 1$ matrix whose sole entry is $\lambda$. Note that for any column vectors $a,b$, we have that $a^\top b=[a\cdot b]$ and $a[\lambda]=\lambda a$. The matrix at hand has the form $A=vw^\top$. For any $u$, we have that $$Au=(vw^\top)u=v(w^\top u)=v[w\cdot u]=(w\cdot u)v.\tag1$$ This means that there are […]

That you can list $K $ does not mean you can list its complement. Perhaps the thing to note to build your intuition is that the program is not listing the elements of $K $ in increasing order. Indeed, maybe program 20 halts on input 20 but only does it after several million steps, while program 19 doesn't halt on input 19 and program 21 halts on input 2 […]

A reasonable follow-up question is whether there are some natural algebraic properties that the class of cardinals satisfies (provably in $\mathsf{ZF}$ or in $\mathsf{ZF}$ together with a weak axiom of choice). This is a natural problem and was investigated by Tarski in the 1940s, see MR0029954 (10,686f). Tarski, Alfred. Cardinal Algebras. With an Appendix: […]

Yes. Lev Bukovský proved a very general theorem that deals precisely with this problem: MR0332477 (48 #10804). Characterization of generic extensions of models of set theory, Fundamenta Mathematica 83 (1973), pp. 35–46. Bukovský characterizes when, for a given regular cardinal $\lambda$, $V$ is a $\lambda$-cc generic extension of a given inner model $W$. For […]

This is a question with a long history. As I mentioned in a comment, I think the best reference to get started on these matters is MR0373902 (51 #10102). Marek, W.; Srebrny, M. Gaps in the constructible universe. Ann. Math. Logic 6 (1973/74), 359–394. The paper does not require knowledge of fine structure, it is directly concerned with the question, provides […]