現代数学の系譜カントル超限集合論他 3

[過去ﾛｸﾞ] 現代数学の系譜カントル超限集合論他 3 (548ﾚｽ)
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42: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [sage] 2020/07/27(月) 21:41:57.91 ID:slbIBvLt

>>39 補足
https://arxiv.org/pdf/1212.5740.pdf
Filters and Ultrafilters in Real Analysis 2012
Max Garcia Mathematics Department California Polytechnic State University

Abstract
We study free filters and their maximal extensions on the set of natural numbers.
We characterize the limit of a sequence of real numbers in terms of the Fr´echet filter, which involves only one quantifier as opposed to the three non-commuting quantifiers in the usual definition.
We construct the field of real non-standard numbers and study their properties.
We characterize the limit of a sequence of real numbers in terms of non-standard numbers which only requires a single quantifier as well.
We are trying to make the point that the involvement of filters and/or non-standard numbers leads to a reduction in the number of quantifiers and hence, simplification, compared to the more traditional ε, δ-definition of limits in real analysis.

Contents
Introduction .  . 1
1 Filters, Free Filters and Ultrafilters 3
1.1 Filters and Ultrafilters . . .. 3
1.2 Existence of Free Ultrafilters . . . . . . 5
1.3 Characterization of the Ultrafilter . . . . . . 6
2 The Fr´echet Filter in Real Analysis 8
2.1 Fr´echet Filter . . . . .  . . . . 8
2.2 Reduction in the Number of Quantifiers . . .. . . 10
2.3 Fr´echet filter in Real Analysis . . . . . . . 11
2.4 Remarks Regarding the Fr´echet Filter . .  . . . 12
3 Non-standard Analysis 14
3.1 Construction of the Hyperreals *R . . . . . 14
3.2 Finite, Infinitesimal, and Infinitely Large Numbers . . . . . . . 16
3.3 Extending Sets and Functions in *R . . . . . . . . . . . . . . . 20
3.4 Non-Standard Characterization of Limits in R . . . . . . . . . 23
A The Free Ultrafilter as an Additive Measure 25

http://rio2016.5ch.net/test/read.cgi/math/1595034113/42

54: １３２人目の素数さん [sage] 2020/07/28(火) 11:04:39.20 ID:U9fCF8yb

>>42
補足

下記PDFで
”This new system would be constructed in a manner similar to Cauchy’s construction of the real numbers”
”Let us consider the factor ring
R~^N = R^N/ 〜Fr
where 〜Fr is the equivalence relation defined by
(an)〜Fr(bn) if and only if {n : an = bn} ∈ Fr.
This is no different to saying that (an) is equivalent to (bn) if and only if
an = bn for all sufficiently large n. ”

ここに、Frは、フレシェ・フィルターです。
なるほど、なるほど、フレシェ・フィルターを使って、”similar to Cauchy’s construction of the real numbers”をやる
”where 〜Fr is the equivalence relation defined by
(an)〜Fr(bn) if and only if {n : an = bn} ∈ Fr.”
数列のシッポの同値を使ってね

そうすると、”Non-standard Analysis 3.1 Construction of the Hyperreals *R ”
が出る！

全部、きっちりと、論文として書いてありますなぁ〜！ｗｗｗ（＾＾；
2012年の論文に〜！！ ｗｗｗｗ(゜ロ゜;

https://arxiv.org/pdf/1212.5740.pdf
Filters and Ultrafilters in Real Analysis 2012
Max Garcia Mathematics Department California Polytechnic State University
(抜粋)
P12
2.4 Remarks Regarding the Fr´echet Filter

This new system would be constructed in a manner similar to Cauchy’s construction of the real numbers
from rational sequences. The elements in this new system would be equivalence classes of real numbered sequences, which take into account sequence
convergence (divergence) as well as the rate of convergence (divergence).
Ideally, the resulting system will contain elements that can be used to characterize convergence in such a manner that we can do away with the limits
of standard analysis or the set constructions from the Fr´echet approach.
Let us consider the factor ring
R~^N = R^N/ 〜Fr
where 〜Fr is the equivalence relation defined by
(an)〜Fr(bn) if and only if {n : an = bn} ∈ Fr.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1595034113/54

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