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We show that the bounded proper forcing axiom BPFA implies that there is a well-ordering of P(ω 1) which is Δ 1 definable with parameter a subset of ω 1. Our proof shows that if BPFA holds then any inner model of the universe of sets that correctly computes ℵ 2 and also satisfies BPFA must contain all subsets of ω 1. We show as applications how to build… (More)

Assume AD + and that either V = L(P(R)), or V = L(T, R) for some set T ⊂ ORD. Let (X, ≤) be a pre-partially ordered set. Then exactly one of the following cases holds: (1) X can be written as a well-ordered union of pre-chains, or (2) X admits a perfect set of pairwise ≤-incomparable elements, and the quotient partial order induced by (X, ≤) embeds into (2… (More)

Mitochondrial activity is central to tissue homeostasis. Mitochondria dysfunction constitutes a hallmark of many genetic diseases and plays a key role in tumor progression. The essential role of mitochondria, added to their recently documented capacity to transfer from cell to cell, obviously contributes to their current interest. However, determining the… (More)

We show that the bounded proper forcing axiom BPFA implies that there is a well-ordering of P(ω 1) which is ∆ 1 definable with parameter a subset of ω 1. Our proof shows that if BPFA holds then any inner model of the universe of sets that correctly computes ℵ 2 and also satisfies BPFA must contain all subsets of ω 1. We show as applications how to build… (More)

We show that if X is a set and A is a non-empty family of n-element subsets of X that does not contain a pairwise disjoint family of cardinality strictly greater than k, then a non-empty subset of X of cardinality strictly less than kn 2 is definable from A. We show that this result is nearly best possible, and investigate analogous questions for families… (More)

Say that an elementary embedding j : N → M is cardinal preserving if CAR M = CAR N = CAR. We show that if PFA holds then there are no cardinal preserving elementary embeddings j : M → V. We also show that no ultrapower embedding j : V → M induced by a set exten-der is cardinal preserving, and present some results on the large cardinal strength of the… (More)