Mathematical Intuitionism and Intersubjectivity by Tomasz Placek download in ePub, pdf, iPad
Then membership of the following two sets is undecidable. For the law of trichotomy we have just shown that it is not intuitionistically true. One can, for example, given a statement A that does not contain any reference to time, i.
Only after Brouwer's introduction of choice sequences did intuitionism obtain its particular flavor and became incomparable with classical mathematics. The two examples above are characteristic for the way in which the continuity axioms are applied in intuitionistic mathematics.
And it is this common substratum, this empty form, which is the basic intuition of mathematics. Functional interpretations such as realizability as well as interpretations in type theory could also be viewed as models of intuitionistic mathematics and most other constructive theories. They are the only axioms in intuitionism that contradict classical reasoning, and thereby represent the most colorful as well as the most controversial part of Brouwer's philosophy. This is the reason that the bar theorem is also referred to as the bar principle. This is particularly visible in descriptive set theory, which emerged as a reaction to the highly nonconstructive notions occurring in Cantorian set theory.
The typical axioms with which one wishes theorems to compare are the fan principle and the bar principle, Kripke's schema and the continuity axioms. Extensions of this principle in which the decidability requirement is weakened can be extracted from Brouwer's work but will be omitted here.