- 1
- 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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- 2
- 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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