The use of 0123456789 by Brouwer and Heyting

For completeness we list three of the `classical' examples:

1. The sequence is a Cauchy sequence. Let the sequence be defined as follows: If the nth digit after the decimal point in the decimal expansion of is the 9 of the first sequence 0123456789 in this expansion, , in every other case . b differs from a in at most one term, so b is classically a Cauchy sequence, but as long as we do not know whether such a sequence 0123456789 occurs in , we are not able to find n such that for every p; we have no right to assert that b is a Cauchy sequence in our sense. ([5], page 16.)

2. A proof of the impossibility of the impossibility of a property is not in every case a proof of the property itself. It will be instructive to illustrate this by an example. [L. E. J. Brouwer 1925, p. 252] ([4]). I write the decimal expansion of and under it the decimal fraction , which I break off as soon as a sequence of digits 0123456789 has appeared in . If the 9 of the first sequence 0123456789 in is the kth digit after the decimal point, . Now suppose that could not be rational; then would be impossible and no sequence could appear in ; but then , which is also impossible. The assumption that cannot be rational has lead to a contradiction; yet we have no right to assert that is rational, for this would mean that we could calculate integers p and q so that ; this evidently requires that we can either indicate a sequence 0123456789 or demonstrate that no such sequence can occur. ([5], page 17.)

3. However, many other classical theorems are no longer valid. I state an example that a bounded monotone sequence need not be convergent. A simple counterexample is the sequence which is defined as follows: if among the first n digits in the decimal expansion of no sequence 0123456789 occurs; if among these n digits such a sequence does occur. Nobody knows if the limit of this sequence, if it exists, will be 1 or 2; so we are not allowed to say that this limit exists as a well defined real number generator. ([5], page 31.)

The sequence predictably continues to occur, next at the 26,852,899,245th place.

We note also that 09876543210 begins at the 42,321,758,803th digit of and that the digits ending at 50 billion are

At any rate, sometime in July 1997, the three examples above began to behave well. Needless-to-say, the strength of the intuitionist argument is in no way diluted by the destruction of these specific examples.