Gerry Myerson wrote: > In article <firstname.lastname@example.org>, > "Gene Ward Smith" <email@example.com> wrote: > > > Gerry Myerson wrote: > > > > > I think Norm would say, 1) you can't form the set (and Norm's reasons > > > are given in the article), and 2) you don't need to - there's no good > > > mathematics you can do with the completed infinite set that you can't > > > do without it. > > > > Norm is also opposed to axioms. Without axioms, how do we know when we > > are "forming" an infinte set? If I state Euclid's theorem on the > > infinitude of primes, am I "forming" a set? Am I forming a set just by > > referencing the integers at all? If Norm won't give a set of axioms he > > finds acceptable, we can't very well say that measureable cardinals > > contradict his foundations for mathematics, because he hasn't really > > given a foundation. He has, in fact, claimed that infinite sets are > > metaphysics; but if they are metaphysics, he's not talking mathematics > > at all, but metaphysics. In which case, so what? What do his > > metaphysical beliefs have to do with mathematics? > > All these questions are better directed to Norm than to me, > but I'll make believe I know what he is on about, and answer > thus: > > If I remember right, Euclid never said "there's an infinitude of > primes." He just said, "given any prime, there's a bigger one." > You and I are accustomed to interpreting the second as meaning > the same thing as the first, but I think that until quite recent > times, mathematicians didn't. They didn't accept "the completed > infinity," and they were still able to develop the theory of > numbers, prove the quadratic reciprocity theorem, the four squares > theorem, etc. If you state the theorem the way Euclid did, > you are not forming an infinite set, and you can get on perfectly > well that way. > [...]
Well, the Greeks were not able to resolve Zeno's paradoxes, were they? The point is that the Greek way of thinking leads to a paradox that Achilles can never catch up with a tortoise (moving ahead of him in a straight line at a slower velocity). This is the problem with the Greek concept of "potential infinity" -- which will not permit Achilles to "complete" the infinitely many conscious acts needed to catch up with the tortoise (first Achilles has to reach a point where the tortoise was previuously located, but now has moved ahead -- this step repeats itself ad infinitum). I don't think classical real analysis resolves Zeno's paradoxes with the concept of limits either -- this leads to the issue of how infinitely many finite, non-zero and non-infinitesimal intervals of reals can sum to a finite interval.
Norman Wildberger, like the Greeks, has the problem of explaining what he means by an "arbitrary" natural number or prime number, etc. For example, when NW says that some proposition holds for an arbitrary natural number, is that not an unverifiable assertion about infinitely many naturals that cannot be proven unless one makes postulates (or axioms) to which he is so opposed? And such axioms *must* take for granted the existence of an infinite class of natural numbers for which the said proposition holds -- otherwise we don't have a proof of this proposition. And NW does oppose the concept of an "arbitrary" real number-- here is a quote from his paper "Set theory: should you believe?" at <http://web.maths.unsw.edu.au/~norman/views2.htm>:
"But here is a very important point: we are not obliged, in modern mathematics, to actually have a rule or algorithm that specifies the sequence r1, r2, r3, · · · . In other words, 'arbitrary' sequences are allowed, as long as they have the Cauchy convergence property. This removes the obligation to specify concretely the objects which you are talking about. Sequences generated by algorithms can be specified by those algorithms, but what possibly could it mean to discuss a 'sequence' which is not generated by such a finite rule? Such an object would contain an 'infinite amount' of information, and there are no concrete examples of such things in the known universe. This is metaphysics masquerading as mathematics."
I agree with NW's conclusion that an arbitrary real number, in the classical sense, is problematic. But definitely not with his stated reason for this conclusion -- that such an arbitrary real number (not specified by a finite rule) contains an "infinite amount of information". The fact is that even, say, a Cauchy sequence converging to Pi, specified by a finite rule, does contain an infinite amount of information -- no human mind can complete the task of running through all the terms of this sequence generated by this finite rule. And NW will have to explain how he can accept statements like "For any real number x>0, (1/x)>0", if he does not accept the concept of an arbitrary Cauchy sequence of rationals. For x is an arbitrary real number, is it not?
My point of view, explained in my arxiv paper <http://arxiv.org/abs/math.LO/0506475>, is that the problem is not with the existence of an inifnite class of naturals -- one can accept that an arbitrary prime can only be defined as that belonging to an infinite class of primes, for example. What consititutes infinitary reasoning (in my view) is quantifying over these infinite classes, i.e., formally referring to infinitely many such infinite classes in a single formula. Thus an arbitrary real number x, in the classical sense, is not allowed in my proposed logic NAFL-- because x can only be defined as belonging to an infinite class of reals, which requires quantification over reals -- not permitted in NAFL because each real is an infinite object. Infinite sets do not exist in NAFL, but infinite classes can exist (in fact MUST exist whenever the infinitely many finite objects belonging to that class exist). Infinite classes, like real numbers, are proper classes-- so the reals do not consititute a class and you can't quantify over reals in NAFL.
I believe that the correct resolution of Zeno's paradoxes is in the version of real analysis proposed in the above-cited arxiv paper <math.LO/0506475>. In particular, see Sec. 4, which is more or less self-contained.