Date: Dec 8, 2012 9:06 PM
Author: fom
Subject: Re: fom - 01 - preface

On 12/8/2012 1:49 PM, WM wrote:
> On 8 Dez., 19:16, fom <> wrote:
>> On 12/8/2012 9:08 AM, WM wrote:
>> There are certain ongoing investigations
>> into the structure of mathematical proofs
>> that interpret the linguistic usage differently
>> from "mathematical logic". You would be
>> looking for various discussions of
>> context-dependent quantification where it
>> is being related to mathematical usage.
>> You will find that a statment such as
>> "Fix x"
>> followed by
>> "Let y be chosen distinct from x"
>> is interpreted relative to two
>> different domains of discourse.
>> This is just how one would imagine
>> traversing from the bottom of a
>> partition lattice.

> A question: Do you believe that there are more than countably many
> finite words?
> Do you believe that you can use infinite words (not finite
> descriptions of infinite sequences).
> Do you believe that you can put in order what you cannot distinguish?
> Regards, WM

There is a certain history here.

As set theory developed, Cantor was confronted
with the notion of "absolute infinity".

I prefer to go with Kant:

"Infinity is plurality without unity"

and interpret the objects spoken of in typical
discussions of set theory as transfinite numbers.

As for "unity", Cantor wrote the
following in his criticism of Frege:

" take 'the extension of a concept' as the
foundation of the number-concept. He [Frege]
overlooks the fact that in general the 'extension
of a concept' is something quantitatively completely
undetermined. Only in certain cases is the 'extension
of a concept' quantitatively determined, then
it certainly has, if it is finite, a definite
natural number, and if infinite, a definite power.
For such quantitative determination of the
'extension of a concept' the concepts 'number'
and 'power' must previously be already given
from somewhere else, and it is a reversal of
the proper order when one undertakes to base
the latter concepts on the concept 'extension
of a concept'."

Cantor's transfinite sequences begin by simply
making precise the natural language references
to the natural numbers as a definite whole. And,
he justifies his acceptance of the transfinite
with remarks such as:

"... the potential infinite is only an
auxiliary or relative (or relational)
concept, and always indicates an underlying
transfinite without which it can neither
be nor be thought."

But the question of existence speaks precisely
to the first edition of "Principia Mathematica"
by Russell & Whitehead. I would love to have
the time to revisit what has been done there.

Russell's first version had been guided in
large part by his views on denotation. So,
the presupposition failure inherent to reference
was to be addressed by his description theory.
Given that, he ultimately would be committed
to the axiom of reducibility.

It is interesting to read what he says
concerning that axiom and set existence,

"The axiom of reducibility is even
more essential in the theory of
classes. It should be observed,
in the first place, that if we assume
the existence of classes, the axiom
of reducibility can be proved. For in
that case, given any function phi..z^
of whatever order, there is a class A
consisting of just those objects which
satisfy phi..z^. Hence, "phi(x)" is
equivalent to "x belongs to A". But,
"x belongs to A" is a statement containing
no apparent variable, and is therefore
a predicative function of x. Hence, if
we assume the existence of classes, the
axiom of reducibility becomes unnecessary."

Personally, I do not think he should
have given it up.

As for my personal beliefs, I reject, for
the most part, the ontological presuppositions
of modern logicians so far as I can discern them
from what I read. Frege made a great achievement
in recognizing how to formulate a deductive
calculus for mathematics. But, I side with
Aristotle on the nature of what roles are played
by a deductive calculus. Scientific demonstration
is distinct from dialectical argumentation that
argues from belief. In turn, that distinction
informs that a scientific language is built up
synthetically. The objects of that language
are individually described using definitions.
The objects of that language are individually
presumed to exist. Consequently, the
names which complete the "incomplete symbols"
exist as references only by virtue of the fact
that the first names introduced for use in the
science are a well-ordered sequence.

Since I cannot possibly defend introducing
more than some finite number of names in
this fashion, the assumption of transfinite
numbers in set theory has a consequence. It
can be reconciled with this position only
if models of set theory are admissible as
such when they have a global well-ordering.

The largest transitive model of ZFC set theory
with these properties is HOD (hereditarily
ordinal definable).