Modern Probability: Competing Semantics
The case [Chap. 4] of limiting frequencies vs. degrees of belief was opened anew in the 20th century. Two major semantic developments occured within ten years around Kolmogorov’s decisive contribution [Chap. 5] to the syntax of probability theory: Von Mises  proposed a strictly frequentist interpretation, which we introduce in Sect. 6.1; and [De Finetti, 1937] offered a degree of belief interpretation [Sect. 6.3]. The semantics of the theory however turned out to be much more controversial than its syntax. Motivated in large part by some of the unresolved problems with von Mises’ approach, logicians were led to the clarification of the concept of randomness. This task was completed only after the notion of algorithmic complexity, pioneered by Kolmogorov , became available and was extended from finite strings to infinite sequences by Martin-Löf  [Sect. 6.2]; in that section, we make use of recursive function theory,the mathematical elements of which are summarized in Appendix C.
KeywordsPure State Binary String Recursive Function Infinite Sequence Iterate Logarithm
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