Abstract
Historically, mathematics has often dealt with the ‘expansion’ of previously accepted concepts and notions. In recent years, Buzaglo (The logic of concept expansion, 2002) has provided a formalisation of concept expansion based on forcing. In this paper, I briefly review Buzaglo’s logic of concept expansion and I apply it to Cantor’s ‘creation’ of the transfinite. I argue that, while Buzaglo’s epistemological considerations fit well into Cantor’s conceptions, Buzaglo’s logic of concept expansion might be unsuitable to justify the creation of the transfinite in terms of a logically rigorous derivation of concepts.
Originally published in C. Ternullo, Remarks on Buzaglo’s Concept Expansion and Cantor’s Transfinite, preprint.
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Notes
- 1.
In more recent times, this ideal seems to have been resurrected by none other than Gödel. Concerning the nature of logic, Gödel says: ‘Logic is the theory of the formal. It consists of set theory and the theory of concepts […] Set is a formal concept. If we replace the concept of set by the concept of concept, we get logic. The concept of concept is certainly formal and, therefore, a logical concept.’ Wang comments thus: ‘It is clear that Gödel saw concept theory as the central part of logic and set theory as a part of logic. It is unclear whether he saw set theory as belonging to logic only because it is, as he believed, part of concept theory, which is yet to be developed.’ Wang [9, p. 247].
- 2.
However, Buzaglo contends that his theory of partially defined functions does not counter Frege’s point of view, but rather expands on it (for discussion of this, see [1], pp. 24–30 and 59–63). Frege opposed concept expansion, as he thought that concepts were rigid constructs, instantiated by a fixed domain of objects. Buzaglo’s theory does not deny this, while, at the same time, conjecturing that concepts qua functions may have undefined values, which can be subsequently somehow ‘filled out’ to produce new concepts.
- 3.
Cf. Hallett, in [7], p. xi: ‘[…] mixed in with Cantor’s prevailing realism are splashes of what could well be called constructivism, and this applies particularly to two crucial elements of his theory, the notion of well-ordering and the set concept itself.’
- 4.
For this, see my [8], pp. 440–443.
- 5.
This argument is briefly introduced and assessed by Hallett in [7], pp. 74–81.
- 6.
- 7.
In fact, he was fiercely against a conception of ‘linear numbers’ from which a violation of the Axiom of Archimedes could be inferred. For this, cf. the thorough discussion in Dauben [4], pp. 33–36, of Cantor’s correspondence with Veronese, Vivanti and Peano concerning the concept of infinitesimal.
- 8.
For instance, he says: ‘But all attempts to force this infinitely small into a proper infinite must finally be given up as pointless. If proper infinitely-small quantities exist at all, that is, are definable, then they certainly stand in no direct relationship to the familiar quantities which become infinitely small.’ ([2], in Ewald [5], p. 888).
- 9.
However, Buzaglo introduces a strengthening of the ‘forced internal expansion’, that is, a ‘strongly forced internal expansion’, whereby, given any set of forcing conditions \(\mathcal {S} \in T\), a specific expansion should take place in a unique way. It is clear, however, that none of the cases examined meets this notion.
Reference
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W. Dauben, Georg Cantor. His Mathematics and Philosophy of the Infinite (Harvard University Press, Harvard, 1979)
W. Ewald (ed.), From Kant to Hilbert: A Source Book in the Foundations of Mathematics, vol. II (Oxford University Press, Oxford, 1996)
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M. Hallett, Cantorian Set Theory and Limitation of Size (Clarendon Press, Oxford, 1984)
C. Ternullo, Gödel’s Cantorianism, in Kurt Gödel: Philosopher-Scientist, ed. by G. Crocco, E.-M. Engelen (Presses Universitaires de Provence, Aix-en-Provence, 2015), pp. 413–442
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Acknowledgements
The writing of this paper has been supported by the John Templeton Foundation grant ID35216 “The Hyperuniverse: Laboratory of the Infinite”.
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Ternullo, C. (2018). Remarks on Buzaglo’s Concept Expansion and Cantor’s Transfinite. In: Antos, C., Friedman, SD., Honzik, R., Ternullo, C. (eds) The Hyperuniverse Project and Maximality. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-62935-3_12
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