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).