Jules Richard - Reactions To Richard's Paradox

Reactions To Richard's Paradox

Georg Cantor wrote in a letter to David Hilbert:

• "Infinite definitions" (i.e., definitions which cannot be done in finite time) are absurdities. If Königs statement was "correct", according to which all "finitely definable" real numbers form a collection of cardinal number, this would imply the countability of the whole continuum; but this is obviously wrong. The question is now what error the alleged proof of his wrong theorem is based upon. The error (which also appears in the note of a Mr. Richard in the last issue of the Acta mathematica, which Mr. Poincaré emphasizes in the last issue of the Revue de Métaphysique et de Morale) is, in my opinion, the following: It is assumed that the system {B} of notions B, which have to be used for the definition of individual numbers, is at most countably infinite. This assumption "must be in error" because otherwise we would have the wrong theorem: "the continuum of numbers has cardinality ".

Here Cantor is in error. Today we know that there are uncountably many real numbers without the possibility of a finite definition.

Ernst Zermelo comments Richard's argument:

• The notion "finitely definable" is not an absolute one but a relative one being always related to the "language" chosen. The conclusion according to which all finitely definable objects are countable is only valid in case that one and the same system of symbols is used; the question whether a single individual can be subject to a finite definition is void because to every thing an arbitrary name can be attached to.

Zermelo points to the reason why Richard's paradox fails. His last statement, however, is impossible to satisfy. A real number with infinitely many digits, which are not determined by some "rule", has an infinitely large contents of information. Such a number could only be identified by a short name if there were only one or few of them existing. If there exist uncountably many, as is the case, an identification is impossible.