Abstract
This chapter discusses the Brouwer's bar theorem and a system of ordinal notations. The chapter constructs a system of ordinal notations. This system of notations has been used to prove the consistency of the formal system obtained by adjoining the axiom of bar induction to elementary intuitionistic analysis. The chapter proves the well-ordering of any proper segment of the notations, but not the whole segment itself, using Brouwer's bar theorem formulated in the form of TI D , the axiom of transfinite induction for decidable relations. A system of ordinal notations and the concept of well-foundedness are discussed in the chapter. The well-ordering of any proper segment of the notations less than Ω by using TI D are proved.
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