Re: Why an inconsistent ZF may be desirable, and should be welcome.

From: Jim Spriggs (jim.sprigs_at_ANTISPAMbtinternet.com.invalid)
Date: 03/07/05 

Date: Mon, 7 Mar 2005 17:10:04 +0000 (UTC)

Babylonian wrote:
>
> Bhupinder Singh Anand wrote:
>
> > 3. ZF can be interpreted intuitively in any such language, L - in the
> > sense that the axioms of ZF interpret as intuitively true in any L,
> and
> > the rules of inference preserve intuitive truth in L.
> >
>
> Good grief. Nobody calls axioms "intuitively true" anymore. They are
> valid for a model, or not. There are other attractive systems besides
> ZF, depending on your purpose or outlook.

The word "axiom" has more than one, and certainly at least two,
meanings. There is what one might call the Euclidean meaning which one
can best make sense of by pretending to be a Platonist. Let us suppose
that there are "real" sets "out there". It makes perfect sense to ask:
what is true of them? Wary of the paradoxes, we lay down some axioms
and make deductions from them. Since we hope that our theorems are true
statements about real sets, it certainly makes sense to start with
intuitively true axioms. Unfortunately our intuitions don't tell us
much about, say, the power set of infinite sets.

Then there is the modern (19th century) view of axioms that I hesitate
to attach a person's name to [1]. This is the view that a set of axioms
serves to define a class of structures, i.e. those structures that are
models of the axioms. The axioms for groups are an example. Groups
were studied, by Lagrange and others, before the modern concept of group
was defined, as concrete groups of permutations of finite sets. As more
general groups were studied the axioms emerged [2] not as intuitive
truths about something existing "out there" but as a means of defining a
certain class of structures.

The matter is confounded by there being more than one meaning of "set".
Putting on our Platonist hat there are the real sets. But those axioms,
mentioned in my first paragraph, define a class of structures <V,
epsilon> such that the members (in the informal sense of that word) of V
are sets. In one (presumably) of those structures the sets (members of
V) are "real" sets. But, as Cohen showed us, there is more than one
structure <V, epsilon> satisfying those axioms.

As long as our theories are first order there will be unintended
interpretations [3]. The intuitively true axioms need to pick out just
the intended one. Presumably if one isn't a Platonist, there isn't an
intended one; there is just the class of interpretations each of which
may, or may not, be of equal interest.

[1]: Boole? who recognized more than one interpretation of Boolean
algebra.

[2]: Dyck 1883 "Gruppentheoretische Studien" Math. Ann.? Roughly 100
years after Lagrange 1771 "R\'eflexions sur la r\'esolution alg\'ebrique
des \'equations".

[3]: Not forgetting that even in the higher order case there can be
unintended interpretations as well. (A result due to Henkin? Which
gives us non-standard analysis for example.)

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