The point of ur-elements


(This post is part of the campaign to save logic in Usenet. You can
contribute by posting sense).

Zuhair here has proposed a formulation of set theory, amounting
essentially to postulating a finite supply of ur-elements and
stipulating that only finite sets obtained by the "set-of" operation
from these exist. What the point to all these theories of zuhair's is
I haven't the foggiest -- he seems to find it somehow fruitful to spew
an endless stream of alternative axiomatisations with no motivation
--, but let's offer some general comments on ur-elements.

Zuhair's stated motivation to his theory was to allow set-theoretic
reasoning about cows, animals, people, and so on. Similarly, in his
_Set Theory and Its Philosophy_ Michael Potter officially adopts
ur-elements, in order for set theory to be applicable to all sorts of
non-mathematical stuff, like physical objects and such like. Indeed,
this is not an uncommon idea; yet, I wonder, what's the point of all
this? We practically never apply any abstract set theoretic reasoning
to cows, apples, oranges, or whatnot. The only instances I can think
of our applying set theoretic reasoning -- in any substantial sense --
to anything outside mathematics is when we're doing, say, theoretical
physics. It is then a pertinent observation that in theoretical
physics we apply, to whatever extent we actually do, set theory to
highly abstract mathematical structures, thought to model some aspect
of physical reality in some way. These structures, in turn, have their
faithful isomorphic counterparts in the hierarchy of pure sets; no
need for ur-elements, then. So to what purpose could we use a
formulation of set theory with ur-elements? The only sensible
motivation seems to be finding out which theorems of set theory depend
or do not depend on the existence of ur-elements, a fairly trivial
task.

This tirade against ur-elements is connected to my now somewhat tired
mantra, that in evaluating a formal theory we must use our good sense
and best judgement to decide whether it adequately captures those
aspects of our reasoning we consider relevant for whatever it is we're
doing. So, the question becomes: what aspect of our reasoning does
allowing ur-elements in stupendously abstract theories of sets
capture, if any?

Now, there is a more practical motivation for ur-elements: modelling
actual mathematical reasoning. As Harvey Friedman, and countless
others, like Alan Turing, have noted, mathematics is, in some sense,
"typed", that is, we find in mathematics not pure sets, but sets of
naturals, graphs, functions on graphs, reals, sets of reals, and so
on. It is thus natural to formulate mathematical arguments, in so far
as we're concerned with formalising /proofs/, in a context where we
have, as "ur-elements", natural mathematical objects, such as the
naturals, the reals, etc. This motivation is wholly unconnected to any
ideas of allowing ur-elements so we can reason about cows, apples, or
electrons.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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