Primitive Recursion and Isaacson's Thesis

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Abstract

Although Peano arithmetic (PA) is necessarily incomplete, Isaacson argued that it is in a sense conceptually complete: proving a statement of the language of PA that is independent of PA will require conceptual resources beyond those needed to understand PA. This paper gives a test of Isaacon's thesis. Understanding PA requires understanding the functions of addition and multiplication. It is argued that the grasping these primitive recursive functions involves grasping the double ancestral, a generalized version of the ancestral operator. Thus we can test Isaacon's thesis by seeing whether when we phrase arithmetic in a context with the double ancestral operator, the result is conservative over PA. This is a stronger version of the test given by Smith, who argued that understanding the predicate "natural number" requires understanding the ancestral operator, but did not investigate what is required to understand the arithmetic functions.
Original languageEnglish
Pages (from-to)4-15
Number of pages12
JournalThought: A Journal of Philosophy
Volume8
Issue number1
Early online date28 Nov 2018
DOIs
Publication statusPublished - 3 Mar 2019

Keywords

  • Arithmetic
  • Double Ancestral
  • Ancestral Logic
  • Incompleteness
  • Nominalism

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