On the methodology of informal rigour: set theory, semantics, and intuitionism

TL;DR

This paper critically examines Georg Kreisel's method of informal rigour, analyzing its historical context, schemas, and applications in logic, set theory, and intuitionism, and compares it with Carnap's explication method.

Contribution

It offers a detailed reconstruction of Kreisel's examples of informal rigour and introduces schemas to unify his various applications in logic and philosophy.

Findings

Reconstructed Kreisel's squeezing argument for model-theoretic validity

Analyzed Kreisel's argument for the determinacy of the Continuum Hypothesis

Compared Kreisel's informal rigour with Carnap's method of explication

Abstract

This paper provides a critical overview of Georg Kreisel's method of informal rigour, most famously presented in his 1967 paper `Informal rigour and completeness proofs'. After first considering Kreisel's own characterization in historical context, we then present two schemas under which we claim his various examples of informal rigour can be subsumed. We then present detailed reconstructions of his three original examples: his squeezing argument in favor of the adequacy of the model theoretic analysis of logical validity, his argument for the determinacy of the Continuum Hypothesis, and his refutation of Markov's principle in intuitionistic analysis. We conclude by offering a comparison of Kreisel's understanding of informal rigour with Carnap's method of explication. In an appendix, we also offer briefer reconstructions of Kreisel's attempts to apply informal rigour to the discovery of set theoretic axioms, the distinction between standard and nonstandard models of arithmetic, and the concepts of finitist proof, predicative definability, and intuitionistic validity.