Understanding preservation theorems: chapter VI of Proper and Improper Forcing, I

Proper forcing and rectangular Ramsey theorems

I prove forcing preservation theorems for products of definable partial orders preserving cofinality of the meager or null ideal. Rectangular Ramsey theorems for the related ideals follow from the proofs.

I. Proper forcing

Chapter 1 2 VI Preservation and Transport

T he emerging field of tissue engineering has identified product storage as a significant obstacle for commercialization of products containing living cells. Traditionally employed methods of tissue banking utilize cryopreservation by freezing. Unfortunately, cryopreservation by freezing may result in loss of tissue function and viability by several recognized mechanisms of which ice formation ...

Preservation theorems for countable support forcing iterations

A major theme of [12] is preservation theorems for iterated forcing. These are theorems of the form “if 〈Pξ : ξ ≤ κ〉 is a countable support forcing iteration based on 〈Q̇ξ : ξ < κ〉 and each Q̇ξ has property such-and-such then Pκ has property thus-and-so.” The archetypal preservation theorem is the Fundamental Theorem of Proper Forcing [12, chapter III], which states that if each Q̇ξ is proper in V...

Proper and Improper Separability

The distinction between proper and improper mixtures is a staple of the discussion of foundational questions in quantum mechanics. Here we note an analogous distinction in the context of the theory of entanglement. The terminology of ‘proper’ versus ‘improper’ separability is proposed to mark the distinction. 1 Proper and Improper mixtures In many discussions of the measurement problem in quant...