Forcing axioms via ground model interpretations

Simple Forcing Notions and Forcing Axioms

Forcing Axioms and Cardinal Arithmetic

We survey some recent results on the impact of strong forcing axioms such as the Proper Forcing Axiom PFA and Martin’s Maximum MM on cardinal arithmetic. We concentrate on three combinatorial principles which follow from strong forcing axioms: stationary set reflection, Moore’s Mapping Reflection Principle MRP and the P-ideal dichotomy introduced by Abraham and Todorčević which play the key rol...

Hierarchies of forcing axioms I

We prove new upper bound theorems on the consistency strengths of SPFA(θ), SPFA(θ-linked) and SPFA(θ-cc). Our results are in terms of (θ,Γ)-subcompactness, which is a new large cardinal notion that combines the ideas behind subcompactness and Γ-indescribability. Our upper bound for SPFA(c-linked) has a corresponding lower bound, which is due to Neeman and appears in his follow-up to this paper....

Hierarchies of forcing axioms II

A Σ21 truth for λ is a pair 〈Q,ψ〉 so that Q ⊆ Hλ, ψ is a first order formula with one free variable, and there exists B ⊆ H λ+ such that (H λ+ ;∈, B) |= ψ[Q]. A cardinal λ is Σ21 indescribable just in case that for every Σ 2 1 truth 〈Q,ψ〉 for λ, there exists λ̄ < λ so that λ̄ is a cardinal and 〈Q ∩ Hλ̄, ψ〉 is a Σ 2 1 truth for λ̄. More generally, an interval of cardinals [κ, λ] with κ ≤ λ is Σ21 in...

Forcing Axioms, Supercompact Cardinals, Singular Cardinal Combinatorics

The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the results so obtai...