From: Simon Gordon (firstname.lastname@example.org)
Date: Sun Apr 27 2003 - 11:02:53 MDT
--- "Perry E. Metzger" <email@example.com> wrote: >
> 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 because of set-theory-envy, if set
theory has uncountable sets why cant the set of all
primitive entities be uncountable? Or even infinite to
the same extent as the set of all sets is infinite
> 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
> Of course, if you're going to bring things down to
> the level of "set"
> you start getting into a much deeper problem. "Sets"
Exactly. Sets dont interact, thats why they make such
good self-contained entities and can be thought of as
> 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: they
are decided by the anthropic principle. But in the
broader context there are "all possible set theories"
and so its just a matter of selecting which one we are
> 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. Surely the most primitive entities
are also the most numerous.
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