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Moving from the finite to the infinite without due care -----------------------------------------------
Some paradoxes of set theory arise when reasoning valid for finite sets is extended to infinite ones. For example, the axiom "the whole is greater than the part", one of the classic self-evident truths of logic, no longer holds for infinite collections: there are "as many" even numbers or prime numbers as there are integers, although there are "more" integers than even numbers (twice as many) or prime numbers (in this case, the exact proportion is unknown). Galileo pondered the paradox involving the infinity of the integers and the even numbers: every integer can be paired with an even number, and yet there are twice as many integers as even numbers…
This paradox can be expressed in terms of information content: in an ordinary object, the whole contains more information than parts of unequal cardinality; in a nested object, it contains as much information as one or more of its parts, which act as attractors; in a fractal object, the whole contains as much information as any one of its parts.