The Philosophers and Mathematics pp 81-100 | Cite as

# Zigzag and Fregean Arithmetic

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## Abstract

In Frege’s logicism, numbers are logical objects in the sense that they are extensions of certain concepts. Frege’s logical system is inconsistent, but Richard Heck showed that its restriction to predicative (second-order) quantification is consistent. This predicative fragment is, nevertheless, too weak to develop arithmetic. In this paper, I will consider an extension of Heck’s system with impredicative quantifiers. In this extended system, both predicative and impredicative quantifiers co-exist but it is only permissible to take extensions of concepts formulated in the predicative fragment of the language. This system is consistent. Moreover, it *proves* the principle of reducibility applied to concepts true of only finitely many objects. With the aid of this form of reducibility, it is possible to develop arithmetic in a thoroughly Fregean way.

## Notes

### Acknowledgements

I would like to thank the invitation of *Argument Clinic* (an association of students of Philosophy at the University of Lisbon) for inviting me to give a presentation at the conference *Principia Mathematica (1913–2013)*, held at the University of Lisbon in February 6–7, 2014. I came with the main idea of this paper while preparing a talk for this conference. Afterwards, I also had the chance to speak about the issues of this paper at the meeting “2014: Abstractionism/Neologicism” (University of Storrs, Connecticut, U.S.A.), at “Journée sur les Arithmétiques Faibles 33” (University of Gothenburg, Sweden) and at the conference “The Philosophers and Mathematics” (University of Lisbon, Portugal). I want to thank the organizers of these meetings (Marcus Rossberg, Ali Enayat and Hassan Tahiri, respectively), as well as the participants for their remarks (specially to Marco Panza, for detailed comments and questions). My final thanks are to an anonymous referee. In a preliminary version of this paper, Section 4.4 used a definition of finiteness due to Paul Stäckel in 1907 (for a reference see Parsons 1987). The referee complained about the artificiality of using this definition. We reformulated the section using a characterization of finiteness given by Frege in volume I of the *Grundgesetze der Arithmetik* (1893). I believe that this change made the paper more natural and tighter.

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