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478: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/08/24(月) 07:43:08.99 ID:+oiN9Lqm

>>476 補足

a)0.999...＝0
b)0.999...≠0

a)とb)と、両方あるんじゃねと、テレンスタオはいう（下記）
そういうことです

https://en.wikipedia.org/wiki/0.999...
0.999...

This number is equal to 1. In other words, "0.999..." and "1" represent the same number.
(In other systems, 0.999... can have the same meaning, a different definition, or be undefined.)
More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is a property of all base representations. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasons?such as rigorous proofs relying on non-elementary techniques, properties, or disciplines?some people can find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education.

Infinitesimals
The standard definition of the number 0.999... is the limit of the sequence 0.9, 0.99, 0.999, ... A different definition involves what Terry Tao refers to as ultralimit, i.e., the equivalence class [(0.9, 0.99, 0.999, ...)] of this sequence in the ultrapower construction, which is a number that falls short of 1 by an infinitesimal amount.

http://rio2016.5ch.net/test/read.cgi/math/1592654877/478

781: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/11/07(土) 11:13:41.99 ID:4jX6N+0z

>>779

追加ご参考

https://en.wikipedia.org/wiki/Transfinite_number
Transfinite number

Definition
Any finite number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify the order of a member within an ordered set[5] (e.g., "the third man from the left" or "the twenty-seventh day of January"). When extended to transfinite numbers, these two concepts become distinct. A transfinite cardinal number is used to describe the size of an infinitely large set,[3] while a transfinite ordinal is used to describe the location within an infinitely large set that is ordered.[5] The most notable ordinal and cardinal numbers are, respectively:

・ω(Omega): the lowest transfinite ordinal number. It is also the order type of the natural numbers under their usual linear ordering.
・アレフ_0 (Aleph-naught): the first transfinite cardinal number. It is also the cardinality of the infinite set of the natural numbers. If the axiom of choice holds, the next higher cardinal number is aleph-one, アレフ_1. If not, there may be other cardinals which are incomparable with aleph-one and larger than aleph-naught. Either way, there are no cardinals between aleph-naught and aleph-one.

The continuum hypothesis is the proposition that there are no intermediate cardinal numbers between アレフ_0 and the cardinality of the continuum (the cardinality of the set of real numbers):[3] or equivalently that アレフ_1 is the cardinality of the set of real numbers. In Zermelo-Fraenkel set theory, neither the continuum hypothesis nor its negation can be proven without violating consistency.

http://rio2016.5ch.net/test/read.cgi/math/1592654877/781

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