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Single Idea 13517

[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis ]

Full Idea

Gödel proved that (if set theory is consistent) we cannot refute the continuum hypothesis, and Cohen proved that (if set theory is consistent) we cannot prove it either.

Clarification

The hypothesis is that the numbers contain gaps

Gist of Idea

If set theory is consistent, we cannot refute or prove the Continuum Hypothesis

Source

report of Kurt Gödel (What is Cantor's Continuum Problem? [1964]) by William D. Hart - The Evolution of Logic 10

Book Ref

Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.273

The 6 ideas from 'What is Cantor's Continuum Problem?'

9942 Gödel proved the classical relative consistency of the axiom V = L [Gödel, by Putnam]
10868 The Continuum Hypothesis is not inconsistent with the axioms of set theory [Gödel, by Clegg]
13517 If set theory is consistent, we cannot refute or prove the Continuum Hypothesis [Gödel, by Hart,WD]
18062 Set-theory paradoxes are no worse than sense deception in physics [Gödel]
8679 We perceive the objects of set theory, just as we perceive with our senses [Gödel]
10271 Basic mathematics is related to abstract elements of our empirical ideas [Gödel]