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Full-Text Articles in Set Theory
Sigma_N-Correct Forcing Axioms, Benjamin P. Goodman
Sigma_N-Correct Forcing Axioms, Benjamin P. Goodman
Dissertations, Theses, and Capstone Projects
I introduce a new family of axioms extending ZFC set theory, the Sigma_n-correct forcing axioms. These assert roughly that whenever a forcing name a' can be forced by a poset in some forcing class Gamma to have some Sigma_n property phi which is provably preserved by all further forcing in Gamma, then a' reflects to some small name such that there is already in V a filter which interprets that small name so that phi holds. Sigma_1-correct forcing axioms turn out to be equivalent to classical forcing axioms, while Sigma_2-correct forcing axioms for Sigma_2-definable forcing classes are consistent relative to …
The Structure Of Models Of Second-Order Set Theories, Kameryn J. Williams
The Structure Of Models Of Second-Order Set Theories, Kameryn J. Williams
Dissertations, Theses, and Capstone Projects
This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of T-realizations of a fixed countable model of ZFC, where T is a reasonable second-order set theory such as GBC or KM, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve …
Joint Laver Diamonds And Grounded Forcing Axioms, Miha Habič
Joint Laver Diamonds And Grounded Forcing Axioms, Miha Habič
Dissertations, Theses, and Capstone Projects
In chapter 1 a notion of independence for diamonds and Laver diamonds is investigated. 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 …