On the interpretation of intuitionistic number theory

Abstract

The purpose of this article is to introduce the notion of “recursive realizability.”Let P be some property of natural numbers. Consider the existential statement, “There exists a number n having the property P.” To explain the meaning which this has for a constructivist or intuitionist, it has been described as a partial judgement, or incomplete communication of a more specific statement which says that a certain given number n, or the number n obtainable by a certain given method, has the property P. The meaning of the existential statement thus resides in a reference to certain information, which it implies could be stated in detail, though the trouble is not taken to do so. Perhaps the detail is suppressed in order to convey a general view of some fact.The information to which reference is made should be thought of as possibly comprising other items besides the value of n or method for obtaining it, namely such items as may be necessary to complete the communication that that n has the property P.

Kolmogorov Complexity and Algorithmic Randomness

“…Как показывает следующая задача, эта разница мо-жет быть сравнима с самими значениями ( : ) и ( : ), если те много меньше KS ( , ). 60 Покажите, что если | слово длины , для которого KS ( | ) , то ( : ) = KS ( ) + (1), в то время как ( : ) = (1).…”

“…332 Докажите, что [60], введённым для интерпретации интуиционистского исчи-сления высказываний (IPC); сложность задачи в описанном выше смысле рассма-тривалась в [157]. Читателя, желающего больше узнать о логических исчислениях, в том числе об IPC, мы отсылаем к книге [174].…”

Lifschitz realizability for intuitionistic Zermelo–Fraenkel set theory

“…Realizability semantics for intuitionistic theories were first proposed by Kleene in 1945 [12]. Inspired by Kreisel's and Troelstra's [14] definition of realizability for higher order Heyting arithmetic, realizability was first applied to systems of set theory by Myhill [18] and Friedman [9].…”

Gödel's Functional Interpretation and Its Use in Current Mathematics

“…Making this precise results in various so-called realizability interpretations. The most important ones are Kleene's realizability ( [54]) and Kreisel's modified realizability ( [78,79]). Let us sketch now the main differences between these two interpretations:…”