Ramified Frege arithmetic
type: Item
instance of: scholarly article
title: Ramified Frege arithmetic
zbMATH Open document ID: 1250.03011
author: Richard G. jun. Heck
published in: Journal of Philosophical Logic
publication date: 2012-03-23T00:00:00Z
review text: This is a contribution to the so-called neologicist program according to which Frege's logicist foundation of arithmetic can be vindicated on the ground of a consistent fragment of Frege's \textit{Grundgesetze der Arithmetik}, basically including second-order logic with full impredicative comprehension and the so-called Hume's principle HP (for a systematic survey and development see, e.g., \textit{J. P. Burgess}'s monograph [Fixing Frege. Princeton, NJ: Princeton University Press (2005; Zbl 1089.03001)]. HP states that two concepts have the same number if and only if they are in a 1-1 correspondence. Frege arithmetic is the system based upon full second-order logic and HP, and Frege's theorem is the statement that Frege arithmetic extends the Dedekind-Peano full second-order impredicative arithmetic. Recent investigations have considered the following problem: What is the weakest fragment of Frege's arithmetic where the axioms on zero, successor and natural numbers can be derived? In particular, is a predicative fragment of second-order comprehension sufficient? By a theorem of \textit{Ø. Linnebo} [Bull. Symb. Log. 10, No. 2, 153--174 (2004; Zbl 1068.03051)] the existence of the successor cannot be derived by means of predicative comprehension and essentially requires \(\Pi^1_1\)-comprehension. Also \(\Pi^1_1\)-comprehension is required for verifying the defining equations of addition and multiplication and in the proof of induction. Now the main result of the paper is that the ramified predicative theory does indeed prove the existence of an appropriate successor. Indeed, it is shown (Theorem 6.1) that the theory of zero and successor based on second-order ramified arithmetic can interpret Robinson's arithmetic Q and hence a significant fragment of first-order Peano arithmetic (i.e. \(\mathrm{I}\Delta_0(\mathrm{exp})\)). To sum up, in the end ``impredicative second-order reasoning is only needed for the proof of mathematical induction''.
reviewed by: Andrea Cantini
Mathematics Subject Classification ID: 03A05
Mathematics Subject Classification ID: 03F35
zbMATH DE Number: 6018336
zbMATH Keywords: Frege arithmetic
zbMATH Keywords: second-order arithmetic
zbMATH Keywords: ramified second-order logic
MaRDI profile type: Publication
full work available at URL: s10992-010-9158-y
OpenAlex ID: W2097441263
cites work: Q4523474
cites work: Q4523473
cites work: Q4523466
cites work: Q4523476
cites work: Q5694788
cites work: On a consistent subsystem of Frege's Grundgesetze
cites work: Q5719494
cites work: Amending Frege's \textit{Grundgesetze der Arithmetik}
cites work: Q4539489
cites work: The development of arithmetic in Frege'sGrundgesetze der arithmetik
cites work: Q4208523
cites work: Finitude and Hume's principle
cites work: Q4266053
cites work: Cardinality, counting, and equinumerosity
cites work: Predicative Fragments of Frege Arithmetic
cites work: Q3037386
DOI: 10.1007/S10992-010-9158-Y
description: scientific article; zbMATH DE number 6018336