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The Sources of Kolmogorov’s Grundbegriffe
, 2006
"... Andrei Kolmogorov’s Grundbegriffe Wahrscheinlichkeitsrechnung put probability’s modern mathematical formalism in place. It also provided a philosophy of probability—an explanation of how the formalism can be connected to the world of experience. In this article, we examine the sources of these two a ..."
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Andrei Kolmogorov’s Grundbegriffe Wahrscheinlichkeitsrechnung put probability’s modern mathematical formalism in place. It also provided a philosophy of probability—an explanation of how the formalism can be connected to the world of experience. In this article, we examine the sources of these two aspects of the Grundbegriffe—the work of the earlier scholars whose ideas Kolmogorov synthesized.
Choice principles in constructive and classical set theories
 POHLERS (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2002
, 2002
"... The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models ..."
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Cited by 9 (3 self)
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The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models.
From the axiom of choice to choice sequences
 Nordic Journal of Philosophical Logic
, 1996
"... The theory of choice sequences is usually considered to be far from the mainstream of mathematics. In this note we show that it did not start that way. There is a continuous development from discussions around the use of axiom of choice to Brouwer’s introduction of choice sequences. We have tried to ..."
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The theory of choice sequences is usually considered to be far from the mainstream of mathematics. In this note we show that it did not start that way. There is a continuous development from discussions around the use of axiom of choice to Brouwer’s introduction of choice sequences. We have tried to trace this development starting in 1904 and ending in 1914.1 In his book on choice sequences, Troelstra (1977) gives the development after 1914, but does not indicate where Brouwer got his concept. This note is a rst attempt at an answer. Our story starts in August 1904, with Zermelo writing a long letter to Hilbert, who thinks part of the letter deserves a wider audience. So he publishes it directly in Mathematische Annalen (Zermelo 1904). The leisurely style is clear from the title, \Proof that every set can be wellordered, (from a letter sent to Mr. Hilbert)", and the rst sentence:... The following proof comes from conversations that I had last week with Mr. Erhard Schmidt and it is as follows. Zermelo gave the standard argument that the axiom of choice implies the wellordering principle. He argued that the axiom of choice was selfevident. The reactions to the proof came immediately. At the end of a note sent to Mathematische Annalen in December 1904, Borel states: This paper has circulated as an unpublished note since 1979. It is referred to in A. S. Troelstra (1982) On the origin and development of Brouwers’s concept of
c ○ Institute of Mathematical Statistics, 2006 The Sources of Kolmogorov’s
, 2006
"... put probability’s modern mathematical formalism in place. It also provided a philosophy of probability—an explanation of how the formalism can be connected to the world of experience. In this article, we examine the sources of these two aspects of the Grundbegriffe—the work of the earlier scholars w ..."
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put probability’s modern mathematical formalism in place. It also provided a philosophy of probability—an explanation of how the formalism can be connected to the world of experience. In this article, we examine the sources of these two aspects of the Grundbegriffe—the work of the earlier scholars whose ideas Kolmogorov synthesized. Key words and phrases: Axioms for probability, Borel, classical
DETERMINISM VERSUS ALEATORISM
"... Objetivist models are based on the deterministic hypothesis that postulates the existence of probability, which is cognoscible only in an asymptotic manner. On the other hand, subjectivist models consider the aleatoristic hypothesis according to which there is no truth about probability. However, bo ..."
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Objetivist models are based on the deterministic hypothesis that postulates the existence of probability, which is cognoscible only in an asymptotic manner. On the other hand, subjectivist models consider the aleatoristic hypothesis according to which there is no truth about probability. However, both hypotheses may only be compared through stochastic models, which are not strictly falsifiable. Therefore, neither the hypothesis stating the existence of a true value regarding the probability of occurrence of an event nor de Finetti´s postulate which sustains that “probability does not exist ” are strictly verifiable. 1. An introduction to the theory of chance The conceptualization of chance arose associated with the idea of lack of sufficient information about the causal structure that supposedly determines the behaviour of factual phenomena: the observer has a piece of informationwhether it is objective in nature or consists of the knowledge of multiple characteristics data and origins that constitute their personal experience (subjective) on this phenomenonthat is incomplete