What is a forcing extension (of V)?

Talk held by Neil Barton (University of London, UK) at the KGRC Friday seminar on 2014‑12‑19.


Recent debates in the philosophy of set theory often focus on how many universes of sets there are. Absolutists hold that there is just one maximal, definite universe of sets, while Multiversists hold that there are many universes. Often, the practice of forcing over V is regarded as evidence against the Absolutist position. In this paper I clarify this debate by examining Absolutist interpretations of forcing. My strategy is as follows:

Section 1 provides an exposition of the two views, and outlines the main problem for the Absolutist. Section 2 then assesses and provides further criticism of the claim that the Absolutist can simply interpret forcing through the use of countable transitive models or Boolean-valued models. It is argued that these well-known interpretations of forcing are not satisfactory on the Absolutist’s position. Section 3 presents a recent interpretation of forcing (the Boolean ultrapower) and argues that it is satisfactory on the Absolutist’s view. Section 4 argues, however, that a conflict of intuitions between the Absolutist and the Multiversist is highlighted, namely the attitudes of each to the role of simulation in mathematics.

It is concluded that while the Absolutist appears to have an interpretation of forcing over V, this may be challenged by philosophical research into the ontological implications of simulating one kind of mathematical object with another.

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