Kolmogorov's extension theorem for infinite measures

In this paper we introduce a new notion of realizability for intuitionistic arithmetic in all finite types. The notion seems to us to capture some of the intuition underlying both the recursive realizability of Kjeene [5] and the semantics of Kripke [7]. After some preliminaries of a syntactic and recursion-theoretic character in §1, we motivate and define our notion of realizability in §2. In §3 we prove a soundness theorem, and in §4 we apply that theorem to obtain new information about provability in some extensions of intuitionistic arithmetic in all finite types. In §5 we consider a special case of our general notion and prove a kind of reflection theorem for it. Finally, in §6, we consider a formalized version of our realizability notion and use it to give a new proof of the conservative extension theorem discussed in Goodman and Myhill [4] and proved in our [3]. (Apparently, a form of this result is also proved in Mine [13]. We have not seen this paper, but are relying on [12].) As a corollary, we obtain the following somewhat strengthened result: Let Σ be any extension of first-order intuitionistic arithmetic (HA) formalized in the language of HA. Let Σω be the theory obtained from Σ by adding functionals of finite type with intuitionistic logic, intensional identity, and axioms of choice and dependent choice at all types. Then Σω is a conservative extension of Σ. An interesting example of this theorem is obtained by taking Σ to be classical first-order arithmetic.

Download Full-text

The Tietze Extension Theorem and the Open Mapping Theorem

American Mathematical Monthly ◽

10.1080/00029890.1986.11971783 ◽

1986 ◽

Vol 93 (3) ◽

pp. 190-191 ◽

Cited By ~ 2

Author(s):

Sandy Grabiner

Keyword(s):

Extension Theorem ◽

Open Mapping ◽

Open Mapping Theorem

Download Full-text

Hopf Extension Theorem of Measure

Formalized Mathematics ◽

10.2478/v10037-009-0018-6 ◽

2009 ◽

Vol 17 (2) ◽

Author(s):

Noboru Endou ◽

Hiroyuki Okazaki ◽

Yasunari Shidama

Keyword(s):

Extension Theorem

Download Full-text

A biregular extension theorem from a fractal surface in

Georgian Mathematical Journal ◽

10.1515/gmj.2011.0004 ◽

2011 ◽

Vol 18 (1) ◽

pp. 21-29

Author(s):

Ricardo Abreu Blaya ◽

Juan Bory Reyes ◽

Tania Moreno García

Keyword(s):

Continuous Function ◽

Bounded Domain ◽

Extension Theorem ◽

Fractal Surface ◽

Fractal Boundary ◽

Hölder Continuous

Abstract The aim of this paper is to prove the characterization on a bounded domain of with fractal boundary and a Hölder continuous function on the boundary guaranteeing the biregular extendability of the later function throughout the domain.

Download Full-text

A SUFFICIENT AND NECESSARY CONDITION FOR CREDIBILITY MEASURES

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488506004175 ◽

2006 ◽

Vol 14 (05) ◽

pp. 527-535 ◽

Cited By ~ 124

Author(s):

XIANG LI ◽

BAODING LIU

Keyword(s):

Set Theory ◽

Fuzzy Set ◽

Fuzzy Set Theory ◽

Extension Theorem ◽

The Self ◽

Necessary Condition ◽

Self Duality ◽

Duality Property ◽

Sufficient And Necessary Condition

Possibility measures and credibility measures are widely used in fuzzy set theory. Compared with possibility measures, the advantage of credibility measures is the self-duality property. This paper gives a relation between possibility measures and credibility measures, and proves a sufficient and necessary condition for credibility measures. Finally, the credibility extension theorem is shown.

Download Full-text