Aleph formula

Aleph formulas are mathematical formulas of cardinal number arithmetic and as such doctrines of the mathematical branch of set theory . Not least, important aleph formulas are associated with the names of mathematicians Gerhard Hessenberg , Felix Hausdorff and Felix Bernstein .

The term aleph formula (s) is mainly used by Arnold Oberschelp and Dieter Klaua in their respective monographs General Set Theory , whereby Oberschelp explicitly means the formula presented by Hessenberg in 1906 (see below).

Hessenberg's formula

The formula presented by Hessenberg in 1906 - which is also cited as the Hessenberg theorem - is of fundamental importance for the entire cardinal number arithmetic. It can be specified as follows:

The following applies to every ordinal number ${\displaystyle \alpha }$

${\displaystyle {\aleph _{\alpha }}^{2}\,=\,\aleph _{\alpha }}$.

Inferences

The Hessenberg formula leads to a number of other Aleph formulas.

I.

For every two ordinal numbers and the Hessenberg equation applies ${\displaystyle \alpha }$ ${\displaystyle \beta }$

${\displaystyle \aleph _{\alpha }+\aleph _{\beta }\,=\,\aleph _{\alpha }\cdot \aleph _{\beta }\,=\,\aleph _{\max(\alpha \,,\,\beta )}\,=\,\max(\aleph _{\alpha }\,,\,\aleph _{\beta })}$.

II

Applying the Hessenberg equation also results in the Bernstein formula presented by Felix Bernstein :

For every two ordinal numbers and with applies ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \alpha \leq \beta +1}$

${\displaystyle 2^{\aleph _{\beta }}\,=\,{\aleph _{\alpha }}^{\aleph _{\beta }}}$.

III

Felix Bernstein provided another aleph formula, which Klaua also calls the amber aleph sentence and which goes back to Bernstein's publication in 1905:

The following applies to every ordinal number and all natural numbers ${\displaystyle \beta }$ ${\displaystyle n}$

${\displaystyle \aleph _{n}^{\aleph _{\beta }}\,=\,2^{\aleph _{\beta }}\cdot \aleph _{n}}$.

Hausdorff's formula

Proceeding as of Bernstein Alephsatz is a phrase that of Felix Hausdorff was proven in 1904 and in which he known Hausdorff recursion ( English Hausdorff recursion formula formulated):

For every two ordinal numbers and and all natural numbers applies ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle n}$

${\displaystyle \aleph _{\alpha +n}^{\aleph _{\beta }}\,=\,\aleph _{\alpha }^{\aleph _{\beta }}\cdot \aleph _{\alpha +n}}$.

In particular, the formula applies to every ordinal number that is not a Limes number and every ordinal number ${\displaystyle \alpha >1}$ ${\displaystyle \beta }$

${\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}\,=\,\aleph _{\alpha -1}^{\aleph _{\beta }}\cdot \aleph _{\alpha }}$.

Related formulas

Beyond the classic aleph formulas outlined above, there are a number of related formulas that place the aleph in a wider context.

King's formula

In 1904 Julius König proved a formula that exacerbates the well-known inequality and at the same time provides an upper estimate for the Alephs by means of confinalities . This formula, which is based on König's theorem , says: ${\displaystyle \aleph _{\alpha }\,<\,2^{\aleph _{\alpha }}}$

The inequality applies to every ordinal number ${\displaystyle \alpha }$

${\displaystyle \aleph _{\alpha }\,<\,\operatorname {cf} {\bigl (}2^{\aleph _{\alpha }}{\bigr )}}$.

Relation to the continuum hypothesis

The generalized continuum hypothesis (GCH) formulated by Hausdorff in 1908 can also be understood as an Aleph formula. One therefore speaks of the aleph hypothesis (AH). This says namely:

The equation applies to every ordinal number ${\displaystyle \alpha }$

${\displaystyle 2^{\aleph _{\alpha }}\,=\,{\aleph _{\alpha +1}}}$.

For this one has the following formulas:

I.

