hyperset reference

Onar Aam (onar@HSR.NO)
Thu, 8 Jun 1995 07:19:40 +0100


"Many mathematician's were uncomfortable with [Russell's] rule, since
it was not 'natural', it was simply appended onto the list of rules
in order to avoid contradiction. And this discomfort became
especially accute after Godel came out with his Incompleteness
Theorem. For Russell's Paradox is essentially a variation on the old
Paradox of Epiminides the Cretan: 'This sentence is false.' But
Godel's theorem is an even clearer variation of the same paradox."

"A _labeled graph_ is defined as any collection of dots with a symbol
drawn next to each dot, and arrows drawn between the dots. Aczel's
'AFA Axiom' implies that every finite graph corresponds to some set."

"According to Godel's Theorem, one can never mathematically prove
that a complicated mathematical theory is _consistent_, devoid of
self-contradictions. But Aczel has shown that, if there are
constradictions in hyperset theory, then there are also
contradictions in plain ordinary mathematics, the kind that
scientists use to make calculations. This is as good a consistence
result as one could hope for. One may confidently say: there are
mathematical objects that contain one another as elements."

Onar.