Re: Well Ordering the Reals


Tony Orlow wrote:
> MoeBlee said:
> > Tony Orlow wrote:
> >
> > > Well, it sounds rather inconsistent, doesn't it, an unending set with a
> > > particular size which is infinite, but no infinite values or positions in the
> > > set? It simply doesn't make any more sense to me that the square circle.
> >
> > It's not a round square. Each of the members is finite. The SET is
> > infinite, because it has infinitely MANY members. There is not even an
> > appearence of contradiction in that.
> >
> > I don't think you are unintelligent as some people claim. But you do
> > have a major block. There are infinitely many natural numbers; there is
> > no greatest natural number; and no natural number is infinite. Children
> > understand this, non-mathematician adults understand it, virtually
> > every mathematician for the last hundred years and even for thousands
> > of years has understood it. Forget about the natural numbers, Euclid
> > understood that there are infinitely many primes. Forget Cantor,
> > Galileo understood one-to-one correspondence of an infinite set and a
> > proper subset.
> >
> > Kids will tell you; people who've never heard of set theory will tell
> > you; the study of mathematics will tell you:
> >
> > If there were only finitely many natural numbers, then there would be a
> > greatest natural number. But there is no greatest natural number since
> > you can always add 1 to get a greater natural number. And no natural
> > number is infinite since every natural number comes from adding 1 to
> > previous natrural number that is not infinite. To have an infinite set,
> > it is not required that one of its members be infinite. The set {0 1 2
> > ...} is infinite but no member is infinte. This is clear. But some
> > block prevents you from seeing it.
> >
> > MoeBlee
> >
> >
> The "block" that prevents me from agreeing with this false picture is an
> appreciation for the identity function between element count and value in the
> naturals. For every additional natural, the set grows by one and the element
> value is increased by one. As many elements as are in the set, that is the
> size, and also the value range, given the density of 1 element per unit of
> element value. Is this a block? No, it's an inductively provable fact. Why is
> it not accepted? Because induction is considered only to hold for finite
> values, and therefore this result is considered invalid in the infinite case.
> But, where you have an equality that holds inductively, it holds for all cases,
> finite and infinite. There is no point at which the infinite chain of logical
> implication breaks down, if the difference of zero that forms the basis of the
> equality is maintained in all cases. There is no reason to abandon this
> relationship when there are an infinite number of elements, and if it holds in
> the finite case, then there is no finite element that marks an infinite set
> anyway, and since that's all there is in your set of naturals, the set cannot
> be infinite, even if considered to include the entirety of its members. None is
> in an infinite position in the set. If this is a block, perhaps you should
> consider it a cornerstone.

Even if we accept your nonsense about induction and infinity, you
can at most demonstrate things for sets of the form

{1,2,3,...,N}

where N is a TO natural number. You cannot use induction to
demonstrate
anything about a set without a largest element.

-William Hughes

.

Relevant Pages

  • Re: Well Ordering the Reals
    ... One of the very root ... >> have in induction, we have an implication that always implies another ... >> the infinite linear set, generally considered the naturals. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... naturals, the nth naturals will ALWAYS be equal to n. ... We claim that induction on peano sets only works according to the ... There is nothing in the Peano axioms which allows TO's "infinite case" ...
    (sci.math)
  • Re: Orlow cardinality question
    ... >> basically the entire system of maths education of the modern world has ... The set is infinite. ... >>> difference between any two successive members. ... > naturals is the number of them, the size of the set, minus 1. ...
    (sci.math)
  • Re: Orlow cardinality question
    ... > using induction, but I can't prove things about BEING infinite. ... Not the set size of the reals. ... For each finite natural, n in N, let n* represent the set of naturals up ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... One of the very root ... > have in induction, we have an implication that always implies another ... > the infinite linear set, generally considered the naturals. ...
    (sci.math)