# A theorem of infinity for Principia Mathematica

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Abstract I prove a theorem of infnity 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 infnitely-descending types just as there are infnitely-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 implicitly suggested by various remarks of Whitehead and Russell. By so-adjusting *Principia*, a core objection to Logicism--namely, that Logicism cannot recover Peano arithmetic without an axiom of infnity--dissipates.