College of Liberal Arts & Sciences
Graduate & Undergraduate Student Seminar (GAUSS)
Abstract:
We prove a theorem of infinity for Principia Mathematica. The proof requires altering the meta-theory of Principia. In Principia we have a simple type theory with a lowest type (call this 'simple N-type theory'). Our key idea is to allow for infinitely-descending types just as there are infinitely-ascending types; that is, we allow our simple type theory to be not well-founded (call this 'simple Z-type theory'). Given the acceptableness of Principia's (well-founded) simple type theory, this adjustment is minor. This adjustment is also supported by Principia's conventions concerning types in Volume II. By so-adjusting Principia, a core objection to Logicism--namely, that Logicism cannot recover Peano arithmetic without an axiom of infinity--dissipates.