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... consistent1
Technically Gödel's result is more general because it applies to formal systems that have an infinite number of axioms and thus cannot have their behavior modeled by a finite computer program. However, for formal systems that finite mathematicians can write down and use as opposed to philosophize about, the results are equivalent.
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... properties2
Set theory uses the concept of an ordinal to characterize all levels of induction and does not explicitly construct a hierarchy of levels of abstraction. The power of a system is determined by the strength of the ordinals defined in the system. Principia Mathematica[16] developed mathematics by explicitly giving a hierarchy of levels of abstraction. These are each equivalent to an ordinal in set theory. Set theory implicitly defines much larger ordinals than those derivable from the explicit hierarchy in Principia. It accomplishes this on a single page in contrast to the three large volumes of Principia. However I suspect that at some point we will only be able to extend mathematics by understanding the explicit hierarchy of types implicitly defined in set theory. With the aid of computers that project is vastly easier than at the time of Principia.
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Mountain Math Software
home consulting videos book QM FAQ contact
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