116c- Lecture 20

June 6, 2008

We showed that $\mathsf{AD}$ implies a weak version of choice, $\mathsf{AC}_\omega({}^\omega\omega)$, namely, every countable family of non-empty sets of reals admits a choice function. This implies that $\omega_1$ is regular and suffices to develop classical analysis in a straightforward fashion (in particular, to construct Lebesgue measure and to prove its basic properties).

Coupled with the fact that all sets of reals have the perfect set property, this implies that $\omega_1>\omega_1^{L[x]}$ for any real $x$ and therefore $\omega_1^V$ is strongly inaccessible in $L[x]$ for any real $x$.

We closed the course by showing that, in fact, $\omega_1$ is a measurable cardinal. We proved this result of Solovay by showing Martin’s result that the “cone measure” is indeed a non-atomic measure on the structure ${\mathcal D}$ of the Turing degrees and then “pulling back” this measure to $\omega_1$.

Finally, given any measurable cardinal $\kappa$, let $\mu$ be a ($\kappa$-complete, non-principal) measure on $\kappa$. Then $L[\mu]$ is a model of choice in which $\kappa$ is measurable. In particular, ${\rm Con}({\sf ZF}+{\sf AD})\Rightarrow{\rm Con}({\sf ZFC}+\mbox{There is a measurable cardinal}).$

Since, under choice, any measurable cardinal is strongly inaccessible and the limit of strongly inaccessible cardinals, this shows that ${\sf AD}$ has significant consistency strength.

116c- Lecture 19

June 4, 2008

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 $A=\mathsf{ZFC}+$“There is a weakly inaccessible cardinal” and $B=\mathsf{ZFC}+$“There is a strongly inaccessible cardinal” are equiconsistent. This means that a weak theory (much less than $\mathsf{PA}$ suffices) can prove $\mathrm{Con}(A)\Leftrightarrow\mathrm{Con}(B)$. Namely: $A$ is a subtheory of $B$, so its inconsistency implies the inconsistency of $B$. Assume $B$ is inconsistent and fix a (say, Hilbert-style) proof $\phi_0,\dots,\phi_n$ of an inconsistency from $B$. Then a proof $\psi_0,\dots,\psi_m$ of an inconsistency from $A$ can be found by showing (by induction on $i\le n$) that each $\phi_i^L$ is a theorem of $A$, and this argument can be carried out in a theory (such as $\mathsf{PA}$) 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 $\mathsf{ZFC}$ by large cardinals. One direction (that models with large cardinals generate models of the hypothesis under study) typically involves the method of forcing and will not be further discussed here. The other direction, just as in the very simple example of weak vs strong inaccessibility, typically requires showing that certain transitive classes (such as $L$) 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 $\mathsf{AD}$. It contradicts choice but it relativizes to the model $L(\mathbb{R})$. This is actually the natural model to study $\mathsf{AD}$ and, in fact, from large cardinals one can prove that $L(\mathbb{R})\models\mathsf{AD}$.

We illustrated basic consequences of $\mathsf{AD}$ 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 $\mathsf{CH}$ is true under $\mathsf{AD}$). Similar arguments give that $\mathsf{AD}$ 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 $\mathsf{AD}$ implies the consistency of strongly inaccessible cardinals. (Though this is beyond the scope of this course, by using more sophisticated ideas, one can prove the optimal stronger result that the consistency of $\mathsf{AD}$ implies the consistency of $\mathsf{ZFC}+$ “there exist infinitely many Woodin cardinals”.)

116c- Lecture 18

May 30, 2008

We briefly discussed relative constructibility and compared the models $L[x]$ where $x$ is exclusively treated as a predicate with the models $L(x)$ where $x$ is an element. In particular, $L[x]$ is a model of choice but $L(x)$ may fail to be.

An amusing application of the fact that $L[x]\models\mathsf{AC}$ is that the result of Exercise 3 from Homework 7 holds in $\mathsf{ZF}$, although the proof I wrote there uses choice. Namely, work in $\mathsf{ZF}$ and consider two well-orderings of a set $X$. We can assume that $X$ is an ordinal $\alpha$ and the first well-ordering is $\in$. Let $\prec$ be the second well-ordering. Then $\prec\in L[\prec]$ (since $\prec$ is a set of ordered pairs of ordinals). In $L[\prec]$, where choice holds, (and therefore also in $V$) there is a subset of $\alpha$ of the same size as $\alpha$ and where $\prec$ coincides with $\in$.

Question. Find a choice-free’ argument for Exercise 3.

(See here.)

The main example we will consider of a model of the form $L(x)$ is $L(\mathbb{R})$, due to its connection with determinacy.

We introduced the setting to discuss determinacy, namely infinite 2-person games with perfect information. We proved the Gale-Stewart theorem that open games are determined and discussed Martin’s extension to Borel games. A nice reference for the proof of Martin’s result (using the idea of unraveling‘, which reduces any Borel game to an open game in a different space) is Kechris’s book on descriptive set theory:

MR1321597 (96e:03057). Kechris, Alexander S. Classical descriptive set theory. Graduate Texts in Mathematics, 156. Springer-Verlag, New York, 1995. xviii+402 pp. ISBN: 0-387-94374-9.

116c- Homework 8

May 29, 2008

Homework 8

Due Thursday, June 5 at 2:30 pm.

116c- Lecture 17

May 29, 2008

We verified that the sets $L_\alpha$ form a continuous increasing sequence and are transitive. It follows that the reflection theorem holds for the $L_\alpha$ and $L$. Arguing in $\mathsf{ZF}$, we proved that $L$ is a model of $\mathsf{ZF}$, and the reflection theorem allowed us to simplify the proof in a few points.

