Borel and the Intrinsic Properties of the Reals

Foundations of Mathematics#

Richard's paradox. This is similar to Cantor's diagonal argument for the nondenumerable infinity of the reals. Consider all real numbers of the unit interval definable by a finite number of words, these definitions can be organized into a sequence according to length and within the same length alphabetically. Define a new number by putting 1 if the \(n\)th binary digit of the \(n\)th number is 0 (and 0 for 1). Thus we defined a real number of the unit interval in a finite number of words that is different from each and every single real number of the unit interval definable by a finite number of words.

Two routes out. Bite the bullet and accept some contradiction, or adopt a constructivist resolution, e.g., Borel's: He holds firmly that the only mathematical objects we might ever encounter are all definable in a finite number of words, but we are not able to decide in general whether something is a definition of a real number in finite words. (We may make the suggested definition depend on the solution of an unsolved mathematical problem.)

Remark. Consider also the Halting problem. Is there a Turing machine that determines whether a Turing machine \(T\) halts on a given input \(I\)? Suppose there is one, say, \(K\), that outputs \(1\) if \((T,I)\) halts, and \(0\) if it doesn't. Then design a Turing machine \(P\) that outputs \(0\) for input \(I\) if \(K(I,I)\) outputs \(0\) and loop forever otherwise. Now consider the possible output of \(K(P,P)\); if \((P,P)\) halts, \(K(P,P)\) should output \(1\), which means \((P,P)\) should loop forever, contradiction.

In a suggestive terminology, Borel says there is a distinction between denumerable infinity in the classical sense, and effective enumerability. But he didn't create any precise theory of what it means for a procedure to be effective. Borel developed some insights into the concepts of computable analysis,. He saya that a representation of a real number as a decimal, in contrat to its rational approximation, is not theoretically important, since that particular representation is not “invariant” under arithmetic operations. Later it was noticed the argument in Richard's paradox assumes real numbers to have decimal expansions, but Brouwer noticed that this would not always be the case since one can, say, calculate arbitrarily long an expansion \(a=0.000…\) without knowing  whether some time a nonzero digit appears, and for numbers like \(0.5-a\) one cannot even determine the first decimal before \(a\) is expanded into infinity. Borel then defines a real function as calculable if its value is calculable for a calculable argument, where calculability means one knows how to obtain arbitrarily good approximations to the function value. Therefore calculable functions have to be continuous.

Calculable real numbers are too meager for many purposes. Borel considered widening his criteria of mathematical definability, since the sequences describing results from random experiments needs to be put into the continuum; the classical continuum contains them, but that's unavailable for Borel. Then he says that

a noncalculable value can only be conceived as defined by chance; the properties of the function are represented by coefficients of probability.

Notice that infinities transcending the denumerable seem like “purely negative notions”, and also randomness is also purely negative.

Philosophy of Probabilities#

Borel's strongly empiricist philosophy is untouched by the possible existence of objective chance. It says instead that there is indeterminism whenever Nature's actions are unpredictable. Indeterminism is aliased to unpredictability. Studies of Borel's never denied the theoretical possibility of determinism, they denied its scientific meaningfulness instead. It was still in the style of the times that the idea of spontaneous chance in nature did not occur to him at this stage.