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Two formal vindications of logicism. (English) Zbl 0838.03004

Czermak, Johannes (ed.), Philosophy of mathematics. Proceedings of the 15th international Wittgenstein-Symposium, 16-23 August 1992, Kirchberg am Wechsel, Austria. Part I. Wien: Hölder-Pichler-Tempsky. Schriftenreihe der Wittgenstein-Gesellschaft. 20/I, 243-250 (1993).
The author presents two infinitary systems of type theory, the cumulative system \(K^\infty_\Delta\) and the infinitary system \(\Sigma_{(\kappa)}\), claiming that the notions and rules involved have purely logical character. A type theory is infinitary if it has finitely long formulae, but an infinitary number of rules of proof, or if it has infinitely long formulae. The discussion of \(\Sigma_{(\kappa)}\) results in the following vindication of logicism: “If a mathematical statement can be proven in \(PM\), then its translation can be proven without cuts …in an infinitary type logic (without nonlogical axioms, and without nonlogical inferences)” (p. 249 f.).
For the entire collection see [Zbl 0836.00022].

MSC:

03A05 Philosophical and critical aspects of logic and foundations
00A30 Philosophy of mathematics
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