Intuitionistic logic. (English) Zbl 1002.03053

van Dalen, Dirk

Goble, Lou (ed.), The Blackwell guide to philosophical logic. Oxford: Blackwell Publishers. Blackwell Philosophy Guides. 224-257 (2001).

The paper under review is a survey of the basic principles of intuitionism, as well as of formalizations and semantical constructions of intuitionistic theories.
The reader must be forewarned about the following two inaccuracies in van Dalens’s text. First, contrary to the statement on page 239, lines 5 and 6 from below, Kripke did not show that Beth models are convertible into his models. In fact, in view of a theorem of E. G. K. López-Escobar [“Equivalence between semantics for intuitionism. I”, J. Symb. Logic 46, 773-780 (1981; Zbl 0497.03047)], we cannot expect to find such a transformation. Secondly, the transformation is into topological models, and not for Beth models, as stated in the text.
The final section of the paper presents the following converse of Brouwer’s indecomposability theorem: If the continuum is not decomposable into two inhabitants, then there are no discontinuous real functions. This result seems to be new.
For the entire collection see [Zbl 0983.03001].

Reviewer: Victor N.Krivtsov (Moskva)

MSC:

03F55	Intuitionistic mathematics
03B47	Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03B20	Subsystems of classical logic (including intuitionistic logic)
03F50	Metamathematics of constructive systems
03-02	Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations

Keywords:

survey; intuitionism; intuitionistic theories

Citations:

Zbl 0497.03047