Re: self-producing

Bruce Edmonds (B.Edmonds@MMU.AC.UK)
Tue, 22 Aug 1995 08:54:10 GMT

> Bruce writes:
> > Your choice of formal tool does not effect the answers
> > to big philosophical questions.
>
> Would this mean that some philosophical questions cannot be answered from
> the perspective of formal systems at all?
>
> I have a couple of related questions: Conventional set theory, as I
> understand it, can lead to "logical paradox" (as exposed in Godels theorem).
> The hyper-set formalism, as I understand it, (and correct me if Im wrong)
> starts with accepting this "paradox" as not being a problem (it claims that
> that sets can be members of themselves), and then proceeds to establish
> theorems. In the special case where the sets arent members of themselves,
> hyperset theory degenerates into conventional set theory. Now, wouldn't
> hyperset theory be "bigger" than conventional set theory and therefor
> applicable to more philosophical questions (than conventional set theory)?

Conventional set theory does not lead to a real logical paradox, (as
far as anyone knows, we don't know it doesn't either). Early
versions of it were shown to be inconsistent by version of Russel's
paradox (and others), some of which involved very strange sets (like
the 'set' of all sets that aren't members of themselves). This was
fixed in several alternative ways, the set theory way was to reduce
the power of the axiom of comphrension, which allowed the
construction of sets from statements (i.e. if you had a statement
P(x) whether you can construct a set of all the objects that satisfy
this; i.e. is {x | P(x)} a set or not). This did not rule out
hypersets, which were members of themselves. It was possible that
there were sets like A={A}, but you could never prove that they
existed - hence you could never use them as a basis for a proof or
construction.

Later the axiom of foundation was added, which did rule out
hypersets. This was not added to avoid paradox, but to add more
structure to the universe of sets (it also made many proofs much
easier).

Peter Aczel showed that a system hypersets could be consistently
built by mapping hypersets into graph theory, so in that system sets
like A={A} are allowed. Graph theory can be formalised in standard
(ZF) set theory, and thus hyperset theory can as well, via Aczel's
mapping.

Just becuase one formal system seems immediately more expressive, it
does not mean that using some clever encoding another, apparently
more expressive system, can not be simulated by it (just imagine a
simple microprocessor, using an interpreter/compiler simmulating what
seems to be a much more expressive language, like smalltalk).

Many systems can do this for each other. For example trying to tile
the plane or solving equations for whole number solutions are both
equivalent to woking out whether a software program will ever
actually finish. One more example: simple arithmetic (PA) can
reduce (via Godel numbers) questions of proof about full ZF set
theory to arithmetic questions!

----------------------------------------------------------
Bruce Edmonds
Centre for Policy Modelling,
Manchester Metropolitan University, Aytoun Building,
Aytoun Street, Manchester M1 3GH. UK.
e-mail b.edmonds@mmu.ac.uk
Tel no. +44 161 247 6479 Fax no +44 161 247 6802
WWW. http://bruce.edmonds.name/bme_home.html