Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe consistency strength of long projective determinacy
72
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We determine the consistency strength of determinacy for projective games of length $omega^2$. Our main theorem is that $boldsymbolPi^1_{n+1}$-determinacy for games of length $omega^2$ implies the existence of a model of set theory with $omega + n$ Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals $A$ such that $M_n(A)$, the canonical inner model for $n$ Woodin cardinals constructed over $A$, satisfies $A = mathbb{R}$ and the Axiom of Determinacy. Then we argue how to obtain a model with $omega + n$ Woodin cardinal from this. We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length $omega^2$ with payoff in $Game^mathbb{R} boldsymbolPi^1_1$ or with $sigma$-projective payoff.
rate research
Read More
We prove that various classical tree forcings -- for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing -- preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes classes are added to thin projective transitive relations by these forcings.
We prove a number of results on the determinacy of $sigma$-projective sets of reals, i.e., those belonging to the smallest pointclass containing the open sets and closed under complements, countable unions, and projections. We first prove the equivalence between $sigma$-projective determinacy and the determinacy of certain classes of games of variable length ${<}omega^2$ (Theorem 2.4). We then give an elementary proof of the determinacy of $sigma$-projective sets from optimal large-cardinal hypotheses (Theorem 4.4). Finally, we show how to generalize the proof to obtain proofs of the determinacy of $sigma$-projective games of a given countable length and of games with payoff in the smallest $sigma$-algebra containing the projective sets, from corresponding assumptions (Theorems 5.1 and 5.4).
In this paper we continue the study, from Frittaion, Steila and Yokoyama (2017), on size-change termination in the context of Reverse Mathematics. We analyze the soundness of the SCT method. In particular, we prove that the statement any program which satisfies the combinatorial condition provided by the SCT criterion is terminating is equivalent to $mathrm{WO}(omega_3)$ over $mathsf{RCA_0}$
We undertake the study of size-change analysis in the context of Reverse Mathematics. In particular, we prove that the SCT criterion is equivalent to $Sigma^0_2$-induction over RCA$_0$.
The Gratzer-Schmidt theorem of lattice theory states that each algebraic lattice is isomorphic to the congruence lattice of an algebra. We study the reverse mathematics of this theorem. We also show that the set of indices of computable lattices that are complete is $Pi^1_1$-complete; the set of indices of computable lattices that are algebraic is $Pi^1_1$-complete; the set of compact elements of a computable lattice is $Pi^{1}_{1}$ and can be $Pi^1_1$-complete; and the set of compact elements of a distributive computable lattice is $Pi^{0}_{3}$, and there is an algebraic distributive computable lattice such that the set of its compact elements is $Pi^0_3$-complete.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
