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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Continuum hypothesis | 5/5 | https://en.wikipedia.org/wiki/Continuum_hypothesis | reference | science, encyclopedia | 2026-05-05T09:59:15.765779+00:00 | kb-cron |
(occasionally called Cantor's aleph hypothesis). The beth numbers provide an alternative notation for this condition:
ℵ
α
=
ℶ
α
{\displaystyle \aleph _{\alpha }=\beth _{\alpha }}
for every ordinal
α
{\displaystyle \alpha }
. The continuum hypothesis is the special case for the ordinal
α
=
1
{\displaystyle \alpha =1}
. GCH was first suggested by Philip Jourdain. For the early history of GCH, see Moore. Like CH, GCH is also independent of ZFC, but Sierpiński proved that ZF + GCH implies the axiom of choice (AC) (and therefore the negation of the axiom of determinacy, AD), so choice and GCH are not independent in ZF; there are no models of ZF in which GCH holds and AC fails. To prove this, Sierpiński showed GCH implies that every cardinality n is smaller than some aleph number, and thus can be ordered. This is done by showing that n is smaller than
2
ℵ
0
+
n
{\displaystyle 2^{\aleph _{0}+n}}
which is smaller than its own Hartogs number—this uses the equality
2
ℵ
0
+
n
=
2
⋅
2
ℵ
0
+
n
{\displaystyle 2^{\aleph _{0}+n}\,=\,2\cdot \,2^{\aleph _{0}+n}}
; for the full proof, see Gillman. Kurt Gödel showed that GCH is a consequence of ZF + V=L (the axiom that every set is constructible relative to the ordinals), and is therefore consistent with ZFC. As GCH implies CH, Cohen's model in which CH fails is a model in which GCH fails, and thus GCH is not provable from ZFC. W. B. Easton used the method of forcing developed by Cohen to prove Easton's theorem, which shows it is consistent with ZFC for arbitrarily large cardinals
ℵ
α
{\displaystyle \aleph _{\alpha }}
to fail to satisfy
2
ℵ
α
=
ℵ
α
+
1
{\displaystyle 2^{\aleph _{\alpha }}=\aleph _{\alpha +1}}
. Much later, Foreman and Woodin proved that (assuming the consistency of very large cardinals) it is consistent that
2
κ
>
κ
+
{\displaystyle 2^{\kappa }>\kappa ^{+}}
holds for every infinite cardinal
κ
{\displaystyle \kappa }
. Later Woodin extended this by showing the consistency of
2
κ
=
κ
+
+
{\displaystyle 2^{\kappa }=\kappa ^{++}}
for every
κ
{\displaystyle \kappa }
. Carmi Merimovich showed that, for each n ≥ 1, it is consistent with ZFC that for each infinite cardinal κ, 2κ is the nth successor of κ (assuming the consistency of some large cardinal axioms). On the other hand, László Patai proved that if γ is an ordinal and for each infinite cardinal κ, 2κ is the γth successor of κ, then γ is finite. For any infinite sets A and B, if there is an injection from A to B then there is an injection from subsets of A to subsets of B. Thus for any infinite cardinals A and B,
A
<
B
→
2
A
≤
2
B
{\displaystyle A<B\to 2^{A}\leq 2^{B}}
. If A and B are finite, the stronger inequality
A
<
B
→
2
A
<
2
B
{\displaystyle A<B\to 2^{A}<2^{B}}
holds. GCH implies that this strict, stronger inequality holds for infinite cardinals as well as finite cardinals.
