Construct · 1 fields · 4 citations

continuum hypothesis: a standalone construct

One name. One construct. A landmark on the map.

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continuumhypothesis1 CONSTRUCTContinuum hyp.CANTOR · CONTINUUM

A single standalone construct; nearby constructs are located against it in the Collider. Tap a field for its definition, unit, and measures.

Set theory (mathematical logic)

No infinite size lies strictly between the whole numbers and the real numbers

Georg Cantor's 1878 paper Ein Beitrag zur Mannigfaltigkeitslehre ("A contribution to the theory of manifolds") first stated the continuum hypothesis: that there is no set whose size falls strictly between the size of the whole numbers and the size of the real number line (the "continuum"). In Cantor's own terms the smallest infinite size is that of the counting numbers, written aleph-null, and the hypothesis claims that the next size up is exactly the size of the reals, with nothing in between. Kurt Gödel showed in 1938 that the hypothesis cannot be disproved from the standard axioms of set theory (the Zermelo-Fraenkel axioms together with the axiom of choice, abbreviated ZFC), and Paul Cohen showed in 1963 and 1964, using a technique he invented called forcing, that it also cannot be proved from them, so the hypothesis is independent of ZFC: those axioms settle neither its truth nor its falsity. The construct is a landmark because it stood as the first of David Hilbert's 1900 list of open problems and because its resolution established that a plainly stated question about the sizes of infinite sets can be formally undecidable within the accepted foundations of mathematics.[1][2][3][4]

Unit
A proposition of set theory about the possible sizes of infinite sets
Recovery means
Not applicable
Measured by
Evaluated by formal proof from axioms; established to be independent of ZFC, hence neither provable nor refutable from those axioms

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Nearby constructs

forcing3 fields · 4 citationsjingle · one word, three constructsset3 fields · 5 citationsjingle · one word, three constructsaxiom of choice1 fields · 3 citationsanchor · a foundational axiom

References

  1. Cantor, G. (1878). Ein Beitrag zur Mannigfaltigkeitslehre. Journal für die reine und angewandte Mathematik, 1878(84), 242–258. https://doi.org/10.1515/crll.1878.84.242
  2. Gödel, K. (1938). The consistency of the axiom of choice and of the generalized continuum-hypothesis. Proceedings of the National Academy of Sciences, 24(12), 556–557. https://doi.org/10.1073/pnas.24.12.556
  3. Cohen, P. J. (1963). The independence of the continuum hypothesis. Proceedings of the National Academy of Sciences, 50(6), 1143–1148. https://doi.org/10.1073/pnas.50.6.1143
  4. Cohen, P. J. (1964). The independence of the continuum hypothesis, II. Proceedings of the National Academy of Sciences, 51(1), 105–110. https://doi.org/10.1073/pnas.51.1.105

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