Abstract
In 1898, the German Association of Mathematicians (DMV) commissioned Schoenflies to write a report on “Curves and point manifolds”. The first part of the report was published in 1900 under the title “The development of the science of point manifolds”. The report was updated in 1908 and 1913 but the subject grew by that time out of the scope of any single report. In 1900, however, the report was the only text book on set theory and its applications, outside Cantor’s 1895/7 Beiträge.
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.
Cantor notes that Schröder was first to have found a proof and that then Bernstein had independently found a proof which is independent of logical calculations.
- 2.
Theorem U is CBT in its two-set formulation.
- 3.
Cantor is using his notation from 1895 Beiträge for union.
- 4.
We translate Cantor’s “eindeutige” into 1–1. Cantor is using “projection” as did Borel.
- 5.
Cantor defines K by the property required for the proof rather than as the intersection of nesting sets or the residue after removal of frames.
- 6.
Schoenflies is using Cantor’s sign ‘~’ for equivalence which he defines (p 5) as Cantor by: “two sets M and N are called equivalent or of equal power (M ~ N), in case it is possible, according to some law, to put them in one-one correlation.”
- 7.
Schoenflies follows Cantor’s notation for union (§3 of Cantor’s 1895 Beiträge). Note that Schoenflies notation for the sets generated in the proof is idiosyncratic for he uses signs derived from M, N for the frames and not for the nested sets, which are equivalent to M, N. In Schoenflies 1913 (p 34f) the notation was set straight.
References
Borel E. Leçons sur la théorie des functions, 1950 edition, Gauthiers-Villars: Paris; 1898
Cantor G. Beiträge zur Begründung der transfiniten Mengenlehre, (‘1895 Beiträge’). Cantor 1932;282–311. English translation: Cantor 1915.
Meschkowski H, Nilsen W. Georg Cantor: briefe. Berlin: Springer; 1991.
Schröder E. Über Zwei Defitionen der Endlichkeit und G. Cantorsche Sätze, Nova Acta. Abhandlungen der Kaiserlichen Leopold-Carolinschen deutchen Akademie der Naturfoscher. 1898;71:301–62.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2013 Springer Basel
About this chapter
Cite this chapter
Hinkis, A. (2013). Schoenflies’ 1900 Proof of CBT. 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_12
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0224-6_12
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)