現代数学の系譜工学物理雑談古典ガロア理論も読む77

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119: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2019/09/13(金) 23:21:51.22 ID:Ct8Lh9wH

>>118
つづき

https://en.wikipedia.org/wiki/Urelement
Urelement
(抜粋)
In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial') is an object that is not a set, but that may be an element of a set. Urelements are sometimes called "atoms" or "individuals."

Contents
1 Theory
2 Urelements in set theory
3 Quine atoms

Urelements in set theory
The Zermelo set theory of 1908 included urelements, and hence is a version we now call ZFA or ZFCA (i.e. ZFA with axiom of choice).[1]
It was soon realized that in the context of this and closely related axiomatic set theories, the urelements were not needed because they can easily be modeled in a set theory without urelements.[2]
Thus, standard expositions of the canonical axiomatic set theories ZF and ZFC do not mention urelements. (For an exception, see Suppes.[3])
Axiomatizations of set theory that do invoke urelements include Kripke?Platek set theory with urelements, and the variant of Von Neumann?Bernays?Godel set theory described by Mendelson.[4]
In type theory, an object of type 0 can be called an urelement; hence the name "atom."

つづく

http://rio2016.5ch.net/test/read.cgi/math/1568026331/119

120: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2019/09/13(金) 23:23:14.89 ID:Ct8Lh9wH

>>119

つづき

Adding urelements to the system New Foundations (NF) to produce NFU has surprising consequences.
In particular, Jensen proved[5] the consistency of NFU relative to Peano arithmetic; meanwhile, the consistency of NF relative to anything remains an open problem, pending verification of Holmes's proof of its consistency relative to ZF.
Moreover, NFU remains relatively consistent when augmented with an axiom of infinity and the axiom of choice.
Meanwhile, the negation of the axiom of choice is, curiously, an NF theorem. Holmes (1998) takes these facts as evidence that NFU is a more successful foundation for mathematics than NF.
Holmes further argues that set theory is more natural with than without urelements, since we may take as urelements the objects of any theory or of the physical universe.[6]
In finitist set theory, urelements are mapped to the lowest-level components of the target phenomenon, such as atomic constituents of a physical object or members of an organisation.
(引用終り)
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http://rio2016.5ch.net/test/read.cgi/math/1568026331/120

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