=== Implications of GCH for cardinal exponentiation === Although the generalized continuum hypothesis refers directly only to cardinal exponentiation with 2 as the base, one can deduce from it the values of cardinal exponentiation
ℵ
α
ℵ
β
{\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}}
in all cases. GCH implies that for ordinals α and β:
ℵ
α
ℵ
β
=
ℵ
β
+
1
{\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}=\aleph _{\beta +1}}
when α ≤ β+1;
ℵ
α
ℵ
β
=
ℵ
α
{\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}=\aleph _{\alpha }}
when β+1 < α and
ℵ
β
<
cf
(
ℵ
α
)
{\displaystyle \aleph _{\beta }<\operatorname {cf} (\aleph _{\alpha })}
, where cf is the cofinality operation; and
ℵ
α
ℵ
β
=
ℵ
α
+
1
{\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}=\aleph _{\alpha +1}}
when β+1 < α and
ℵ
β
≥
cf
(
ℵ
α
)
{\displaystyle \aleph _{\beta }\geq \operatorname {cf} (\aleph _{\alpha })}
. The first equality (when α ≤ β+1) follows from:
ℵ
α
ℵ
β
≤
ℵ
β
+
1
ℵ
β
=
(
2
ℵ
β
)
ℵ
β
=
2
ℵ
β
⋅
ℵ
β
=
2
ℵ
β
=
ℵ
β
+
1
{\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}\leq \aleph _{\beta +1}^{\aleph _{\beta }}=(2^{\aleph _{\beta }})^{\aleph _{\beta }}=2^{\aleph _{\beta }\cdot \aleph _{\beta }}=2^{\aleph _{\beta }}=\aleph _{\beta +1}}
while:
ℵ
β
+
1
=
2
ℵ
β
≤
ℵ
α
ℵ
β
.
{\displaystyle \aleph _{\beta +1}=2^{\aleph _{\beta }}\leq \aleph _{\alpha }^{\aleph _{\beta }}.}
The third equality (when β+1 < α and
ℵ
β
≥
cf
(
ℵ
α
)
{\displaystyle \aleph _{\beta }\geq \operatorname {cf} (\aleph _{\alpha })}
) follows from:
ℵ
α
ℵ
β
≥
ℵ
α
cf
(
ℵ
α
)
>
ℵ
α
{\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}\geq \aleph _{\alpha }^{\operatorname {cf} (\aleph _{\alpha })}>\aleph _{\alpha }}
by Kőnig's theorem, while:
ℵ
α
ℵ
β
≤
ℵ
α
ℵ
α
≤
(
2
ℵ
α
)
ℵ
α
=
2
ℵ
α
⋅
ℵ
α
=
2
ℵ
α
=
ℵ
α
+
1
{\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}\leq \aleph _{\alpha }^{\aleph _{\alpha }}\leq (2^{\aleph _{\alpha }})^{\aleph _{\alpha }}=2^{\aleph _{\alpha }\cdot \aleph _{\alpha }}=2^{\aleph _{\alpha }}=\aleph _{\alpha +1}}
== See also == Absolute infinite Beth number Cardinality Ω-logic Second continuum hypothesis Wetzel's problem
== References ==
Maddy, Penelope (June 1988). "Believing the axioms, [part I]". Journal of Symbolic Logic. 53 (2). Association for Symbolic Logic: 481–511. doi:10.2307/2274520. JSTOR 2274520.
== Sources == This article incorporates material from Generalized continuum hypothesis on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. Archived 2017-02-08 at the Wayback Machine
== Further reading == Cohen, Paul Joseph (2008) [1966]. Set theory and the continuum hypothesis. Mineola, New York City: Dover Publications. ISBN 978-0-486-46921-8. Dales, H.G.; Woodin, W.H. (1987). An Introduction to Independence for Analysts. Cambridge. Enderton, Herbert (1977). Elements of Set Theory. Academic Press. Gödel, K.: What is Cantor's Continuum Problem?, reprinted in Benacerraf and Putnam's collection Philosophy of Mathematics, 2nd ed., Cambridge University Press, 1983. An outline of Gödel's arguments against CH. Martin, D. (1976). "Hilbert's first problem: the continuum hypothesis," in Mathematical Developments Arising from Hilbert's Problems, Proceedings of Symposia in Pure Mathematics XXVIII, F. Browder, editor. American Mathematical Society, 1976, pp. 81–92. ISBN 0-8218-1428-1 McGough, Nancy. "The Continuum Hypothesis". Wolchover, Natalie (15 July 2021). "How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer".
== External links == Quotations related to Continuum hypothesis at Wikiquote
Szudzik, Matthew & Weisstein, Eric W. "Continuum Hypothesis". MathWorld.