In <GbydnfG1Xq3Nu5HMnZ2dnUVZ_jqdnZ2d@giganews.com>, on 02/01/2013 at 02:32 PM, fom <fomJUNK@nyms.net> said:
>So, for example, there are "gaps" in the system of rational >numbers. One can, assuming completed infinities, define infinite >sets of rational numbers corresponding to the elements of a >Cauchy sequence.
What would be the point? You're introducing extra machinery into what is a very simple construction.
>When the limit of the sequence is, itself, a rational number, >that infinite set becomes a representation of that rational number >in the complete space whose "numbers" are equivalence classes of >Cauchy sequences sharing the same limit.
No; the set of values taken by the sequence is irrelevant. If a Cauchy sequence in the rationals converges then the *sequence* is a representative of its limit.
>When the limit of a Cauchy sequence does not exist as a rational >number, that Cauchy sequence becomes a representative of the >equivalence class of Cauchy sequences that cannot be >differentiated from that representative using the order relation >between the rational numbers of the underlying set.
What are you trying to say? The definition of the equivalence relation is the same whether the Cauchy sequences converge or not; two sequences are equivalent if their difference converges to zero. Any Cauchy sequence is a representative of its equivalence class, by definition.
>These "numbers" have no corresponding rational number as a limit >and are, therefore, distinguished as a different logical type in >the *new*, completed space.
Non sequitor, and false. There is nothing logically special about equivalence classes of Cauchy sequences that converge.
>To call a subset of a complete space a dense subset is to say that >such a logical type construction could be made from that subset >to recover the original space.
Completeness only applies to metric spaces, while denseness applies to subsets of arbitrary topological spaces. You can't in general reconstruct a topological space from only a dense subset, not even if the space is compact.
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