Abstract
Abstract Hamkins and Löwe proved that the modal logic of forcing is S4.2. In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra H of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras.
Highlights
We shall deal with three different languages, the language L of propositional logic, the language L of modal propositional logic, and the language LZFC of set theory
If Λ is any modal logic extending S4, we can say “an interior algebra B, satisfies Λ” and mean that we identify each theorem φ of Λ with a term tφ in the interior algebra, and B, |= tφ = 1
We call a Heyting algebra fatal if every prime filter is contained in only one maximal filter
Summary
We shall deal with three different languages, the language L of propositional logic, the language L of modal propositional logic, and the language LZFC of set theory (i.e., first-order logic with a binary relation symbol ∈). The Lindenbaum algebra of ZFC is the Boolean algebra BZFC of classes of provably equivalent LZFC-sentences (i.e., formulæ with no free variables). We call a Heyting algebra fatal if every prime filter is contained in only one maximal filter. The class of fatal Heyting algebras will be denoted by fHA. For a Heyting algebra H the following are equivalent: 1. F and G are incomparable as ¬p ∈ F \G and ¬¬p ∈ G\F They extend to two different maximal filters y and z. There are two different maximal filters containing x, and so H is not fatal. (2) ⇒ (1): Suppose there is a prime filter x contained in two different maximal filters y and z. As ¬¬p ∧ (p ∨ ¬p) ≤ p, it follows that p ∨ ¬p ≤ ¬¬p → p in any Heyting algebra H. The left hand side is obviously 1, so (2) follows
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