Abstract: In his 1907 dissertation Brouwer sketched the construction of the lawlike real numbers, showing that they had the same order type as the rationals. He then claimed that a (denumerable) ``scale'' of this kind could be completed to a measurable continuum by using a primitive notion of continuity or ``fluidity.''
By 1918 however, he was willing to accept infinitely proceeding sequences of integers or rationals, and structured sets of these sequences, as admissible mathematical objects. Arbitrary Cauchy sequences of rationals represented (uncountably many) real numbers. Continuity, no longer primitive, was eventually retained as a necessary attribute of every total function on the continuum.
Lawlike sequences are surely intensional objects. Arbitrary choice sequences should be purely extensional. Brouwer's notions of ``spread'' and ``fan,'' the basic structures of intuitionistic analysis, enabled successive restrictions on choices; but Brouwer avoided higher-order restrictions.
Beginning around 1950 Kleene, Vesley, Kreisel, Troelstra, and Veldman clarified, interpreted, and adapted Brouwer's notions. The axiomatic systems they proposed to characterize arbitrary, lawlike, ``lawless'' and other varieties of choice sequence became objects of study in their own right. In this talk I will focus on Troelstra's many contributions to the theory of choice sequences.
The talk will be held on Zoom. Contact Laura.Crosilla @ ifikk.uio.no for a Zoom-invite.