現代数学の系譜　カントル　超限集合論

[過去ﾛｸﾞ] 現代数学の系譜　カントル　超限集合論 (1002ﾚｽ)
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473: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2019/10/27(日) 23:44:36.69 ID:EUeYkluT

>>472 補足

https://scholarpublishing.org/sse/global-set-theory/
Services for Science and Education Ltd
Global Set Theory
Satoko Titani
Professor Emeritus of Chubu University, Japan.

Our reasoning is based on dichotomous logic. That is, we are naturally convinced that a statement is either true or false exclusively. We accept the dichotomy as axiom.
The dichotomous logic is a language of mathematics in which science is described. Set theory, which is the base of mathematics, is formulated into a formal system of a logic with set theoretical axioms. A logic provided with globalization or basic implication is called a global logic.
A set theory based on the global logic is called a global set theory. Global set theory comprehends the meta-theory of set theory.
By introducing the globalization into a set theory, we can express the truth value set and also express the universe of the set theory in the set theory. That is, global set theory is nested in the global set theory.
It follows that we can prove the completeness of global set theories such as lattice valued set theory and quantum set theory.
Logical science is founded on the base of global classical logic. Each classical statement is either true or false, and these outcomes are mutually exclusive. Thereby, the global classical theory determines true-or-false definitely.
The logical science gives us very fruitful information, even though it covers only a bounded aspect of nature that falls within the realm of logic. We see that our logic is not absolute and is in fact determined by the establishment of a truth value set, which depends on object world.
Nature in its entirety is far beyond the scope of logic. Therefore, logical science is unable to describe the entire natural universe, in the same way as a net is unable to scoop up everything in its path.

DOI: 10.14738/tnc.092018.1
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617: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2019/12/07(土) 08:42:52.69 ID:H2e5WMAT

>>614
無理するな（＾＾

（>>612より）
https://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0065
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Untersuchungen uber die Grundlagen der Mengenlehre. I. Von E. ZERMELO in Gottingen. P261
(抜粋英訳)
P263
Axiom I. If every element of a set M is simultaneously an element of N and vice versa, that is, if M = E N and N = E M at the same time, then M = N is always M or shorter: every set is determined by its elements.

P266
But in order to secure the existence of "infinite" sets, we still need the following axiom, which derives from its essential content by Mr. R. Dedekind.
Axiom VII. The domain contains at least a set Z which contains the null set as an element and is such that each of its elements a is another element of the form {a}, or which with each of its elements a is also the corresponding set {a } as an element.
(Axiom of the infinite.)
14 VII. *) If Z is an arbitrary set of the properties required in VII, then for each of its subsets Z1 it is definite whether it possesses the same property. For if a is any element of Z1 ', it is definite whether {a} ∈ Z1,
and all the elements a of Z1 thus constituted form the elements of a subset Z1' for which it is definite whether Z1 '= Z1 or Not. Thus, all subsets Z1 of the considered property form the elements of a subset T = E UZ,
and the average corresponding to them (# 9) Z0 = DT is an amount of the same nature.
つづく

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