Re: abundance of irrationals!)

Randy Poe said:
>
> aeo6 Tony Orlow wrote:
> > Randy Poe said:
> > > If you accept all that, where in the world did you hear
> > > a statement that you think is equivalent to "infinitely
> > > many marbles in a finite sized bag"? Do you think somebody
> > > said N is a finite set, somebody besides you?
> >
> > You can't get the idea that there is a finite difference between each
>
> > successive pair of naturals can you?
>
> There is a difference of 1 between each pair of
> successive naturals.
>
> > This means in the set each takes a finite
> > space on the number line. So, and infinite number of them takes an
> infinite
> > amount of space.
>
> Yes.
"There is no limit to how far you'd have to go to find the last one."
>
> > What is space between two points on the number line? The
> > difference between the values of those two points. If the points are
> > infinitely far away from each other on this infinite line,
>
> Stop. Which points are infinitely far away? You just jumped
> to the nonexistent "last natural".
Incorrect. See your quoted statement above. No limit to how far you need to go.
It is infinitely far from the first one, an infinite number of unit differences
separate them.

What is the difference between the 10th and 50th natural? Is it 40? What is the
difference between the 1st and the ooth? Is it finite?

>
> "The" points re not infinitely far away. 5 and 2 are not
> infinitely far away. 10^100 and 0 are not infinitely
> far away.
If there are an infinite number of them, some pair of them has an infinite
number of steps separating them, and infinite number of elements whose values
are between their two values.
>
> But up to this point you were talking about successive
> naturals, and suddenly you refer to "the points are
> infinitely far away". What is "the points"? Nothing
> prior in this paragraph.
The naturals are each represented by a point on the number line. Ever heard of
the number line? Every distinct value is a distinct point on the number line,
each a distinct distance form the origin, where the value is 0. I am surprised
you have never heard of this construct. Perhaps you should try school or a
book.

>
> > then there is an infinite
> > difference between them. Since a finite minus a finite is always
> finite, one of
> > those point MUST have infinite value.
>
> WHICH points? When did you start talking about a value
> at the end of the number line?
The two points (whole numbers) that have an infinite number of points (whole
numbers) between them.
>
> You started talking about successive naturals. Then
> you started talking about the extent of the whole set,
> but you aren't stopped talking about any particular pair at
> this point.
"What is space between two points on the number line? The difference between
the values of those two points. " Remember?
>
> Then you say "the points", after the moment in the
> conversation when you stopped referring to any specific
> points.
>
> You have no idea what I'm talking about, do you?
Are you talking to the mirror?

>
> Humor me. Since you're talking about an infinite difference
> between two particular values a and b, tell me what values
> a and b have in tracing through your argument above.
a is the origin, b is an element in the infinite set with an infinite index in
the set, that is, with an infinite number of elements between it and the
origin.
>
> - Randy
>
>

--
Smiles,

Tony
.

Relevant Pages

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