The Continuum Hypothesis - Logic at Harvard

More documents

Recommendations

Info

an axiom V=L Ω with the following features: First, V=L Ω is “effectivelycomplete.” Second, V=L Ω is compatible with all large cardinal axioms.Thus, on this scenario, the ultimate theory would be the (open-ended) theoryZFC+V=LΩ +LCA, where LCAis a schema standing for“largecardinalaxioms.” The large cardinal axioms will catch instances of Gödelian independenceand the axiom V=L Ω will capture the remaining instances of independence.This theory would imply CH and settle the remaining undecidedstatements. Independence would cease to be an issue.It turns out, however, that there are other candidate axioms that sharethese features, and so the spectre of pluralism reappears. For example, thereare axioms V=L Ω S and V=LΩ (∗). These axioms would also be “effectivelycomplete” and compatible with all large cardinal axioms. Yet they wouldresolve various questions differently than the axiom V=L Ω . For example,the axiom, V=L Ω (∗)would imply ¬CH. How, then, is one to adjudicatebetween them?Further Reading: For an introduction to inner model theory see Mitchell(2010) and Steel (2010). For more on the recent developments at the level ofone supercompact and beyond see Woodin (2011b).7 The Structure Theory of L(V λ+1 )This brings us to the second global approach, one that promises to select thecorrect axiom from among V=L Ω , V=L Ω S , V=LΩ (∗), and their variants. Thisapproach is based on the remarkable analogy between the structure theoryof L(R) under the assumption of AD L(R) and the structure theory of L(V λ+1 )under the assumption that there is an elementary embedding from L(V λ+1 )into itself with critical point below λ. This embedding assumption is thestrongest large cardinal axiom that appears in the literature.The analogy between L(R) and L(V λ+1 ) is based on the observation thatL(R) is simply L(V ω+1 ). Thus, λ is the analogue of ω, λ + is the analogueof ω 1 , and so on. As an example of the parallel between the structure theoryof L(R) under AD L(R) and the structure theory of L(V λ+1 ) under theembedding axiom, let us mention that in the first case, ω 1 is a measurablecardinal in L(R) and, in the second case, the analogue of ω 1 —namely, λ + —isa measurable cardinal in L(V λ+1 ). This result is due to Woodin and is justone instance from among many examples of the parallel that are contained34
in his work.Now, we have a great deal of information about the structure theory ofL(R) under AD L(R) . Indeed, as we noted above, this axiom is “effectivelycomplete” with regard to questions about L(R). In contrast, the embeddingaxiom on its own is not sufficient to imply that L(V λ+1 ) has a structure theorythat fully parallels that of L(R) under AD L(R) . However, the existence ofan already rich parallel is evidence that the parallel extends, and we can supplementthe embedding axiom by adding some key components. When onedoes so, something remarkable happens: the supplementary axioms becomeforcing fragile. This means that they have the potential to erase independenceand provide non-trivial information about V λ+1 . For example, thesesupplementary axioms might settle CH and much more.The difficulty in investigating the possibilities for the structure theory ofL(V λ+1 ) is that we have not had the proper lenses through which to view it.Thetroubleis thatthemodel L(V λ+1 )contains alargepiece oftheuniverse—namely, L(V λ+1 )—and the theory of this structure is radically underdetermined.The results discussed above provide us with the proper lenses. Forone can examine the structure theory of L(V λ+1 ) in the context of ultimateinner models like L Ω , L Ω S , LΩ (∗), and their variants. The point is that thesemodels can accommodate the embedding axiom and, within each, one willbe able to compute the structure theory of L(V λ+1 ).This provides a means to select the correct axiom from among V=L Ω ,V=L Ω S , V=LΩ (∗) , and their variants. One simply looks at the L(V λ+1) ofeach model (where the embedding axiom holds) and checks to see which hasthe true analogue of the structure theory of L(R) under the assumption ofAD L(R) . Itisalreadyknownthatcertainpieces ofthestructuretheorycannothold in L Ω . But it is open whether they can hold in L Ω S .Let us consider one such (very optimistic) scenario: The true analogueof the structure theory of L(R) under AD L(R) holds of the L(V λ+1 ) of L Ω Sbut not of any of its variants. Moreover, this structure theory is “effectivelycomplete” for the theory of V λ+1 . Assuming that there is a proper classof λ where the embedding axiom holds, this gives an “effectively complete”theory of V. And, remarkably, part of that theory is that V must be L Ω S .This (admittedly very optimistic) scenario would constitute a very strongcase for axioms that resolve all of the undecided statements.One should not place too much weight on this particular scenario. It isjust one of many. The point is that we are now in a position to write down a35
Page 1 and 2: The Continuum HypothesisPeter Koell
Page 3 and 4: and Determinacy”, which contains
Page 5 and 6: infinite cardinals: Assuming GCH, i
Page 7 and 8: Theorem 1.6 (Shelah, 1978). If ℵ
Page 9 and 10: this version of CH holds for all se
Page 11 and 12: more generally, all universally Bai
Page 13 and 14: arithmetic (orabout L(R)) inthepres
Page 15 and 16: 3.2 Ω-LogicWe will now recast the
Page 17 and 18: We will need a strong form of this
Page 19 and 20: and T A ≠ T B .How then shall one
Page 21 and 22: Now to arrive at this conclusion on
Page 23 and 24: Theorem 4.1 (Woodin). Assume ZFC an
Page 25 and 26: Theorem 4.6 (Woodin). Assume ZFC an
Page 27 and 28: forced to fail. The challenge for t
Page 29 and 30: So, of the various theories T A inv
Page 31 and 32: Theorem 5.8 (Koellner and Woodin).
Page 33: the fact that under (∗) the theor
Page 37 and 38: Koellner, P. (2010). Strong logics

The Continuum Hypothesis - Logic at Harvard

Create successful ePaper yourself

Delete template?

Save as template?