Because of the finiteness of our world, there are certain difficulties to treat the infinite. One difficulty is the question whether one shall define the number of elements of the set of algebraic numbers finite or infinite. In the infinite case, one obtains as complement finitely transcendental numbers, in the finite case transcendental numbers. It is also shown that the set of natural, integer, rational, algebraic, real and complex numbers is not closed. Hence, the difference of (finitely) algebraic numbers is no longer necessarily (finitely) algebraic, what complicates the theory of the transcendental numbers.

A trans-rational and transcendental number can also consist of a finite continued fraction, in which the last partial denominator is infinitely large. Would it be identified with a rational number by setting the last partial fraction equal to zero, it would simultaneously solve a linear equation with integer as with trans-integer coefficients. This identification leads to contradictions, if the first equation is subtracted several times from the second one and the solution of every newly emerged equation is determined, and is therefore incorrect. Correct is working with approximate fractions.

One can show by (trans-natural) induction that one, starting with the set of natural numbers, can diagonalise up to any power, so that all (!) infinite sets are equipotent to the set of natural numbers, if one uses Hilbert's translations as an aid. This contradictoriness is met with the proposition that there is for no set a bijection to its proper subset. Hence, Dedekind-infinity and Hilbert's hotel are destroyed, since the image sets of translations of every set lead out of this.

Every number of elements of a set can be specified by reference to the set of natural numbers. Only from the exact construction instruction it can be specified precisely. Thus, it makes an important difference if one treats all multiples of five and constructs thereby the associated set in such a way that one times each natural number with five or that one deletes all numbers, up to each fifth, from the set of the natural numbers. Cantor regarded such sets as of the same cardinal number. If one considers, however, bijections correctly, one receives on the one hand the number of the set of the natural numbers or on the other hand a fifth of it.

Cantor's distinction between merely countable and uncountable sets is too undifferentiated. The correct treatment of bijections yields the statement that, concerning the number, there are between infinite many sets between the set of natural numbers and the one of real numbers. Thus, the continuum hypothesis gets a new answer. Furthermore, the asymptotic function of the number of algebraic numbers is determined.

Below transcendental numbers necessary and sufficient criteria for (finite) transcendence are stated. There are some numbers determined as (finitely) transcendental from that one previously did not know this. (Finitely) transcendental numbers are essentially the sum of a (finitely) algebraic and a sufficiently small or sufficiently large number. The transcendence of a number can be furthermore characterised via an infinite product of complex-rational numbers and an algebraic number. (Finitely) algebraic numbers, however, are determined by the zeros of polynomials with integer or rational coefficients. From this, criteria for (finite) transcendence can be directly derived from the coefficients of the polynomials.

© 06.09.2009 by Boris Haase

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