Assuming the generalized continuum hypothesis (GCH), for ordinal numbers and in the case that it is regular : ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \aleph _{\alpha }}$

${\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}\,=\,\aleph _{\alpha }}$if ${\displaystyle \beta <\alpha }$

${\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}\,=\,\aleph _{\beta +1}}$if ${\displaystyle \beta \geq \alpha }$

II

Assuming the generalized continuum hypothesis (GCH) applies to ordinal numbers and in the case that is singular : ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \aleph _{\alpha }}$

${\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}\,=\,\aleph _{\alpha }}$if ${\displaystyle \aleph _{\beta }<\operatorname {cf} {\bigl (}\aleph _{\alpha }{\bigr )}}$

${\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}\,=\,\aleph _{\alpha +1}}$if ${\displaystyle \operatorname {cf} {\bigl (}\aleph _{\alpha }{\bigr )}\leq \aleph _{\beta }\leq \aleph _{\alpha }}$

${\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}\,=\,\aleph _{\beta +1}}$if ${\displaystyle \aleph _{\beta }\geq \aleph _{\alpha }}$

Explanations and Notes

• The Alephs are as ordinal characterized in that they infinite and - with respect to those on the Ordinalzahlenklasse given well-ordering relation - with no real smaller ordinal equipotent are. ${\displaystyle On}$
• Dieter Klaua does not explicitly define in his general set theory what he means by aleph formulas . However, it is clear from the context what is meant.
• Hessenberg's formula (apparently) includes the theorem, which was already proven by Georg Cantor with the help of his pairing function , according to which and are sets of equal power. ${\displaystyle {\mathbb {N} }_{0}\times {\mathbb {N} }_{0}}$ ${\displaystyle {\mathbb {N} }_{0}}$
• The Hessenberg formula was rediscovered by Philip Jourdain in 1908 .
• The term aleph hypothesis goes back to Felix Hausdorff and his work from 1908. Hausdorff even uses the term Cantor's Alef hypothesis there .
• Some authors - such as Walter Felscher in Naive Sets and Abstract Numbers III - differentiate between the Generalized Continuum Hypothesis (GCH) and the Aleph Hypothesis (AH). According to Felscher: “In a set theory with a foundation axiom (GCH) and (AH) are equivalent; in any case it follows from (GCH) also (AH). ”As Ulrich Felgner showed in 1971, the generalized continuum hypothesis (GCH) and the aleph hypothesis (AH) are not equivalent to each other in a set theory without a choice axiom and without a foundation axiom.

See also

• Cardinal number arithmetic
• Ordinal number arithmetic
• Starting number

literature

• Felix Bernstein: Investigations from set theory . In: Mathematical Annals . tape 61 , 1905, pp. 117-155 ( MR1511337 ).
• Heinz-Dieter Ebbinghaus : Introduction to set theory (=  university pocket book . Volume 141 ). 4th edition. Spektrum Akademischer Verlag , Heidelberg, Berlin 2003, ISBN 3-8274-1411-3 .
• Walter Felscher: Naive sets and abstract numbers III . Transfinite methods. Bibliographisches Institut , Mannheim, Vienna, Zurich 1979, ISBN 3-411-01553-5 ( MR0536486 ).
• Felix Hausdorff: The power concept in set theory . In: Annual report of the German Mathematicians Association . tape 13 , 1904, pp. 569-571 .
• Felix Hausdorff: Fundamentals of a theory of ordered sets . In: Mathematical Annals . tape 65 , 1908, pp. 435-505 .
• Felix Hausdorff: Fundamentals of set theory . Reprinted, New York, 1965. Chelsea Publishing Company , New York, NY 1965.
• Gerhard Hessenberg: Basic concepts of set theory . In: Treatises of the Friesschen Schule, New Series . tape 1 . Vandenhoeck and Ruprecht , Göttingen 1906, p. 478-706 .
• Karel Hrbacek , Thomas Jech : Introduction to Set Theory (=  Monographs and Textbooks in Pure and Applied Mathematics . Volume 220 ). 3. Edition. Marcel Dekker, Inc. , New York, Basel 1999, ISBN 0-8247-7915-0 ( MR1697766 ).
• Philip EB Jourdain : The multiplication of Alephs . In: Mathematical Annals . tape 65 , 1908, pp. 506-512 ( MR1511479 ).
• Erich Kamke : Set theory (=  Göschen Collection . 999 / 999a). 7th edition. Walter de Gruyter , Berlin, New York 1971.
• Dieter Klaua : General set theory . A foundation of mathematics (=  mathematical textbooks and monographs, I. Department, mathematical textbooks . Volume X ). Akademie-Verlag , Berlin 1964 ( MR0175791 ).
• J. König : On the continuum problem . In: Mathematical Annals . tape 60 , 1905, pp. 177-180 ( MR1511296 ).
• J. König: Correction . In: Mathematical Annals . tape 60 , 1905, pp. 462 ( MR1511318 ).
• Kazimierz Kuratowski , Andrzej Mostowski : Set Theory . With an Introduction to Descriptive Set Theory. Translated from the 1966 Polish original (=  Studies in Logic and the Foundations of Mathematics . Volume 86 ). 2nd Edition. North-Holland Publishing Company , Amsterdam, New York, Oxford 1976 ( MR0485384 ).
• Azriel Lévy : Basic Set Theory (=  Perspectives in Mathematical Logic ). Springer-Verlag , Berlin, Heidelberg, New York 1979, ISBN 3-540-08417-7 ( MR0533962 ).
• Arnold Oberschelp : General set theory . BI Wissenschaftsverlag , Mannheim, Leipzig, Vienna, Zurich 1994, ISBN 3-411-17271-1 ( MR0536486 ).
• Wacław Sierpiński : Cardinal and Ordinal Numbers . Panstwowe Wydawnictwo Naukowe, Warsaw 1958 ( MR0095787 ).
• Alfred Tarski: Sur quelques théorèmes sur les alephs . In: Fundamenta Mathematicae . tape 7 , 1925, pp. 1-14 .

