From: Perry E. Metzger (email@example.com)
Date: Sun Apr 27 2003 - 16:46:31 MDT
Simon Gordon <firstname.lastname@example.org> writes:
> > All formal systems are expressable as finite strings
> > of symbols (by
> > definition), and thus trivially mappable to
> > integers. QED.
> Thanks Perry, thats all i need to know. Clearly i want
> my systems to be a little more generic than just
> formal systems
Well, sadly the Church-Turing Thesis has yet to be disproven, so I'm
afraid that for now your wants may remain unfulfilled.
> because of set-theory-envy, if set
> theory has uncountable sets why cant the set of all
> primitive entities be uncountable?
Why? Because mathematics isn't infinitely mutable. We have this notion
of proof, which is a means of going from a set of axioms to a theorem
via a sequence of finitely many productions of an unrestricted
grammar. That's the core of our idea of what mathematics means, and
set theory is easily contained within that idea. The number of axioms
for simple forms of set theory is pretty low.
By the way, we've also done a lot of work for about the last century
tightening up what we know about formal systems, proof and computation
-- and they're all inextricably linked. So far, we haven't found
anything more potent than our formal systems, Turing machines and
> Or even infinite to
> the same extent as the set of all sets is infinite
What is the "set of all sets"? Many kinds of set theory can't express
such a thing. Again, you keep throwing out ideas about sets as though
they somehow had a life outside a particular flavor of set
theory. They don't, any more than the integers have a life outside of
theories like Peano's axioms.
> > It is literally the set of all formal systems,
> > because it is the set
> > of all consistent mathematical systems.
> I consider formal systems to be a subset of the set of
> consistent mathematical systems but you have equated
What other sort of mathematics is there outside of formal systems?
> > Of course, if you're going to bring things down to
> > the level of "set"
> > you start getting into a much deeper problem. "Sets"
> > don't
> > interact.
> Exactly. Sets dont interact,
And as a result, in Tegmark's sense, they aren't capable of supporting
self aware substructures, so there isn't an operational way of
assessing their reality...
> > And if you were willing to accept that we aren't in
> > a universe and
> > move on to "sets" as your fundamental universes,
> > what sort of set
> > theory are you going to adopt as "fundamental"?
> > After all, in some set
> > theories, you get the axiom of choice, in some you
> > don't. In some, the
> > continuum hypothesis is an axiom, in some it isn't.
> > The list goes on
> > and on.
> Axioms are not something i worry about too much:
Could you tell me a bit about your background? Do you know much about
set theory or metamathematics to begin with? If not, perhaps it would
be more profitable to have this discussion after you've read a bit
more on notions like formal systems and set theory...
> > Sets, you see, aren't as primitive as formal systems
> > at all, even
> > though they are much less powerful, just as integers
> > aren't as
> > primitive as formal systems although they are much
> > less powerful.
> I dont get this.
> If sets are a subset of the set of
> all formal systems why are there more sets than there
> are formal systems.
I said that set theory is a subset of the set of formal systems, not
that sets are a subset of the set of all formal systems. As for sets
themselves, well, there are far more entities talked about by formal
systems than there are formal systems -- just the reals outnumber the
formal systems even though a fairly simple formal system is all you
need to capture the reals.
I think you need to get a better background in the area before having
this discussion -- it will make it less frustrating for both sides.
> Surely the most primitive entities are also the most numerous.
Nope. Not at all, for the same reason that the Chaitin-Kolmogrov
information of the set of all integers is tiny compared to that of
almost all individual integers.
-- Perry E. Metzger email@example.com
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