[FOM] Founders of set theory and natural language

Aatu Koskensilta aatu.koskensilta at xortec.fi
Thu Oct 23 03:30:21 EDT 2003

Dean Buckner wrote:

>On Friedman's claim that Cantor had no "natural language concerns", well
>yes, but Cantor was not the only founder of set theory.  My rough
>understanding of events is: Frege, Cantor,  Peano and Dedekind are all
>contenders for being founding fathers.  Russell, Zermelo & others were
I don't think this is correct. Cantor's role in founding set theory lies 
in his two discoveries,
and their subsequent development: the discovery of the transfinite 
ordinal sequence and
the discovery of a scale of cardinal numbers. While Dedekind, Frege, 
Peano and others
produced much that is relevant today, their accomplishments do not come 
anywhere near
Cantor's with regards to influence and importance to what we know 
consider set theory.

The distinction between Cantor's work and those of Frege and Peano lies 
in the fact
that while Frege did much to formalise and analyse the notion of a 
logical class, i.e.
an extension of a concept, he did not share Cantor's significant idea of 
a set as a
distinctly *mathematical* object. Peano's main contributions lie in his 
(and his followers')
showing how to render ordinary mathematics into a precise and convenient 
symbolism. Dedekind contributed greatly to the "reduction" of 
mathematical objects
to sets, but these are not in the same league as the truly fundamental 
discoveries of

>Frege is universally revered as the founder of natural language philosophy,
>as we know.  
Not where I'm coming it's not.

>The matter is complicated by Zermelo's contribution, of a nearly unflawed
>set theory in 1908.  The first part of the paper is devoted to addressing
>"Russell's antinomy", however he claims in an earlier paper that he
>discovered the antinomy earlier than Russell (and there is some evidence
>from Hilbert & Husserl to support this).
Zermelo's system, which was produced to secure his proof of 
well-ordering theorem
against the criticism - which included claims from Cambridge and from 
school that the proof implicitly fell victim to Burali-Forti's paradox - 
is strikingly
*mathematical*, i.e. directly in Cantorian tradition. If set theory if a 
discipline, as it is for Zermelo and others in Göttingen, there is no 
reason to seek
refuge from the paradoxes by formulating complex alternative explications of
logic, but simply to axiomatise the theory, i.e. find axioms which 
capture the
essential of the use set theory has been put in the hands of 
mathematicians. Zermelo
himself says this much in his 1908 article, if I recall correctly.

>Zermelo was the first teacher of "mathematical logic" in Germany.  However
>he leaves the logic of his paper implicit.   He uses a mixture of
>Peano's symbols, Frege's notion of assertion and "truth-values", plus
>(according to Volker Peckhaus) a technique of first order quantification
>derived from Schroder.  He uses the vague adjective "definite" to
>characterise a propositional function in Axiom III - Russell noticed this
>defect at once (in a letter to Jourdain 8 March 1908).  Later Weyl (Uber die
>Definitionen der mathematiscen Grundbegriffe, Math-naturw. Blatter 7, 93-5,
>1910) made it clear that "definite" means the function is constructed of a
>finite number of logical connectives, quantifiers and set-theoretic
This was by no means the last word on the subject, and there seems to be 
no reason
to exlude Fraenkel and Skolem here. And as late as in 1930's Zermelo 
includes second order quantification in his definite properties.

There's actually much to say for Zermelo's position and his refusal to 
accept the
restriction of properties to first order definable ones (from 
parameters). Certainly
the intuition behind allowing unrestricted quantification in forming 
subsets is
based on the idea that we aren't "defining" them into existence, but 
only that
however we define an extensional collection, be it by means of arbitrary 
over the whole universe or by some more predicative means, it already exists
in "extension" at the level of cumulative hierarchy at which we are applying
the axiom of separation. This is remiscent of F.P.Ramsey's 
simplification of the
ramified theory of types: the predicative/non-predicative distinction 
loses validity
if we think of the "classes" or predicative properties in a realist 
fashion: even if something is only definable with reference to the whole of
set theoretic universe (or propositions of some higher type), surely the 
is simply a means we have to resort to because of our limited 
capabilities to
pick out one of the already existing possible combinatorical subcollections.

This sort of intuitive thinking about the cumulative hierarchy (which is 
for example in Scott's axiomatisation of the cumulative hierarchy: here 
the axiom
of separation is taken as basic, not as something that is derived from 
his axioms
for the cumulative hierarchy) is close to Zermelo's idea, at least much 
than any first order variant of the axiom of separation.

>So the question is, whether Zermelo's system owes what rigour it has to the
>work of philosophers of language such as Frege and Russell?  Or not?  I
>don't know.  Finally, on the relevance of the philosophy of language to
>logic itself:
I believe the proper way to view Zermelo's system is to view it on par 
with other
axiomatisations of disciplines of mathematics, especially Hilbert's 
of geometry.

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

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

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