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Axioms 2016, 5(2), 18;

Potential Infinity, Abstraction Principles and Arithmetic (Leśniewski Style)

Centre for Logic and Philosophy of Science, University of Ghent, 9000 Ghent, Belgium
Institute of Philosophy, Sociology and Journalism, University of Gdańsk, 80-309 Gdańsk, Poland
Academic Editor: Urszula Wybraniec-Skardowska
Received: 19 March 2016 / Revised: 2 June 2016 / Accepted: 2 June 2016 / Published: 15 June 2016
(This article belongs to the Special Issue Lvov—Warsaw School)
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This paper starts with an explanation of how the logicist research program can be approached within the framework of Leśniewski’s systems. One nice feature of the system is that Hume’s Principle is derivable in it from an explicit definition of natural numbers. I generalize this result to show that all predicative abstraction principles corresponding to second-level relations, which are provably equivalence relations, are provable. However, the system fails, despite being much neater than the construction of Principia Mathematica (PM). One of the key reasons is that, just as in the case of the system of PM, without the assumption that infinitely many objects exist, (renderings of) most of the standard axioms of Peano Arithmetic are not derivable in the system. I prove that introducing modal quantifiers meant to capture the intuitions behind potential infinity results in the (renderings of) axioms of Peano Arithmetic (PA) being valid in all relational models (i.e. Kripke-style models, to be defined later on) of the extended language. The second, historical part of the paper contains a user-friendly description of Leśniewski’s own arithmetic and a brief investigation into its properties. View Full-Text
Keywords: arithmetic; potential infinity; finite models; Leśniewski’s systems; Leśniewski’s Ontology; neologicism; abstraction principles arithmetic; potential infinity; finite models; Leśniewski’s systems; Leśniewski’s Ontology; neologicism; abstraction principles
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Urbaniak, R. Potential Infinity, Abstraction Principles and Arithmetic (Leśniewski Style). Axioms 2016, 5, 18.

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