PHYSICAL CONSTRAINTS OF NUMBERS

independent of all of them, but only based upon logic. This conservative concept, however, needs to be revised because, contrary to Platonic idealism (frequently called "realism " by mathematicians), mathematical ideas, notions, and, in particular, numbers are not at all independent of physical laws and prerequisites. ARE THERE ALL THE NATURAL NUMBERS? Though this question seems to be too absurd to dwell upon, we will attempt it. The answer to the question obviously depends on the meaning of the notions "natural number " and "there is". The natural numbers are defined by a set of axioms as given by G. Peano or by E. Schmidt, the most important ingredient of which is the principle of induction: If n is a natural number, then n + 1 is a natural number too. The smallest sequence 1, 2, 3,... which contains the unit 1 and obeys this principle is called "the sequence of natural numbers". Set theory gives a meaning to "there is " by the axiom of infinity. There is a set M tha