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Ordinals and cardinals form the backbone of mathematics. Ordinal numbers describe orderings. Cardinal numbers are the measure of size in mathematics.
They first ordinals are the integers. , the set containing all integers, is the ordinal of the integers. The next ordinal is the successor of . This set has one members just as the successor of 0 has one set, the set containing the empty set. The successor of is the set containing and the successor of .
Ordinals can be thought of as general way of representing induction or iteration. corresponds to iteration that can be characterized by a loop up to any integer or induction on all the integers. contains and all finite successors to . contains all finite successors to . contains for all integers n. corresponds to two nest loops each with a limit of any integer. corresponds to loops on the integers nested n deep. corresponds to loops on the integers nested n deep where n is a parameter. All forms of iterations and all induction can be characterized by some ordinal number. By using infinite sets to describe iteration one masks over the rich combinatorial structures that are required to define higher levels of iteration. Recursive iteration is characterized by the recursive ordinals but there is no recursive algorithm to describe the structure of all recursive ordinal s although there is such an algorithm for any recursive ordinal.
Each integer is a finite cardinal number. Different integers have different sizes. is the cardinal number of all integers. No one knows what the next largest cardinal is. Cantor proved that the Cardinal of the reals is larger than the cardinal of the integers by showing that there could not exist a one to one map between the integers and reals. Gödel constructed a model for set theory in which the first cardinal larger than the integers is the cardinal of the real numbers. Gödel proved this model was consistent of ZF set theory was consistent[19,20]. Cohen proved that if ZF set theory is consistent that it is consistent to assume there exists cardinals greater than the cardinal of the integers but less than the cardinal of the reals. This question is known as the Continuum Hypothesis and is undecidable in ZF.
If there do not exist completed infinite totalities than there does not exist a single real number let alone the set of all numbers. In that case the set of all real numbers only make sense relative to a given axiom system. The continuum hypothesis is true, false or unprovable in a given formal system, but it has no absolute truth value.
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