Abstract
This chapter offers two constructions of arithmetic inspired by Gottlob Frege.
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Notes
- 1.
Informality will not be our only departure from Frege. So be warned: this chapter does not pretend to offer a close reading of Frege’s own exposition. It draws freely on later reconstructions such as the one in George and Velleman [7]. For Frege’s own version, see [4] and [5]. For some helpful commentary, see Boolos and Heck [1].
- 2.
- 3.
See Burgess [2], pp. \(147-150\).
- 4.
- 5.
- 6.
For some of the history, see Quine [8] (http://www.jstor.org/stable/2251464), Rang & Thomas [9], and van Heijenoort [11], pp. 124–128.
- 7.
Transitivity : if \(F\equiv G\) and \(G\equiv H\), then \(F\equiv H\). Symmetry : if \(F\equiv G\), then \(G\equiv F\). Reflexivity : \(F\equiv F\).
- 8.
For one noteworthy appraisal of this program of “neo-Fregeanism,” see Weir [13] (available via www.projecteuclid.org). See also Shapiro [10].
- 9.
Yet again, this is just our name for the theory.
- 10.
That is, the function \(\#\) cannot be one-to-one:
$$-\forall F,G(\#F=\#G\rightarrow F=G).$$This is a version of a result known as “Cantor’s theorem”. Georg Cantor (1845–1918) was a pioneer set theorist. The basic problem here is that you cannot use objects to supply each set of objects with an avatar exclusive to that set—or, rather, you cannot do so while also making all our comprehension axioms true.
- 11.
Note that you will not be able to complete the remaining step in the recipe I just gave. Why? Given one object and two equivalence classes, you will not be able to pair each equivalence class with exactly one object and each object with no more than one equivalence class. You are facing a crippling shortage of objects.
- 12.
As another ink-saving measure, we write
$$\ulcorner \exists \mu \in \varGamma \;\phi \urcorner $$instead of
$$\ulcorner \exists \mu (\mu \in \varGamma \wedge \phi )\urcorner .$$We will also write
$$\ulcorner \forall \mu \in \varGamma \;\phi \urcorner $$instead of
$$\ulcorner \forall \mu (\mu \in \varGamma \rightarrow \phi )\urcorner .$$We treat ‘\(\notin \)’ similarly. For example, we write
$$\ulcorner \exists \mu \notin \varGamma \;\phi \urcorner $$instead of
$$\ulcorner \exists \mu (\mu \notin \varGamma \wedge \phi )\urcorner .$$ - 13.
The following results are sufficient: Theorems 6.1–6.5, 6.8, 6.9, 6.11 and Exercises 6.7, 6.9, 6.12, 6.13, 6.15.
- 14.
Cf. the definition of multiplication in §\(5\) of Visser [12].
References
Boolos, G. S., & Heck, R. G. (1998). Die Grundlagen der Arithmetik §§82-83. In M. Schirn (Ed.), Philosophy of mathematics today (pp. 407–428). Oxford: Oxford University Press.
Burgess, J. P. (2005). Fixing frege. Princeton NJ: Princeton University Press.
Frege, G. (1884). Die Grundlagen der Arithmetik. Breslau: Wilhelm Koebner.
Frege, G. (1893). Die Grundgesetze der Arithmetik (Vol. 1). Jena: Hermann Pohle.
Frege, G. (1903). Die Grundgesetze der Arithmetik (Vol. 2). Jena: Hermann Pohle.
Frege, G. (1980). The foundations of arithmetic. Evanston, IL: Northwestern University Press.
George, A., & Velleman, D. J. (2002). Philosophies of mathematics. Oxford: Blackwell.
Quine, W. V. O. (1955). On Frege’s way out. Mind, 64, 145–159.
Rang, B., & Thomas, W. (1981). Zermelo’s discovery of the ‘Russell paradox’. Historia Mathematica, 8, 15–22.
Shapiro, S. (2011). The company kept by cut abstraction (and its relatives). Philosophia Mathematica, 19, 107–138.
van Heijenoort, J. (Ed.). (1967). From Frege to Gödel. Cambridge MA: Harvard University Press.
Visser, A. (2011). Hume’s principle, beginnings. Review of Symbolic Logic, 4, 114–129.
Weir, A. (2003). Neo-Fregeanism: An embarrassment of riches. Notre Dame Journal of Formal Logic, 44, 13–48.
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Pollard, S. (2014). Frege Arithmetic. In: A Mathematical Prelude to the Philosophy of Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-05816-0_6
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