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 <fomJ...@nyms.net> 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:

"...to 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).