Principia Mathematica, Negative Types, and a theorem of infinity for Z-Principia Mathematica

Abstract: I here develop a new, foundationless simple-type grammar to replace Principia Mathematica's well-founded simple-type grammar. Rewriting the axiom schemata of Principia in foundationless simple-types, or Z-types, gives us a new system, ZPM. Adding to ZPM a plausible new axiom schema, Z*107, allows us prove Infinity in every type. Z*107 is a plausible new axiom schema because, as I will argue, it is a logical truth of ZPM. Further, using Z*107 to prove Infinity is not circular: the new axiom alone does not secure a proof of Infinity, but crucially relies on heterogeneous relations. So using Z*107 to prove Infinity is not question-begging. In this talk I also relate this system to earlier discussions of Wang's Negative Types (and its extension by Specker's TA).

logic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic