Transfinite Induction

2019 ◽  
pp. 73-87
Author(s):  
Henryk Kotlarski
Keyword(s):  
Transfinite Induction

Correction to a definition of negation

1984 ◽  
Vol 49 (1) ◽  
pp. 47-50 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Fixed Point ◽  
Personal Communication ◽  
The Other ◽  
Transfinite Induction ◽  
Original Form ◽  
Basic Logic ◽  
Double Negation ◽  
Original Letter ◽  
The Law ◽  
Definition Of

In [3] a definition of negation was presented for the system K′ of extended basic logic [1], but it has since been shown by Peter Päppinghaus (personal communication) that this definition fails to give rise to the law of double negation as I claimed it did. The purpose of this note is to revise this defective definition in such a way that it clearly does give rise to the law of double negation, as well as to the other negation rules of K′.Although Päppinghaus's original letter to me was dated September 19, 1972, the matter has remained unresolved all this time. Only recently have I seen that there is a simple way to correct the definition. I am of course very grateful to Päppinghaus for pointing out my error in claiming to be able to derive the rule of double negation from the original form of the definition.The corrected definition will, as before, use fixed-point operators to give the effect of the required kind of transfinite induction, but this time a double transfinite induction will be used, somewhat like the double transfinite induction used in [5] to define simultaneously the theorems and antitheorems of system CΓ.

scholarly journals Transfinite methods in metric fixed-point theory

2003 ◽  
Vol 2003 (5) ◽  
pp. 311-324 ◽  
Author(s):  
W. A. Kirk
Keyword(s):  
Fixed Point ◽  
Fixed Point Theory ◽  
Nonexpansive Mappings ◽  
Point Theory ◽  
Transfinite Induction ◽  
Caristi’S Theorem ◽  
Countable Compactness ◽  
Recent Extension

This is a brief survey of the use of transfinite induction in metric fixed-point theory. Among the results discussed in some detail is the author's 1989 result on directionally nonexpansive mappings (which is somewhat sharpened), a result of Kulesza and Lim giving conditions when countable compactness implies compactness, a recent inwardness result for contractions due to Lim, and a recent extension of Caristi's theorem due to Saliga and the author. In each instance, transfinite methods seem necessary.

On The Extension Of Measure By The Method Of Borel

1956 ◽  
Vol 8 ◽  
pp. 516-523 ◽  
Author(s):  
L. LeBlanc ◽  
G. E. Fox
Keyword(s):  
Outer Measure ◽  
Transfinite Induction ◽  
Boolean Ring

Introduction. This paper concerns the problem of extending a given measure defined on a Boolean ring to a measure on the generated σ-ring. Two general methods are familiar to the literature, that of Lebesgue (outer measure) and a method proposed by Borel using transfinite induction (4, 49-134; 2, 228-238).

scholarly journals Arithmetic transfinite induction and recursive well-orderings

1985 ◽  
Vol 56 (3) ◽  
pp. 283-294 ◽  
Author(s):  
Harvey M Friedman ◽  
Andrej S̆čedrov
Keyword(s):  
Transfinite Induction

scholarly journals A system of axiomatic set theory. Part V. General set theory (continued)

1943 ◽  
Vol 8 (4) ◽  
pp. 89-106 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Transfinite Induction ◽  
Recursive Definition ◽  
Axiomatic Set Theory

We have still to consider the extension of the methods of number theory to infinite ordinals—or to transfinite numbers as they may also, as usual, be called.The means for establishing number theory are, as we know, recursive definition, complete induction, and the “principle of the least number.” The last of these applies to arbitrary ordinals as well as to finite ordinals, since every nonempty class of ordinals has a lowest element. Hence immediately results also the following generalization of complete induction, called transfinite induction: If A is a class of ordinals such that (1) ΟηA, and (2) αηA → α′ηA, and (3) for every limiting number l, (x)(xεl → xηA) → lηA, then every ordinal belongs to A.

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