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Topic: § 531 What does § 529 show us?
Replies: 51   Last Post: Jul 29, 2014 10:02 AM

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 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
§ 531 What does § 529 show us?
Posted: Jul 23, 2014 10:21 AM

1) The sets s_n of the sequence (s_n) tell us that always (for all n in |N) infinitely many positive rationals =< n remain without index =< n. So much is irrefutable. The proof holds for all natural numbers. Nothing can index further rationals. But the sets s_n are never empty.
2) If all natural numbers n could be used and all sets s_n could be constructed, then the finally remaining rationals without index could be indentified. But that is not possible. This is also irrefutable.
Both points taken together show that not all natural numbers n can be used and, therefore, not all s_n can be constructed. Therefore it is not a logical problem that always something remains. It is simply the exclusive property of infinity, namely to be never finished.
What is the advantage of this idea over set theory with its finished infinity, besides that it is the truth? It gets along without finishing the infinite, without exhausting |N such that an empty set lim(n-->oo) |N\{1,2,3,...,n} remains. It gets along without undefinable "real" numbers, without the necessity to distinguish between finite positiv cardinal numbers and natural numbers, without paradoxes of Löwenheim-Skolem or Banach-Tarski. It gets along without an inexplicable discontinuity of the cardinality functions from Tristram Shandy (cp. § 077 and 200) to McDuck (cp. § 515) or |lim s_n| (cp. § 529) that unavoidably always would strike the not initiated thinker. It gets along without an empty limit of the sequence (s_n) of, for all n, infinite sets. Everybody not toughened up in a long study of set theory would ask: "How can the infinite sequence have an empty limit? What is the reason? What causes this vacuum?" I think my answer will be accepted by 99 % of all intelligent thinkers. In fact, I have enjoyed this releasing and satisfying experience for many times.

Regards, WM