The Continuum Hypothesis - Logic at Harvard

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