On Dec 8, 6:06 pm, fom <fomJ...@nyms.net> wrote: > 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).
Aleph_0 is a cardinal, lengths are scalar.
Quite erudite, and of interest, I find your post there.
Wouldn't the HOD's order type as model be an ordinal and thus irregular, i.e. Burali-Forti?
It's of interest to read of a space-filling curve and the general position, do you see any curve really fill space? What of a spiral real-space-filling curve?
Some have geometry first, others integers first, I'd agree they're separate domains of discourse, but of the same domain of discourse. Some have points then lines, others points then space, some have zero then one, others zero then infinity, and the latters are to an appreciable extent more fundamental or primitive.
You denote earlier the simple conjunctions of binary truth tables, and I'd appreciate myself a better understanding of the impredicative: what do you think of: an or the axiomless system of natural deduction?