116c- Lecture 19

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”.)


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