List of statements undecidable in ZFC について

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List of statements undecidable in ZFC ：ウィキペディア英語版

List of statements undecidable in ZFC
The mathematical statements discussed below are undecidable in ZFC (the Zermelo–Fraenkel axioms plus the axiom of choice, the canonical axiomatic set theory of contemporary mathematics), assuming that ZFC is consistent. A statement is undecidable in ZFC (a.k.a. independent of ZFC) if it can neither be proven nor disproven from the axioms of ZFC.
==Axiomatic set theory==
In 1931, Kurt Gödel proved the first ZFC undecidability result, namely that the consistency of ZFC itself was undecidable in ZFC (Gödel's second incompleteness theorem).
Moreover, the following statements are undecidable in ZFC:
* The continuum hypothesis (CH); (Gödel produced a model of ZFC in which CH is true, showing that CH cannot be disproven in ZFC; Paul Cohen later invented the method of forcing to exhibit a model of ZFC in which CH fails, showing that CH cannot be proven in ZFC. The following four undecidability results are also due to Gödel/Cohen.)
* The generalized continuum hypothesis (GCH);
* The axiom of constructibility (''V'' = ''L'');
* The diamond principle (◊);
* Martin's axiom (MA);
* MA + ¬CH. (Undecidability shown by Solovay and Tennenbaum.)
We have the following chains of implication:
:''V'' = ''L'' → ◊ → CH.
:''V'' = ''L'' → GCH → CH.
:CH → MA
Another statement that is undecidable in ZFC is:
:If the set ''S'' has fewer elements than ''T'' (in the sense of cardinality), then ''S'' also has fewer subsets than ''T''.
Several statements related to the existence of large cardinals cannot be proven in ZFC (assuming ZFC is consistent). These are undecidable in ZFC provided that they are consistent with ZFC, which most working set theorists believe to be the case. These statements are strong enough to imply the consistency of ZFC. This has the consequence (via Gödel's second incompleteness theorem) that their consistency with ZFC cannot be proven in ZFC (assuming ZFC is consistent). The following statements belong to this class:
* Existence of inaccessible cardinals
* Existence of Mahlo cardinals
* Existence of measurable cardinals (first conjectured by Ulam)
* Existence of supercompact cardinals
The following statements can be proven to be undecidable in ZFC assuming the consistency of a suitable large cardinal:
* Proper forcing axiom
* Open coloring axiom
* Martin's maximum
* Existence of 0^#
* Singular cardinals hypothesis
* Projective determinacy

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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