**From:** Perry E. Metzger (*perry@piermont.com*)

**Date:** Sun Apr 27 2003 - 16:46:31 MDT

**Next message:**Perry E. Metzger: "Re: Infinite universe"**Previous message:**Ben Goertzel: "RE: Infinite universe"**In reply to:**Simon Gordon: "Re: Infinite universe"**Next in thread:**Ben Goertzel: "RE: Infinite universe"**Reply:**Ben Goertzel: "RE: Infinite universe"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

Simon Gordon <sim_dizzy@yahoo.com> 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

lambda calculi.

*> 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
*

*> them.
*

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.
*

Apparently.

*> 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 perry@piermont.com

**Next message:**Perry E. Metzger: "Re: Infinite universe"**Previous message:**Ben Goertzel: "RE: Infinite universe"**In reply to:**Simon Gordon: "Re: Infinite universe"**Next in thread:**Ben Goertzel: "RE: Infinite universe"**Reply:**Ben Goertzel: "RE: Infinite universe"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

*
This archive was generated by hypermail 2.1.5
: Wed Jul 17 2013 - 04:00:42 MDT
*