CALIBRATING DETERMINACY STRENGTH IN LEVELS OF THE BOREL HIERARCHY

AbstractWe analyze the set-theoretic strength of determinacy for levels of the Borel hierarchy of the form$\Sigma _{1 + \alpha + 3}^0 $, forα<ω1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to requireα+ 1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of weak reflection principles, Π1-RAPα, whose consistency strength corresponds exactly to the logical strength of${\rm{\Sigma }}_{1 + \alpha + 3}^0 $determinacy, for$\alpha < \omega _1^{CK} $. This yields a characterization of the levels ofLby or at which winning strategies in these games must be constructed. Whenα= 0, we have the following concise result: The leastθso that all winning strategies in${\rm{\Sigma }}_4^0 $games belong toLθ+1is the least so that$L_\theta \models {\rm{``}}{\cal P}\left( \omega \right)$exists, and all wellfounded trees are ranked”.

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An algebraic characterization of power set in countable standard models of ZF

Journal of Symbolic Logic ◽

10.2307/2271897 ◽

1975 ◽

Vol 40 (2) ◽

pp. 167-170

Author(s):

George Metakides ◽

J. M. Plotkin

Keyword(s):

Standard Model ◽

Boolean Algebra ◽

Classical Result ◽

Algebraic Characterization ◽

Categorical Theory ◽

Power Set ◽

Standard Models ◽

Image Position ◽

Atomic Boolean Algebra

The following is a classical result:Theorem 1.1. A complete atomic Boolean algebra is isomorphic to a power set algebra [2, p. 70].One of the consequences of [3] is: If M is a countable standard model of ZF and is a countable (in M) model of a complete ℵ0-categorical theory T, then there is a countable standard model N of ZF and a Λ ∈ N such that the Boolean algebra of definable (in T with parameters from ) subsets of is isomorphic to the power set algebra of Λ in N. In particular if and T the theory of equality with additional axioms asserting the existence of at least n distinct elements for each n < ω, then there is an N and Λ ∈ N with 〈PN(Λ), ⊆〉 isomorphic to the countable, atomic, incomplete Boolean algebra of the finite and cofinite subsets of ω.From the above we see that some incomplete Boolean algebras can be realized as power sets in standard models of ZF.Definition 1.1. A countable Boolean algebra 〈B, ≤〉 is a pseudo-power set if there is a countable standard model of ZF, N and a set Λ ∈ N such thatIt is clear that a pseudo-power set is atomic.

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Singular perturbations of the unicritical polynomials with two parameters

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2015.114 ◽

2016 ◽

Vol 37 (6) ◽

pp. 1997-2016 ◽

Cited By ~ 1

Author(s):

YINGQING XIAO ◽

FEI YANG

Keyword(s):

Julia Set ◽

Dynamical Behavior ◽

Rational Maps ◽

Topological Properties ◽

Fatou Set ◽

The Family ◽

Image Position ◽

Two Parameters ◽

Unicritical Polynomials

In this paper, we study the dynamics of the family of rational maps with two parameters $$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$ where $n\geq 2$ and $a,b\in \mathbb{C}^{\ast }$. We give a characterization of the topological properties of the Julia set and the Fatou set of $f_{a,b}$ according to the dynamical behavior of the orbits of the free critical points.

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