Abstract
There is a central fallacy that underlies all our thinking about the foundations of arithmetic. It is the conviction that the mere description of the natural numbers as the “successors of zero” (i.e., as what you get by starting at 0 and iterating the operation x ↦ x + 1) suffices, on its own, to characterise the order and arithmetical properties of those numbers absolutely. This is what leads us to suppose that the dots of ellipsis in
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Notes
- 1.
Dedekind discusses this fallacy in (new 1971).
- 2.
- 3.
Newton’s definition of “number” in Newton (1728) runs as follows:
By a Number we understand not so much a Multitude of Unities, as the abstracted Ratio of any Quantity to another Quantity of the same Kind, which we take for Unity.
- 4.
This is Common Notion 5 in Book I of the Elements.
- 5.
I shall allow for sets some or all of whose members are individuals, which are not sets. Sets and individuals together comprise the objects—the subject matter—of arithmetic.
- 6.
If a is an individual it behaves like the empty set, \({\emptyset}\), in set-theoretical contexts: thus \(\emptyset\subseteq a\), \(\mathrm{P}(a)=\{\emptyset\}\), and \(a\cup x=x\), \(a\subseteq x\), etc., for any x.
- 7.
A property is a one place global relation. What it means for a property to be “definite” will be explained later, after Brouwer’s Principle has been laid down.
- 8.
The axioms of Zermelo-Fraenkel corresponding to these global operations give necessary and sufficient conditions for membership in the sets obtained by applying them to arbitrary arguments. I take those conditions to be implicit in my descriptions of the operations.
- 9.
Thus given a three element set \(\{a,b,c\}\), the linear ordering \([b,a,c]\) of its elements in which b comes first followed by a and c in that order is defined by \([b,a,c,]=\{\,\{b\}\,\{b,a\}\,\{b,a,c\}\,\}\).
- 10.
Both Foundation and Extensionality are essential here. See Maberry (2000, sections 4.11 and 8.6).
- 11.
See section 8.4 of Mayberry (2000).
- 12.
It is possible, however, to justify limited recursion in Euclidean arithmetic. See Mayberry (2000, sections 9.2 and 10.3).
- 13.
It can be shown that the arithmetical functions and relations of Euclidean arithmetic suffice to provide an interpretation of \(I\Delta_0+\exp\). This was established in Homolka (1983).
- 14.
We can speak of the species \(\mathcal{N}_{\sigma,a}\), of linear orderings generated by its successor function σ from its initial term a, even when it does not form a natural number system.
- 15.
This system is what is generated when we attempt to specify an isomorphism from \({\mathcal N}\) to \({\mathcal M}\) in the obvious, intuitive way. We don’t in general get an isomorphism but a new natural number system with the properties suggested by its name.
- 16.
A purely set-theoretical definition of ACK is given in section 10.6 of Mayberry (2000).
- 17.
If we add a new global function φ to our list of initial global functions, the notion of global function must be changed accordingly. See Mayberry (2000), section 10.7.
- 18.
The proofs of these claims are straightforward and are to be found in Mayberry (2000), chapter 10.
- 19.
On formalisation, this becomes a Π 1 condition on the free function variable φ, where we must include a rule of substitution for such variables.
- 20.
- 21.
Many such examples are given in Pettigrew (2008).
- 22.
See section 10.4 of Mayberry (2000). It is straightforward to define a set-theoretical analog of log2.
- 23.
Mayberry (2000), 10.4.15.
- 24.
By “predicative” I mean that, comprehension for global relations holds only for formulas free of both first or second order global quantifiers.
- 25.
In these formalised theories we must include a singleton selector, ι, such that \(\iota(S)=x\) if S is the singleton \(\{x\}\) and \(\iota(S)=S\) otherwise.
- 26.
In fact, the logical and philosophical difficulties that arise in the treatment of infinite totalities in Euclidean arithmetic also arise in infinitary set theory in the guise of difficulties surrounding what is sometimes called Cantor’s Absolute (See Mayberry, 2000, section 3.5).
- 27.
This can be formulated as a classically witnessed postulate.
- 28.
Here we exploit the fact that it is possible to have subspecies of a finite set that are not subsets. Vopenka calls these species semi-sets in Vopenka (1979).
References
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Dedekind, R. (new 1971) Letter to Kefferstein. Translated by H. Wang and S. Bauer-Mengelberg in J. van Heijenoort, ed. (1971) From Frege to Gödel. A Source Book in Mathematical Logic, 1879–1931, in J. van Heijenoort, Cambridge, MA: Harvard University Press, 1971.
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Mayberry, J. (2011). Euclidean Arithmetic: The Finitary Theory of Finite Sets. In: Sommaruga, G. (eds) Foundational Theories of Classical and Constructive Mathematics. The Western Ontario Series in Philosophy of Science, vol 76. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0431-2_12
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