We show that several theorems about Polish spaces, which depend on the axiom of choice ), have interesting corollaries that are theorems of the theory \, where \ is the axiom of dependent choices. Surprisingly it is natural to use the full \ to prove the existence of these proofs; in fact we do not even know the proofs in \. Let \ denote the axiom of determinacy. We show also, in the theory \\), a theorem which strenghtens and generalizes (...) the theorem of Drinfeld and Margulis about the unicity of Lebesgue’s measure. This generalization is false in \, but assuming the existence of large enough cardinals it is true in the model \,\in \rangle \). (shrink)

Given that \(L(\mathbb {R})\models {\text {ZF}}+ {\text {AD}}+{\text {DC}}\), we present conditions under which one can generically add new elements to \(L(\mathbb {R})\) and obtain a model of \({\text {ZF}}+ {\text {AD}}+{\text {DC}}\). This work is motivated by the desire to identify the smallest cardinal \(\kappa \) in \(L(\mathbb {R})\) for which one can generically add a new subset \(g\subseteq \kappa \) to \(L(\mathbb {R})\) such that \(L(\mathbb {R})(g)\models {\text {ZF}}+ {\text {AD}}+{\text {DC}}\).

Strong Turing Determinacy, or ${\mathrm {sTD}}$, is the statement that for every set A of reals, if $\forall x\exists y\geq _T x (y\in A)$, then there is a pointed set $P\subseteq A$. We prove the following consequences of Turing Determinacy ( ${\mathrm {TD}}$ ) and ${\mathrm {sTD}}$ over ${\mathrm {ZF}}$ —the Zermelo–Fraenkel axiomatic set theory without the Axiom of Choice: (1) ${\mathrm {ZF}}+{\mathrm {TD}}$ implies $\mathrm {wDC}_{\mathbb {R}}$ —a weaker version of $\mathrm {DC}_{\mathbb {R}}$.(2) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every (...) set of reals is measurable and has Baire property.(3) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every uncountable set of reals has a perfect subset.(4) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that for every set of reals A and every $\epsilon>0$ :(a)There is a closed set $F\subseteq A$ such that $\mathrm {Dim_H}(F)\geq \mathrm {Dim_H}(A)-\epsilon $, where $\mathrm {Dim_H}$ is the Hausdorff dimension.(b)There is a closed set $F\subseteq A$ such that $\mathrm {Dim_P}(F)\geq \mathrm {Dim_P}(A)-\epsilon $, where $\mathrm {Dim_P}$ is the packing dimension. (shrink)

This paper criticizes non-constructive uses of set theory in formal economics. The main focus is on results on preference aggregation and Arrow’s theorem for infinite electorates, but the present analysis would apply as well, e.g., to analogous results in intergenerational social choice. To separate justified and unjustified uses of infinite populations in social choice, I suggest a principle which may be called the Hildenbrand criterion and argue that results based on unrestricted axiom of choice do not meet this criterion. The (...) technically novel part of this paper is a proposal to use a set-theoretic principle known as the axiom of determinacy ), not as a replacement for Choice, but simply to eliminate applications of set theory violating the Hildenbrand criterion. A particularly appealing aspect of \ from the point of view of the research area in question is its game-theoretic character. (shrink)

We show, using the fine structure of K, that the theory ZF + AD + X R[X K] implies the existence of an inner model of ZF + AD + DC containing a measurable cardinal above its Θ, the supremum of the ordinals which are the surjective image of R. As a corollary, we show that HODK = K for some P K where K is the Dodd-Jensen Core Model relative to P. In conclusion, we show that the theory ZF (...) + AD + ¬DCR implies that R† exists. (shrink)

It is proved that in the absence of proper class inner models with Woodin cardinals, for each n ε {1,…,ω}, ∑3 + n1 absoluteness implies there are n strong cardinals in K (where this denotes a suitably defined global version of the core model for one Woodin cardinal as exposed by Steel. Combined with a forcing argument of Woodin, this establishes that the consistency strength of ∑3 + n1 absoluteness is exactly that of n strong cardinals so that in particular (...) projective absoluteness is equiconsistent with the existence of infinitely many strong cardinals. It is also argued how this theorem is to be construed as the first step in the long range program of showing that projective determinacy is equivalent to its analytical consequences for the projective sets which would settle positively a conjecture of Woodin and thereby solve the last Delfino problem. (shrink)

This paper consists of three parts supplementing the papers of K. Hauser 2002 and D. Mumford 2000: There exist regular open sets of points in with paradoxical properties, which are constructed without using the axiom of choice or the continuum hypothesis. There exist canonical universes of sets in which one can define essentially all objects of mathematical analysis and in which all our intuitions about probabilities are true. Models satisfying the full axiom of choice cannot satisfy all those intuitions and (...) they violate them in a very similar way as those models which satisfy also the continuum hypothesis. (shrink)

This paper introduces the real core model K() and determines the extent of scales in this inner model. K() is an analog of Dodd-Jensen's core model K and contains L(), the smallest inner model of ZF containing the reals R. We define iterable real premice and show that Σ1∩() has the scale property when vR AD. We then prove the following Main Theorem: ZF + AD + V = K() DC. Thus, we obtain the Corollary: If ZF + AD +()L() (...) is consistent, then ZF + AD + DC + α < ω2 -AD is also consistent. (shrink)

Jensen's celebrated Covering Lemma states that if 0# does not exist, then for any uncountable set of ordinals X, there is a Y∈L such that X⊆Y and |X| = |Y|. Working in ZF + AD alone, we establish the following analog: If ℝ# does not exist, then L(ℝ) and V have exactly the same sets of reals and for any set of ordinals X with |X| ≥ΘL(ℝ), there is a Y∈L(ℝ) such that X⊆Y and |X| = |Y|. Here ℝ is (...) the set of reals and Θ is the supremum of the ordinals which are the surjective image of ℝ. (shrink)