Proofs of the Cantor-Bernstein Theorem

Hinkis, Arie

doi:10.1007/978-3-0348-0224-6

Cited by 36 publications

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“…A diagrammatic presentation of the consideration underlying Dedekind's proof of the fundamental lemma is given in figure 1. Taking for granted the possibility of making explicit the 1 An account of proofs and their history is given in [5], a 429-page book that is very informative and quite comprehensive. However, Hinkis does not provide a unifying mathematical examination.…”

Section: Dedekind's Fundamental Lemmamentioning

confidence: 99%

“…We have only to observe that, in the first case, a\c = c* ∪ r and, in the second case, a\c* = c ∪ r. h* and h** are the canonical mappings that are obtained also in all the other proofs I have analysed. 5 There are two important and problematic issues in the above arguments; first, we have to find for the informally described sets c and c* an explicit set-theoretic definition and, second, we have to prove the structural identities. If one defines c 'from below' as ∪ [h n [a\d] | n ∈ N] with h 0 [a\d] = a\d and h n+1 [a\d] = h[h n [a\d]], then it is immediate that c = (a\d) ∪ h [c].…”

Section: Dedekind's Fundamental Lemmamentioning

confidence: 99%

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The Cantor–Bernstein theorem: how many proofs?

Sieg

2019

Phil. Trans. R. Soc. A.

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Dedekind's proof of the Cantor–Bernstein theorem is based on his chain theory, not on Cantor's well-ordering principle. A careful analysis of the proof extracts an argument structure that can be seen in the many other proofs that have been given since. I contend there is essentially one proof that comes in two variants due to Dedekind and Zermelo , respectively. This paper is a case study in analysing proofs of a single theorem within a given methodological framework, here Zermelo–Fraenkel set theory (ZF). It uses tools from proof theory, but focuses on heuristic ideas that shape proofs and on logical strategies that help to construct them. It is rooted in a perspective on Beweistheorie that predates its close connection and almost exclusive attention to the goals of Hilbert's finitist consistency programme. This earlier perspective can be brought to life (only) with the support of powerful computational tools. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

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Section: Dedekind's Fundamental Lemmamentioning

confidence: 99%

Section: Dedekind's Fundamental Lemmamentioning

confidence: 99%

The Cantor–Bernstein theorem: how many proofs?

Sieg

2019

Phil. Trans. R. Soc. A.

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“…Although König [1990, p. 171] observes that his proof of the Infinity Lemma still requires the Axiom of Choice when stated in full generality, he also notes this may be avoided in many of its applications. See [Franchella, 1997] and [Hinkis, 2013] for more on König's use of the Lemma in his proof of the Cantor-Schröder-Bernstein theorem.…”

Section: Weak König's Lemma and Related Systemsmentioning

confidence: 99%

“…Although König (1990, p. 171) observes that his proof of the Infinity Lemma still requires the Axiom of Choice when stated in full generality, he also notes this may be avoided in many of its applications. See (Franchella, 1997) and (Hinkis, 2013) for more on König's use of the Lemma in his proof of the Cantor-Schröder-Bernstein theorem. 21 The Covering Lemma states that the unit interval is compact with respect to the standard topology on the reals -i.e., "Every covering of the closed unit interval [0, 1] by a sequence of open intervals has a finite subcovering".…”

Section: The Completeness Theorem For Classical Predicate Calculus Imentioning

confidence: 99%

The Prehistory of the Subsystems of Second-Order Arithmetic

Dean

Walsh

2017

The Review of Symbolic Logic

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This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program of Friedman and Simpson. We look in particular at: (i) the long arc from Poincaré to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak König's Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others.

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“…Thus, by the Theorem, there is a bijection from a to d. Let us now prove the Fundamental Lemma. 8 For the complex and intriguing history of proofs of the theorem we refer to the accounts in (Hinkis, 2013), (Deiser, 2010, sec. 3) and (Kanamori, 2004, sec. 4).…”

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confidence: 99%

Natural Formalization: Deriving the Cantor-Bernstein Theorem in Zf

Sieg

Walsh

2019

The Review of Symbolic Logic

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Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas. To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at http://www.phil.cmu.edu/legacy/Proof_Site/. We must—that is my conviction—take the concept of the specifically mathematical proof as an object of investigation. Hilbert 1918

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Proofs of the Cantor-Bernstein Theorem

Cited by 36 publications

References 0 publications

The Cantor–Bernstein theorem: how many proofs?

The Cantor–Bernstein theorem: how many proofs?

The Prehistory of the Subsystems of Second-Order Arithmetic

Natural Formalization: Deriving the Cantor-Bernstein Theorem in Zf

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