The class forcing theorem, which asserts that every class forcing notion admits a forcing relation , that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory to the principle of elementary transfinite recursion for class recursions of length . It is also equivalent to the existence of truth predicates for the infinitary languages , allowing any class parameter A; to the existence of truth predicates for the language ; to the existence of -iterated truth predicates for first-order set theory ; to the assertion that every separative class partial order has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most . Unlike set forcing, if every class forcing notion has a forcing relation merely for atomic formulas, then every such has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between and Kelley–Morse set theory .
Vaginal pulse amplitude (VPA) has been the most commonly analyzed signal of the vaginal photoplethysmograph. Frequent, large, and variable-morphology artifacts typically have crowded this signal. These artifacts usually were corrected by hand, which may have introduced large differences in outcomes across laboratories. VPA signals were collected from 22 women who viewed a neutral film and a sexual film. An automated, wavelet-based, denoising algorithm was compared against the uncorrected signal and the signal corrected in the typical manner (by hand). The automated wavelet denoising resulted in the same pattern of results as the hand-corrected signal. The wavelet procedure automated artifact reduction in the VPA, and this mathematical instantiation permits the comparison of competing methods to improve signal:noise in the future.
A sequence of Laver diamonds for κ is joint if for any sequence of targets there is a single elementary embedding j with critical point κ such that each Laver diamond guesses its respective target via j. In the case of measurable cardinals (with similar results holding for (partially) supercompact cardinals) I show that a single Laver diamond for κ yields a joint sequence of length κ, and I give strict separation results for all larger lengths of joint sequences. Even though the principles get strictly stronger in terms of direct implication, I show that they are all equiconsistent. This is contrasted with the case of θ-strong cardinals where, for certain θ, the existence of even the shortest joint Laver sequences carries nontrivial consistency strength. I also formulate a notion of jointness for ordinary ♦ κ -sequences on any regular cardinal κ. The main result concerning these shows that there is no separation according to length and a single ♦ κ -sequence yields joint families of all possible lengths.In chapter 2 the notion of a grounded forcing axiom is introduced and explored in the case of Martin's axiom. This grounded Martin's axiom, a weakening of the usual axiom, states that the universe is a ccc forcing extension of some inner model and the restriction of Martin's axiom to the posets coming from that ground model holds. I place the new axiom in the hierarchy of fragments of Martin's axiom and examine its effects on the cardinal characteristics of the continuum. I also show that the grounded version is quite a bit more robust under mild forcing than Martin's axiom itself. I wish to thank my advisor, Joel David Hamkins, for his constant support and guidance through my studies and the writing of this dissertation. It is safe to say that this text would not exist without his keen insight, openness to unusual ideas, and patience. It has been a pleasure and a privilege to work with him and I could not have asked for a better role model of mathematical ingenuity, rigour, and generosity. I would also like to thank Arthur Apter and Gunter Fuchs, the other members of my dissertation committee. Thank you both for your contributions, the many conversations, and for taking the time to read through this text. Thank you as well to Kameryn, Vika, Corey, the whole New York logic community, and the many other friends at the Graduate Center. I will remember fondly the hours spent talking about mathematics, playing games, and doing whatever other things graduate students do. Thank you, Kaethe. You have been (and hopefully remain) my best friend, sharing my happy moments and supporting me in the less happy ones. Thank you for letting me explain every part of this dissertation to you multiple times, and thank you for teaching me about mathematics that I would never have known or understood without you. I am incredibly fortunate to have met you and I can only hope to return your kindness and empathy in the future. Lastly, thank you to my parents and other family members who have stood by me and supported me through my long jo...
In this article I investigate the phenomenon of minimum models of second-order set theories, focusing on Kelley-Morse set theory KM, Gödel-Bernays set theory GB, and GB augmented with the principle of Elementary Transfinite Recursion. The main results are the following. (1) A countable model of ZFC has a minimum GBC-realization if and only if it admits a parametrically definable global well-order. (2) Countable models of GBC admit minimal extensions with the same sets. (3) There is no minimum transitive model of KM. (4) There is a minimum β-model of GB + ETR. The main question left unanswered by this article is whether there is a minimum transitive model of GB + ETR.2010 Mathematics Subject Classification. Primary 03E70; Secondary 03C62. Key words and phrases. Kelley-Morse, Gödel-Bernays, elementary transfinite recursion, minimum model, second-order set theory.I am grateful to the anonymous referee for their many helpful comments. The present article is substantially improved thanks to their time and care in reviewing it. I also want to thank Joel David Hamkins, under whose supervision I wrote my dissertation, written while this paper was under review and consisting in part of the material herein.1 The anonymous referee noted that the Shepherdson-Cohen minimum model Lα is also the minimum model of ZF in the following sense: it is, up to isomorphism, the unique model of ZF which can be isomorphically embedded into every model of ZF. I reproduce their proof of this fact here. If N is well-founded, then by the Shepherdson-Cohen theorem Lα embeds into a transitive submodel of N . In this case, the embedding is moreover ∆ 0 -elementary. If N is not well-founded, it has a countable elementary submodel N 0 which must also be ill-founded. By a theorem of Hamkins [Ham13, corollary 29] there is therefore an embedding of Lα into N 0 , hence an embedding of Lα into N . That Lα is the unique such model follows from the Shepherdson-Cohen theorem combined with the fact that any model which embeds into a well-founded model must also well-founded. 1 2 KAMERYN J. WILLIAMS There is, for instance, a minimum transitive model of ZFC + there is a Mahlo cardinal. It is not difficult to see, however, that this cannot be extended too far up the large cardinal hierarchy. There is no minimum transitive model of any T extending ZFC which proves there is a measurable cardinal. This is because by a well-known theorem of Scott [Sco61], if N |= ZFC has a measurable cardinal then there is an elementary embedding j : N → M to an inner model M , with the measure not in M . 2 So any transitive model of such T contains a strictly smaller transitive model of T .Moreover observe that Scott's theorem can be internalized to any model of ZFC with a measurable cardinal, whether or not it is well-founded. This yields that any M |= ZFC + "there is measurable cardinal" has a proper transitive submodel which satisfies the same first-order theory. Define the pre-order ⊳ end on models of ZFC as M ⊳ end N if M embeds as a transitive submodel of N . Or to use lang...
We present a class forcing notion M(η), uniformly definable for ordinals η, which forces the ground model to be the η-th inner mantle of the extension, in which the sequence of inner mantles has length at least η. This answers a conjecture of Fuchs, Hamkins, and Reitz [FHR15] in the positive. We also show that M(η) forces the ground model to be the η-th iterated HOD of the extension, where the sequence of iterated HODs has length at least η. We conclude by showing that the lengths of the sequences of inner mantles and of iterated HODs can be separated to be any two ordinals you please.
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