Re: abundance of irrationals!)
- From: Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx>
- Date: Mon, 2 May 2005 10:36:41 -0400
Virgil said:
> In article <MPG.1cdc489db5181f50989b60@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>
> > >
> > > If a set of finite objects must be finite, how does one deal with
> > > the
> > > set of rationals between 0 and 1?
> > >
> >
> > The fact is that one cannot have an infinite set of finite numbers,
> > if the difference between any two is at least 1, whiich is true of
> > natural numbers. If the set of the first n naturals always has n
> > elements and a largest element with value n, then a set of the first
> > N naturals, an infinite number, must have a maximum number equal to
> > the set size, N.
>
> You are arguing that adding 1 more thing a set and adding infinitely
> many more things to a set are equivelent. This requires proofs that I do
> not see.
No, I am arguing that if the set size is infinite, then so are most members of the set, if there is a finite
constant difference between each successive pair of members. It's simple. If you have an infinite number of
points on a line, spaced an inch apart, most points will be infinitely far away from any given point. Not
every point will be a finite distance away, if there are an infinite number. The value of the natural number
is measured as its difference from zero, and the value increases by one with the addition of each member.
Now, tell me how you can add an ifninite number of 1's and get less than infinity?
>
> Consider the set of rationals { 1/n : n in N}. If your thesis were true,
> tehn this set would have to contain 0. But that wold require you to
> provide a natural number n such that 1/n = 0.
>
> Provide it!
ha ha! Okay, the number is N if that's what you want.
>
> > If, on the other hand, we insist that the naturals
> > are all finite, then the greatest difference between any two must be
> > finite, which means there can only be a finite number of elements
> > between them. The resolution to this problem
>
>
> What problem?
The problem of thinking you can add oo 1's and call the result finite. That's a serious problem in need of
repair.
>
> I propose axioms as follows
> Existence of a set called P with the following properties
> (i) P is not empty, in particular there is an elemeant e in P.
> (ii) For every element x in P there is a different element x' in P
> (iii) For every x in P, x' =/= e.
> (iv) for x any y in P, if x =/= y then x' =/= y'.
> (v) If S is any subset of P such that e is a memver of S and
> whenever x is a member of S so is x',
> then S = P
>
> To prove your contentions, you must prove from these axioms that there
> exists some element of P other then e which is not of form x' for any x
> in P.
>
> I defy you to do that.
>
I should prove my understanding to you using your axioms? Tell you what, why don't you just show me a finite
line segment divided into an infinite number of concatentated constant finite intervals, and I'll concede.
Just demonstrate that the limit of f(x)=1, as x goes to infinity, is finite.
Also, while you're at it, why don't you comment on the resolution I suggested, instead of clipping it out?
Why can't naturals be taken to include the 10-adics?
--
Smiles,
Tony
.
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