Explicit constructions of models
Explicit constructions of models [ edit ] We shall not prove that any models of the axioms are isomorphic. Such a proof can be found in any number of modern analysis or set theory textbooks. We will sketch the basic definitions and properties of a number of constructions, however, because each of these is important for both mathematical and historical reasons. The first three, due to Georg Cantor / Charles Méray , Richard Dedekind / Joseph Bertrand and Karl Weierstrass all occurred within a few years of each other. Each has advantages and disadvantages. A major motivation in all three cases was the instruction of mathematics students. Construction from Cauchy sequences [ edit ] A standard procedure to force all Cauchy sequences in a metric space to converge is adding new points to the metric space in a process called completion . � is defined as the completion of Q with respect to the metric | x - y |, ...