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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
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] |