Kurt Gödel Research Center

Which cardinality quantifiers are logical?

Talk held by Alexander Paseau (Wadham College, Oxford, UK) at the KGRC Friday seminar on 2018‑03‑16.

Abstract

By a cardinality quantifier, I understand an expression of the form ‘There are at least $X$ ...’ for $X$ a cardinal. The finite cardinality quantifiers are thus all expressions of the form ‘There are at least $n$ ...’ where $n$ is finite (‘There's at least one...’, ‘There are at least two ...’ and so on). Since all these finite quantifiers are definable in first-order logic, they are universally taken to be logical. What about transfinite cardinality quantifiers, the first of which is ‘There are infinitely many’? Which of them is logical? I argue that they all are. The talk will explain the answer's significance and the new light it sheds on the perennial question ‘What is Logic?’.