>O.K. but where does that leave us? Is the claim that hyperset
>theory is "enough" (resolves the paradox, as Onar points out) or
>is it simply another, alternative for us to use along with others?
>Forgive me for being dense, but it is not clear to me that hyperset
>theory and "ordinary" set theory (unextended) can be used simultaneously
>with out glaring contradictions. Am I missing something?
>Here's where I see the multiple formal system idea and complexity
>coming together. If we had just one universal formalism, nothing would seem
>complex anymore.
Which is probably why we will never see one universal formalism. Hyperset theory
and ordinary set theory are not in contradiction as I see it. And if they are it
is surely possible to model it consistently in hyperset theory! In many ways
Hyperset theory is a theory of paradoxes. Hyperset theory sees paradoxes as real
and consistent mathematical objects. It's like it is saying: "sure there are
paradoxes, but these paradoxes behave in a completely consistent mathematical
manner." And that is in fact what Peter Aczel has proved, that paradoxes ARE
mathematical objects that are just as real and consistent as 2+2=4. Let me give
a concrete example, namely the classical paradox "this statement is false". This
reduces to a simple hyperfunction:
A={not A}
This is by definition a paradox, right? But let us see how the paradox unfolds:
A, not A, A, not A, A, not A, A, not A, A, not A, A, not A, A...
In other words, the paradox creates a completely predictable and consistent
oscillating pattern. Similarly, every constructable paradox have a unique such
pattern. So what hyperset theory really does is to formalize these patterns. In
a sense it views the paradox as an EMERGENT PHENOMENON in mathematics rather
than a black hole. Paradoxes are particular mathematical objects. They even have
a geometrical interpretation. We may view paradoxes as knots or tori in the
formal systems. Not only are they consistent, they are beautiful and
marvelling as well.
Onar.