A formalisation of the four statemtments in Onar's e-mail (0 - 3)
could be taken as an axiomatisation of hypersets. It is not written
in the language of hypersets and so consists of a non-self-producing
defnition of it (as does Peter Aczel's original mapping into graph
theory). For example, the fact that sets have exactly three parts
(inside, outside and boundary) is a primitve notion taken before
hyperset theory (despite the fact that 'threeness' can be formalised
within it).
This sort of "groundedness" is in some sense necessary if we are to
communicate ideas to each other. This does not mean that we can not
then conceive of that system as completely self-referential (although
I am unsure of whether this can be done _competely_ for hyperset
theory) - we use the ladder and then throw it away (just as
Wittgenstein suggests in his Tractatus).
Self-production should not be confused with mere circularity, which
is a weaker property. For example, recursive definitions are
circular in that they refer to themselves, but are usually grounded
in order to be computable (by a machine) and as such are not
self-productive.
In fact many well-established systems are circular. Traditional set
theory can be used to define the integers, but the integers are used
in defining set-theory (especially when producing the countably
infinite supply of symbols needed). Many logic systems seek to
formalise the relation of implication but themselves use a rule
exterior to the logic (if a and a->b then b) which is itself a sort
of implication.
Many self-productive systems can be considered as mappings into
non-self-productive systems (as Peter Aczel does with hyperset
theory). This does not mean that all can.
I suggest that we use the terms 'self-production' only along with an
indication of the intended interpretation (e.g. although Onar lists
a non-self-productive definition of hypersets he implies that he
intends for it to be considered from the self-productive
view-point), and also use the terms 'circularity' and 'groundedness'
(the latter borrowed from non-monotonic logics) advisedly to keep the
discussion clear. The differences are important.
----------------------------------------------------------
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