## Foundations of Logic and Mathematics, Volume 3 |

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AB,R binary relation Boolean algebra bound of M,R Cantor sequence A,R cardinal number cardinal type Cartesian product class of cardinal class of elements closed in K,R continuous sequence contraposition cRxr cut of K,R Dedekind cut Dedekind extension DEFINITION denote dense in K,R dense-in-itself discrete sequence distinct elements element of K,R empty equals or precedes finite ordinal follows at once follows by Def follows by Thm follows from Def follows from Thm g(xr greatest lower bound Hence hypothesis immediate successor infinite class initial sequence initial subsequence interval last element least upper bound Lemma Let K,R Lf(x limit-point of M,R lower segment Lrf(x maximal element non-Dedekind cut non-empty class non-empty subclass once from Def once from Thm paired present theorem proper subclass rational numbers real numbers similar single-valued function smallest ordinal subsequence M,R subsequence of K,R succeeds Suppose theorem follows theorem is true transfinite cardinal unique well-ordering theorem whence