(Time in Japan Standard TIme: GMT +9)
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Gaishi Takeuti gave an exposition on Riis' Axiom [2] and on his own Reflection Axiom (which was introduced in his 1971 paper [3]) in the 1999.05 issue of 数学セミナー (Sugaku Seminar) [4]. In my talk, I am going to address to these two axioms and their variations.
References
[1] Paul Cohen, A Large Power Set Axiom, The Journal of Symbolic Logic, Vol.40, No.1, (1975), 48--54.
[2] Søren Riis, FOM: A proof of not-CH, Sun Sep 13 12:24:49 EDT 1998.
[3] Gaishi Takeuti, Hypotheses on power set, Proceedings of Symposia in Pure Mathematics, Vol.13, Part I, American Mathematical Society, Providence, R.I., (1971), 439--446.
[4] 竹内外史，ランダム実数と連続体仮説，数学セミナー，1999年 5月号，(1999), 34--37.
The Scott process of a relational structure \( M \) is the sequence of sets of formulas given by the Scott analysis of \( M \). In a paper from 2017 we presented axioms for the class of Scott processes of structures in a relational vocabulary \( \tau \), and used them to give new proofs of three theorems from the 1970's on counterexamples to Vaught's Conjecture. In this talk we will review some of the main results from this paper, and talk about some more recent developments, one of which is a theorem with Shelah on the number of models of a fixed Scott rank for a counterexample to Vaught's Conjecture.
It is proved that the epsilon collapse (the pure side condition method) preserves a non-meager set. I will give a motivation of this research and the proof.
We consider generically setwise large cardinal, which is a variant of generic large cardinal. We show that the consistency strength of the existences of generically setwise supercompact is very weak, but is strong if it is greater than \( \omega_2 \).
We shall present some recent results about principles of Structural Reflection that correspond to large cardinal notions at the level of superstrong, globally superstrong, and huge cardinals. Some of the results are joint work with Philipp Lücke.
An old result of Isbell characterises measurable cardinals in terms of certain canonical limits in the category of sets. After introducing this characterisation, I will talk about recent work with Adamek, Campion, Positselski and Rosicky teasing out the importance of the canonicity for this and related results. The language will be category-theoretic but the proofs will be quite hands-on combinatorial constructions with sets.
For infinite cardinals \( \theta < \kappa \), colorings of the form \( f: [ \kappa ]^2 \rightarrow \theta \) that exhibit certain strong unboundedness properties have seen a wide variety of applications. In this talk, we will discuss some results about the existence of such strongly unbounded colorings as well as strongly unbounded colorings that satisfy additional closure or subadditivity requirements. We will then present some recent applications of these colorings to questions regarding the infinite productivity of strong chain conditions and the tightness of topological spaces. This is joint work with Assaf Rinot.
Woodin proved that if AD holds, then \( \Theta \) is a Woodin cardinal in HOD. We show that the converse does not necessarily hold in \( L( \mathbb{R} ) \) . In this talk, we will discuss the background & sketch of proof of this result. This is joint work with Nam Trang.
Given a class \( C \) of forcing notions, \( \mathfrak{m}(C) \) denotes the smallest cardinal where MA (Martin's axiom) for \( C \) fails. Several forcing axioms take the form \( \mathfrak{m}(C)=\kappa \) for some cardinal \( \kappa \), and the axiom is weaker when the cardinal \( \kappa\) gets smaller. In this talk I present techniques to preserve inequalities of the form \( \mathfrak{m}(C)\leq\kappa \) via finite support iterations, where \( C \) is a standard class like ccc, \( n \)-Knaster, precaliber \( \omega_1 \) and \( \sigma \)-centered. This means preserving a failure of \( \mathfrak{m}(C) \) at \( \kappa \), which can be forced by either adding Cohen reals or colorings that support this failure. If time allows, I will show consistency results where the values of \( \mathfrak{m}(C) \) are controlled while modifying other cardinal characteristics, like those in Cichon's diagram, and the groupwise density number. This is a joint work with J. Kellner, M. Goldstern and S. Shelah.
We present a way to deal with names for ultrafilters on Boolean algebras in the random forcing. We show that every such name induces naturally a measure on Boolean algebra. We use this approach to prove and reprove some results about the classical random model. The talk represents joint works with Damian Sobota and Katarzyna Cegielka.
I shall discuss senses in which set-theoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for the atomic or elementary diagram of a model of set theory \( \langle M , \in^M \rangle \), for example, one may in various senses compute \( M \)-generic filters \( G \subset P \in M \) and the corresponding forcing extensions \( M[G] \). Meanwhile, no such computational process is functorial, for there must always be isomorphic alternative presentations of the same model of set theory \( M \) that lead by the computational process to non-isomorphic forcing extensions \( M[G]\not\cong M[G'] \). Indeed, there is no Borel function providing generic filters that is functorial in this sense. This is joint work with Russell Miller and Kameryn Williams.