Individual evidence

1. ↑ ^{a b} Dieter Klaua: General set theory. 1964, p. 507 ff.
2. ↑ Heinz-Dieter Ebbinghaus: Introduction to set theory. 2003, p. 127 ff.
3. ^ Walter Felscher: Naive sets and abstract numbers III. 1979, p. 107 ff.
4. ↑ Erich Kamke: Set theory. 1971, p. 176 ff.
5. ^ Kuratowski / Mostowski: Set Theory. 1976, p. 267 ff.
6. ^ Azriel Lévy: Basic Set Theory. 1979, p. 92 ff.
7. ^ ^{A b} Arnold Oberschelp: General set theory. 1994, p. 237 ff.
8. ^ Wacław Sierpiński: Cardinal and Ordinal Numbers. 1958, p. 389 ff.
9. ↑ Ebbinghaus, op.cit., P. 127
10. ↑ Kamke, op.cit., P. 176
11. ↑ Klaua, op.cit., P. 507
12. ↑ Lévy, op.cit., P. 94.
13. ↑ Klaua, op.cit., P. 509
14. ↑ Kamke, op.cit., P. 177.
15. ↑ Lévy, op.cit., P. 95.
16. ↑ Oberschelp, op.cit., P. 239
17. ↑ Sierpiński, op.cit., P. 395.
18. ↑ Klaua, op.cit., P. 510
19. ↑ Felscher, op.cit., P. 109.
20. ↑ Oberschelp, op.cit., P. 241.
21. ↑ ^{a b} Klaua, op.cit., P. 512
22. ↑ ^{a b} Sierpiński, op.cit., P. 402.
23. ↑ Felix Hausdorff: The power concept in set theory. Annual German Math Ver. 13, p. 570
24. ↑ Lévy, op.cit., P. 187.
25. ↑ Oberschelp, op.cit., P. 246.
26. ↑ ^{a b} Hrbacek / Jech: Introduction to Set Theory. 1999, p. 165.
27. ↑ Although the year 1904 is mentioned here, it was only published in the Mathematische Annalen of 1905.
28. ↑ Klaua, op.cit., P. 500
29. ↑ Oberschelp, op. Cit., Pp. 241–242.
30. ↑ Hrbacek / Jech, op. Cit., Pp. 166-167.
31. ↑ Felscher, op.cit., P. 107.
32. ↑ Lévy, op.cit., P. 97.
33. ↑ Felix Hausdorff: Fundamentals of a theory of ordered sets. Math. Ann. 65, p. 494
34. ↑ Felscher, op. Cit., Pp. 173-175.
35. ↑ Felscher, op.cit., P. 174.
36. ↑ Oberschelp, op.cit., P. 242.