Abstract
We review Tarski’s 1949a proof of the inequality-BDT: If k ≠ 0 and km ≤ kn then m ≤ n, where k is a natural number and m, n are cardinal numbers, from which Tarski easily deduced, using CBT, BDT: If k ≠ 0 and km = kn then m = n.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For some n, the Nn, N′n may be empty.
- 2.
- 3.
- 4.
Theorem 5 of Lindenbaum-Tarski 1926 is in the language of sets and mappings in which the above proof is given; theorem 6 there is in the language of cardinal numbers as is theorem 2 above.
- 5.
If ℵ0 q ≤ m then m = m′ + ℵ0 q = m′ + 2ℵ0 q = m′ + ℵ0 q + ℵ0 q = m + ℵ0 q = m + (ℵ0 + 1)q = m + ℵ0 q + q = m + q. However, q ≤ m does not entail m = m + q.
- 6.
For there are three cardinal numbers n, p 1, q 1 such that m = n + p 1 and m′ = n + q 1 and m + p 1 = m = m + q 1. So m′ + p 1 = n + q 1 + p 1 = n + p 1 + q 1 = m + p 1 = m hence m ≥ m′.
- 7.
Our notation here slightly differs from that of Tarski and as a result so does our chain of equalities. In Tarski there seems to be a typo on the third equality from the bottom of the proof: p′ is missing on the left side.
- 8.
Theorem 30 of Lindenbaum-Tarski 1926 says that m + p = m + q iff 2m + p = 2m + q.
- 9.
From which it follows that n = m + s so that n ≥ m and by the given m ≥ n and CBT, m = n.
- 10.
Note that ϕ j and ψ j are chosen from the many equivalences that exist between the partitions but since the number of choices is finite the axiom of choice is not invoked.
- 11.
Bernstein extended ϕ and ψ to include their inverse and he was followed in this by Sierpiński 1922 while Tarski used inverses, as we have already remarked with regard to his proof of Theorem 2. The advantage of Bernstein’s notation is that ϕ and ψ, and thus the χ n , are always defined while the advantage of Tarski’s notation will become apparent in the definition of the G n below. Tarski stressed this difference (p 88 footnote 13) and claimed that the Bernstein-Sierpiński method cannot be generalized, an observation, with regard to BDT, from which we differed (see Sects. 14.3 and 28.1).
- 12.
The necessity in having the χ enumerated will become clear in the proof.
- 13.
The R j (R′ j ) are equivalent by ϕ j (ψ j ).
- 14.
Our notation deviates from Tarski’s mainly in the signs used for entities.
- 15.
Compare this definition with the definition of the sets N n in the proof of Theorem 2 above. This definition is the main metaphor of Tarski’s paper discussed here. A similar definition appeared already in Bernstein’s proof of BDT (see Sect. 14.2) for the sets to be interchanged. We have here perhaps another case of proof-processing.
- 16.
The equivalence mapping is the union of the χ j reduced to G j .
- 17.
Incidentally, in their proof Doyle-Conway ignore the lemma kn = kn + kr → kn = kn + r but use the lemma kn = kn + r → n = n + r (lemma 3 p 26).
References
Bernstein F. Untersuchungen aus der Mengenlehre. Mathematische Annalen. 1905;61:117–55.
Doyle PG, Conway JH. Division by three. 1994. http://www.math.dartmouth.edu/~doyle/docs/three/three.pdf.
Kőnig D. Über Graphen und ihre Anwendung auf Determinanttheorie und Mengenlehre, Mathematische Annalen 77:453–65. Partial English translation: Biggs NL, Lloyd EK., Wilson RJ, editors. Graph theory 1736–1936. Oxford: Clarendon press; 1986. pp. 203–5.
Lindenbaum A, Tarski A. Communication sur les recherches de la thèorie des ensembles. Comptes rendu des séances de la société Polonaise de Mathematique section Varsovie Annales de la Societe Polonaise Mathematique. 1926;19:299–330.
Sierpiński W. Sur l’ègalitè 2m=2n pour les nombres cardinaux. Fund Math. 1922;3:1–6.
Sierpiński W. Demonstration de l’egalite 2m-m=2m pour les nombres cardinaux transfini. Fund Math. 1947c;34:113–8.
Sierpiński W. Sur l’implication (2m≤2n)→(m≤n) pour les nombres cardinaux. Fund Math. 1947d;34:148–54.
Sierpiński W. Sur la division des types ordinaux. Fund Math. 1948;35:1–12.
Tarski A. Über Äquivalenz der Mengen in Bezug auf eine beliebige Klasse von Abbildungen. International Congress of Mathematicians. 1928;2:243–52.
Tarski A. Axiomatic and algebraic aspects of two theorems on sums of cardinals. Fund Math. 1948;35:79–104.
Tarski A. Cardinal algebras. New York: Oxford University Press; 1949a.
Zermelo E. Über die Addition transfiniter Cardinalzahlen, Nachrichten von der Kőniglich Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse aus dem Jahre 1901;34–8.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2013 Springer Basel
About this chapter
Cite this chapter
Hinkis, A. (2013). Tarski’s Proofs of BDT and the Inequality-BDT. In: Proofs of the Cantor-Bernstein Theorem. Science Networks. Historical Studies, vol 45. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0224-6_34
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0224-6_34
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0223-9
Online ISBN: 978-3-0348-0224-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)