A proof for Cantor-Schröder-Bernstein Theorem using the diagonal argument

2021 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Bernstein Theorem ◽  
Diagonal Argument

We prove Cantor-Schröder-Bernstein theorem using the diagonal argument.

Undecidability and Recursive Inseparability

1993 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Decision Procedure ◽  
Arbitrary System ◽  
System P ◽  
Diagonal Argument ◽  
Diagonal Function ◽  
Recursive Set

§1. Some Preliminary Theorems. we continue to let S be an arbitrary system, P be the set of Gödel numbers of the provable formulas of S and R be the set of Gödel numbers of the refutable formulas of S. Theorem 1. The set P̃* is not representable in S. Proof. This is the diagonal argument all over again. If H(v1) represents P̃* and h is the Gödel number of H(v1), the H[h̅] is provable in S iff h Ï p* iff d(h) ÏP iff H[h̅] is not provable in S, which is a contradiction. Theorem 1.1. If S is consistent, then P* is not definable in S. Proof. Suppose P* is definable in S. If S were consistent, then P* would be completely representable in S (cf. §3.1, Ch. 0). Hence P̃* would be representable in S, contrary to Theorem 1. Therefore, if S is consistent, then P* is not definable in S. Theorem 1.2. If the diagonal function d(x) is strongly definable in S and S is consistent, then P is not definable in S. Proof. Suppose d(x) is strongly definable in S. Since P* = d -1(P), then if P were definable in S, P* would be definable in S (by Th. 11.2, Ch. 0). Hence S would be inconsistent by Theorem 1.1. Exercise 1. Show that if S is consistent, then R* is not definable in S. Exercise 2. Show that if S is consistent, then no superset of R* disjoint from P* is definable in S, and no superset of P* disjoint from R* is definable in S. Exercise 3. Prove that if S is consistent and if the diagonal function is strongly definable in S, then no superset of P disjoint from R is definable in S. [This is stronger than Theorem 1.2.] §2. Undecidable Systems. A system S is said to be decidable (or to admit of a decision procedure) if the set P of Gödel numbers of the provable formulas of S is a recursive set. It is undecidable if P is not recursive. This meaning of ‘undecidable’ should not be confused with the meaning of ‘undecidable’ when applied to a particular formula (as being undecidable in a given system S).

scholarly journals A simple visual proof of the Schröder-Bernstein theorem

10.4171/em/65 ◽  
2007 ◽  
pp. 118-120
Author(s):  
Dongvu Tonien
Keyword(s):  
Bernstein Theorem

scholarly journals Input Pattern Classification Based on the Markov Property of the IMBT with Related Equations and Contingency Tables

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 245
Author(s):  
István Finta ◽  
Sándor Szénási ◽  
Lóránt Farkas
Keyword(s):  
Binary Tree ◽  
Contingency Tables ◽  
Markov Property ◽  
Input Pattern ◽  
Statistical Characteristics ◽  
Bernstein Theorem ◽  
Data Packets ◽  
Efficient Data ◽  
Search Operation ◽  
Fibonacci Sequences

In this contribution, we provide a detailed analysis of the search operation for the Interval Merging Binary Tree (IMBT), an efficient data structure proposed earlier to handle typical anomalies in the transmission of data packets. A framework is provided to decide under which conditions IMBT outperforms other data structures typically used in the field, as a function of the statistical characteristics of the commonly occurring anomalies in the arrival of data packets. We use in the modeling Bernstein theorem, Markov property, Fibonacci sequences, bipartite multi-graphs, and contingency tables.

scholarly journals Complete Self-Shrinking Solutions for Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Ruiwei Xu ◽  
Linfen Cao
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Indefinite Metric ◽  
Bernstein Theorem ◽  
Strictly Convex ◽  
Convex Solution

Letf(x)be a smooth strictly convex solution ofdet(∂2f/∂xi∂xj)=exp(1/2)∑i=1nxi(∂f/∂xi)-fdefined on a domainΩ⊂Rn; then the graphM∇fof∇fis a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean spaceRn2nwith the indefinite metric∑dxidyi. In this paper, we prove a Bernstein theorem for complete self-shrinkers. As a corollary, we obtain if the Lagrangian graphM∇fis complete inRn2nand passes through the origin then it is flat.

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