Eastonʼs theorem and large cardinals from the optimal hypothesis

AbstractThe equiconsistency of a measurable cardinal with Mitchell order o(κ)=κ++ with a measurable cardinal such that 2κ=κ++ follows from the results by W. Mitchell (1984) [13] and M. Gitik (1989) [7]. These results were later generalized to measurable cardinals with 2κ larger than κ++ (see Gitik, 1993 [8]).In Friedman and Honzik (2008) [5], we formulated and proved Eastonʼs (1970) theorem [4] in a large cardinal setting, using slightly stronger hypotheses than the lower bounds identified by Mitchell and Gitik (we used the assumption that the relevant target model contains H(μ), for a suitable μ, instead of the cardinals with the appropriate Mitchell order).In this paper, we use a new idea which allows us to carry out the constructions in Friedman and Honzik (2008) [5] from the optimal hypotheses. It follows that the lower bounds identified by Mitchell and Gitik are optimal also with regard to the general behavior of the continuum function on regulars in the context of measurable cardinals

Global singularization and the failure of SCH

We say that κ is µ-hypermeasurable (or µ-strong) for a cardinal µ ≥ κ + if there is an embedding j: V → M with critical point κ such that H(µ) V is included in M and j(κ)> µ. Such j is called a witnessing embedding. Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then there exists a cardinal-preserving forcing extension V ∗ where F is realised on all V-regular cardinals and moreover: all F(κ)-hypermeasurable cardinals κ, where F(κ)> κ +, with a witnessing embedding j such that either j(F)(κ) = κ + or j(F)(κ) ≥ F(κ), are turned into singular strong limit cardinals with cofinality ω. This provides some partial information about the possible structure of a continuum function with respect to singular cardinals with countable cofinality. As a corollary, this shows that the continuum function on a singular strong limit cardinal κ of cofinality ω is virtually independent of the behaviour of the continuum function below κ, at least for continuum functions which are simple in that 2 α ∈ {α +, α ++} for every cardinal α below κ (in this case every κ ++-hypermeasurable cardinal in the ground model is witnessed either by a j with either j(F)(κ) ≥ F(κ) or j(F)(κ) = κ +)

The Hyperuniverse Project and maximality

This collection documents the work of the Hyperuniverse Project which is a new approach to set-theoretic truth based on justifiable principles and which leads to the resolution of many questions independent from ZFC. The contributions give an overview of the program, illustrate its mathematical content and implications, and also discuss its philosophical assumptions. It will thus be of wide appeal among mathematicians and philosophers with an interest in the foundations of set theory. The Hyperuniverse Project was supported by the John Templeton Foundation from January 2013 until September 2015