Conjectures
Continuum Hypothesis
kun-TIN-yoo-um hy-POTH-uh-sis
Also Known As CH
Logic and Foundations
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Proposed by Georg Cantor in 1878 as he worked out the theory of infinite sets, the continuum hypothesis asks whether there is a size of infinity strictly between the countable infinity of the integers and the larger infinity of the real numbers. It is a different kind of open problem from Goldbach or the twin primes: it was eventually shown that the axioms mathematicians actually use cannot settle the question either way, so it is not merely unproven but proven unprovable from the usual starting points.
Facts
StatementThere is no set whose size is strictly between that of the integers and that of the real numbers: every infinite subset of the real numbers can be matched one to one either with the integers or with the real numbers themselves, with nothing of intermediate size in between. 1 Proposed Year Prize StatusNot a Millennium Prize Problem. It is the rare open question that was settled not by being proved true or false but by being proved independent of the standard ZFC axioms of set theory. 1 Progress Toward ResolutionIn 1940 Kurt Godel showed the continuum hypothesis cannot be disproved from the standard Zermelo-Fraenkel axioms with choice, ZFC; in 1963 Paul Cohen showed it cannot be proved from them either, using the technique of forcing. Together these results mean the continuum hypothesis is independent of ZFC: the usual axioms of set theory are simply silent on the question. Mathematicians remain divided over whether it should be considered true, false, or neither; Godel himself believed it was false and that ZFC fails to fully capture the true universe of sets, a minority position Paul Cohen also leaned toward despite his own formalist inclinations. 1 Cross-Tradition Connections
Associated With
Godel proved in 1940 that the negation of CH cannot be proved from standard ZFC set theory, the first half of the independence result completed by Cohen in 1963 (Cohen has no live entity in this atlas to link to).
In Branch
Posed By
In the Other Atlases
- Also in Philosophy Atlas: Platonism, influenced this subject there.
Sources
1. Continuum Hypothesis (Wikipedia)
Wikimedia FoundationIntroduction sectionQuote, Introduction section
This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940.
1. Continuum Hypothesis (Wikipedia)
Wikimedia FoundationArguments for and against sectionQuote, Arguments for and against section
Gödel believed that CH is false, and that his proof that CH is consistent with ZFC only shows that the Zermelo–Fraenkel axioms do not adequately characterize the universe of sets.
1. Continuum Hypothesis (Wikipedia)
Wikimedia FoundationIn Branch: Logic and Foundations, Opening paragraphQuote, In Branch: Logic and Foundations, Opening paragraph
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis
1. Continuum Hypothesis (Wikipedia)
Wikimedia FoundationPosed By: Georg Cantor, History sectionQuote, Posed By: Georg Cantor, History section
The continuum hypothesis was first introduced by Georg Cantor in his 1878 paper
1. Continuum Hypothesis (Wikipedia)
Wikimedia FoundationAssociated With: Kurt Godel, Independence from ZFC section, second paragraphQuote, Associated With: Kurt Godel, Independence from ZFC section, second paragraph
Kurt Godel proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate cardinality, could not be proved in standard set theory.
Open Questions (1 open question)
Is there really a size of infinity strictly between the integers and the real numbers, and if not, does that fact hold absolutely or only relative to which further axioms mathematicians choose to accept?
Godel (1940) and Cohen (1963) proved that the continuum hypothesis can be neither proved nor disproved from the standard ZFC axioms of set theory. That result closes the question of what ZFC alone can say, but it opens a harder one: whether CH has a determinate truth value at all, and if so what further axiom would reveal it. Mathematicians who accept that set theory describes a single true universe of sets, Godel among them, hold this is a real unanswered question rather than a dead end; committed formalists tend to hold there is nothing further to ask.
What would resolve this Either a broadly accepted new axiom for set theory, beyond ZFC, that settles CH one way or the other and gains the kind of consensus ZFC itself enjoys, or a philosophical argument persuasive enough to convince most set theorists that the independence result is the end of the matter rather than the start of a harder question.
Mathematical logic and set theoryContinuum Hypothesis (Wikipedia)
Dissenting Readings (1 dissenting reading)
Progress Toward Resolution
Godel did not treat the independence of the continuum hypothesis from ZFC as the end of the question. He held that CH is false, on the ground that ZFC's axioms are an incomplete description of a single, determinate universe of sets, so a further true axiom, not yet found, would settle CH the way new evidence settles an empirical question. On this view, independence exposes a gap in ZFC rather than showing that mathematics has no fact of the matter here. Paul Cohen, the mathematician who proved the other half of the independence result, leaned toward rejecting CH as well despite describing himself as a formalist, for whom Godel's kind of Platonist appeal to an underlying true universe of sets is normally out of bounds.
A dissenting reading, from Kurt GodelContinuum Hypothesis (Wikipedia), Wikimedia Foundation
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