On the Algebraic Closure of Two
نویسنده
چکیده
In each case, 0~’ and /Y range over all ordinals smaller than 01 and p, respectively. Conway has shown, inter alia, that a suitable beginning segment of Onz is an algebraic closure of the two-element subfield (0, 11, cf. section 1. The purpose of this paper is to prove that, in this beginning segment, the field operations can be performed in an effective manner. Following Conway we distinguish the ordinary ordinal operations from those in On2 by the use of square brackets [ ]-that is, all sums, products and powers appearing inside square brackets are meant in the sense of classical ordinal arithmetic, cf. Bachmann [2], and all others represent operations in Ona. A single decimal digit between square brackets refers to the bibliography at the end of this paper. We denote by o the least infinite ordinal, and we identify each ordinal number with the set of all previous ones. In particular, 2 = {0, 1).
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