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## Cardinality of Fib

Theorem: The set of possible Fib objects is countable infinite.

Sketch of proof for "countable":

Every Fib object can be represented with a finite number of letters, and thus bits or numbers, and the amount of these is countable. This follows from the fact, that the number of Fib elements in a Fib object is always countable and every Fib element consists of countably many parts, which are themselves countable, there are for example only integer or rational numbers, and also the number of variables is countable.

Sketch of proof for "infinite":

All natural numbers can be represented by Fib objects. In the following a possible representation of any natural number with a Fib object is described.

A point object itself is the natural number 0 . If a Fib object is inserted in a new function element the resulting Fib object is the successor of the original Fib object. In this manner the 0 and the successor function can be reproduced in the Fib multimedia language, and all natural numbers can be represented. Since the set of natural numbers is infinite, the amount of Fib objects is also infinite.

Even every multimedia object can be represented by a countable infinite set of Fib objects, because to a Fib object, that represents a multimedia object, any Fib object can be attached with the help of a list element, as long as the multimedia object representation is not changed. For example, to a Fib object a copy of itself using a list element can be as often attached as needed, without changing the multimedia object.

Next: Any complete Fib object Up: Theoretical statements for the Previous: Power of the Fib   Contents   Index
Betti Ã–sterholz 2013-02-13