An uncountable countable set
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From: Virgil on 20 Aug 2006 14:15 In article <ec9rfa$7tb$2(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > > And the next in the sequence is {0} = 1, followed by {0, 1} = 2... > > That's the von Neumann ordinals. I am talking about {}, {1}, {1,2}, ... Which is no more that the von Neumann ordinals in drag.
From: Virgil on 20 Aug 2006 14:31 In article <ec9s3c$8qv$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > > Then these infinite numbers would not deserve the name "natural number". > > Why is that? If they are whole numbers, each with successor, does that > not fit the bill? What is your definition of "whole number", TO, that allows it to be applied to objects not members of the minimal of the inductive sets required by the axiom of infinity? One of TO's faults is that he will not allow any definition to mean only what it states, but always has to suppose it to mean something else. This is a very anti-mathematical anti-logical attitude, and explains much of TO's difficulty in comprehending things logical and things mathematical. In ZF, ZFC andr NBG, a set, S, is called an inductive set if and only if (1) {} is a member of S, and (2) Whenever x is a member of S, then union {x,{x}} is a member of S. The axioms of infinity in those systems declare that there exists inductive sets, and other axioms guarantee that there must be a minimal inductive set which is called the set of natural numbers and its members, and only its members, are natural numbers in those systems. If TO wants to call anything else natural numbers, he has put himself outside the pale.
From: Virgil on 20 Aug 2006 14:44 In article <ec9ssu$9km$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <ec9nrs$44k$2(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > >> Virgil wrote: > >>> In article <ec8h1q$prm$1(a)ruby.cit.cornell.edu>, > >>> Tony Orlow <aeo6(a)cornell.edu> wrote: > >>> > >>>> It would seem to me that any good theory > >>>> of infinite sets could be applied to infinite sets of points, such as > >>>> the reals in (0,1] or those in (0,2], and be able to draw conclusions > >>>> such as that there is twice as many points and twice as much space in > >>>> the second as in the first > >>> That would require each point to occupy enough space that only a finite > >>> number of them could fit into any finite interval. > >> WRONG! > >> > >> It requires that some relationship be set up between a particular > >> infinity of points and a finite length. What you suggest would be a > >> finite set of finite elements, not any kind of infinite set. > > > > If TO insists that infinitely many points can be compressed into a > > finite space, then one can just as easily compress twice as many points > > into the same space. > > > > If one takes the points of (0,2] and places each point from x at > > position x/2 in (0,1], one has compressed "twice as many" points into > > (0,1] as were originally there. Thus there cannot be any fixed > > proportionality between the "number of points" in an interval and its > > length for intervals of positive length. > > There is when one declares it. The problem being that one can declare as many different ones as one chooses to declare, and each is just as valid as any other. > Big'un is the number of reals in the unit > interval, and the number of unit intervals on the infinite real line. I declare Card(P(N)) to be the number of reals in the unit interval and Card(N) to be the number of unit intervals with disjoint interiors in the infinite real line. TO's definition is either wrong (if his unit intervals are to have disjoint interiors) or he has corrupted his own theory by letting the umber of points in the unit interval equal the number of points in the entire real line (if overlapping intervals are allowed, the number of unit intervals in the real line equals the umber of points in the real line). > Where you claim to compress or stretch the inherent density of the real > points on the line, you make proper measure impossible. Is TO going to pontificate on measure theory on the real line? If so how is he going to deal with the inevitable non-measurable sets? Cardinality has no problem with them.
From: Virgil on 20 Aug 2006 14:57 In article <ec9tb3$a4n$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <ec9o8p$4ne$1(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > >> We have the prior definitions of all those operations with regards to > >> variables in general. > > > > On the contrary, each operation has been defined only for particular > > domains, none of which domains include any infinite quantities. > > In fact, what we call a single operation is often several operations. > > For example addition has one definition for naturals, a different > > definition for rationals, yet a different one for reals and another > > different one for complex numbers. > > Balderdash! What TO disclaims as balderdash is, in fact, proper mathematics. Mathematics requires precision where TO requires ambiguity. That TO does not understand logical and mathematical precision or the need for them, is merely a measure of his ineptitude at both logic and mathematics, not any inherent fault in the logic or the mathematics. >Addition is defined geometrically It cannot be defined geometrically for non-geometrical objects. And sets are non-geometrical until some geometrical interpretation has been defined for them. There is no unique geometrical interpretation, so there can be no unique geometrical definition. > > > > > So definitions of operations are valid only for their original domains > > of definition and new definitions are required for different domains of > > definition. > > Bull. The rules for arithmetic manipulation can be applied without > problems (for anything but transfinite mathology) to variables of > infinite value. Not in mathematics, they can't. What happens in the isolated island of TOmania only TO knows, or cares. > > > > > Anyone wishing to extend an operation only defined on one domain to a > > larger domain must give a clear definition of what that extended > > definition is to mean. TO has not done this. > > > > The domain is the real line. Which specifically does not include any points at infinity, and at best only defines arithmetic on the finite numerical values associated with geometrical points, and on nothing else. I > > > >> Yeah, that would the the lst natural. We all know THAT doesn't exist. > >> Or, is it aleph_0? Same thing anyway. ;) > > > > > > And here I though the first natural was 0 or 1, depending on whose > > model one was following.
From: Virgil on 20 Aug 2006 15:01
In article <ec9th3$a4n$2(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Well, then, since the size of the set in width is always equal to the > max of the set in height, then it would seem that aleph_0 is the LUB on > the set size as well, but that the set never actually achieves this > size. Once it does have some aleph_0th element, that element would have > to be.....aleph_0! But that's not allowed in the set, so how can it be > the size of the set? The same way that the size of the open interval from a to b, with a < b, is of length equal to the difference b-a even though neither a nor b is a member of the set. |