## 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

### Keywords:

infinitary type theory; logicism