We then proceeded to argue that $L$ is also a model of choice. In fact, there is a globally definable well-ordering of $L$. It is worth emphasizing that the well-ordering is a very natural one, as we simply proceed to enumerate the sets in $L$ in the order in which their membership is verified. The definitions of the sequence of sets $L_\alpha$ and of this well-ordering are absolute, and we used this to prove that $L$ is a model of the statement “$V=L$,” and so is any $L_\alpha$, for $\alpha$ limit. Moreover, the well-ordering of $L$, when restricted to $L_\alpha$, coincides with its interpretation inside $L_\alpha$.

An easy induction shows that for $\alpha$ infinite, $|L_\alpha|=|\alpha|$. An argument using the Mostowski collapsing theorem allowed us to prove Gödel’s condensation lemma: If $X\prec L_\alpha$ for $\alpha$ a limit ordinal, then $X$ is isomorphic to some $L_\beta$. These two facts combine to provide a proof that $\mathsf{GCH}$ holds in $L$.

Remark. These arguments prove that $\mathrm{Con}(\mathsf{ZF})$ implies $\mathrm{Con}(\mathsf{ZFC})$, but they also indicate that showing that $\mathrm{Con}(\mathsf{ZF})$ implies $\mathrm{Con}(\mathsf{ZF}+\lnot\mathsf{AC})$ ought to be more complicated. The reason is that the absoluteness of the construction of $L$ implies that if $M$ is a transitive proper class model of $\mathsf{ZF}$, then $L\subseteq M$ and in fact $L^M=L$, i.e., the result of running the construction of $L$ from the point of view of $M$ is $L$ itself. But, since $V=L$ holds in $L$, we cannot prove in $\mathsf{ZF}$ that there is a non-constructible set. If we tried to establish the consistency of $\mathsf{ZF}$ with the negation of choice by a similar method, namely, the construction of a transitive class model $M$ of $\mathsf{ZF}+\lnot\mathsf{AC}$, then running the construction inside $L$ would give us that $L=L^{M^L}\subseteq M^L\subseteq L$, so $M^L=L$, which would be a contradiction, since we are assuming that (provably in $\mathsf{ZF}$) $M$ is a model of $\lnot\mathsf{AC}$ but $L$ is a model of choice.

This also suggests that in order to show that $\mathsf{AC}$ is independent of $\mathsf{ZF}$, one should try first to show that $V\ne L$ is consistent with $\mathsf{ZF}$. The remarkable solution found by Paul Cohen in 1963, the method of forcing, allows us to prove the consistency of both statements, and also to do this while working with transitive models. The method of forcing is beyond the scope of this course, but good explanations can be found in a few places, there is for example a book by Cohen himself, or look at Kunen’s book mentioned at the beginning of the course. Richard Zach has compiled in his blog a list of papers providing an introduction to the method (search for forcing’).

116c- Lecture 16

May 24, 2008

We showed that $\Delta_1$ formulas are absolute among transitive models of (enough) set theory, and used this to prove that satisfiability for transitive sets is absolute. More precisely, let $\mathrm{Sat}(a,b,c)$ mean that $a$ is a transitive set $M$, $b$ codes a formula $\varphi(\vec x)$$c$ is a tuple $\vec X$ of elements of $M$, and $M\models\varphi(\vec X)$. Then $\mathrm{Sat}(a,b,c)$ is $\Delta_1$. Using this and the reflection theorem we can conclude that $\Delta_1$ is actually the extent of absoluteness in set theory, meaning that whenever there is a finite $S$ such that $\phi$ is absolute for transitive models of $S$, then $\phi$ is (provably equivalent to) a $\Delta_1$ statement.

We exhibited a few formulas that are not absolute. For example, “$x$ is a cardinal” and “$x=V_\alpha$,” although both are $\Pi_1$ and therefore relativize downwards.

The main application of the absoluteness of satisfiability is that it allows us to define the constructible hierarchy and Gödel’s constructible universe $L$.

Remark. On the other hand, we cannot define in general satisfiability for transitive classes, by Tarski’s undefinability of truth theorem. The difference with the case of sets is that with sets the recursive definition of $M\models\dots$ involves several bounded quantifiers, ranging over finite powers of $M$. With general proper classes $M$, these quantifiers would be unbounded. An easy inductive argument shows that we can define partial satisfiability predicates (and therefore partial truth predicates), meaning that for each natural number $n$ and each class $M$ we can find a $\Sigma_n$ formula that defines satisfiability for $\Sigma_n$ formulas with respect to $M$; although we cannot in general find a uniform definition that works simultaneously for all $n$.

116c- Lecture 15

May 21, 2008

We presented a list of statements, definable relations, functions, and constants, that are absolute for transitive models of enough set theory. We showed that absolute functions are closed under composition, although $\Delta_0$ functions are not. We also verified that being a well-ordering is absolute. The same argument actually shows:

Theorem. The statement $\text{`} R$ is a well-founded relation on $A\text{''}$ is absolute for transitive models of $\mathsf{ZF}-\mathsf{Power\,set}$.

This is a key result very useful in a variety of situations. Notice that we are not claiming that being well-orderable is absolute; in fact, it is not. The difference is that in the first case we are given a witness to the well-orderability, and claim that no matter in which transitive model the witness is observed, in all of them it has the property of being a well-ordering. The second case only states that there is a witness, and a given model may very well fail to produce such a witness unless it is a model of (at least a fragment of) the axiom of choice.