Re: The infintely small number b

On Nov 22, 10:16 am, mike3 <mike4...@xxxxxxxxx> wrote:
On Nov 21, 10:14 pm, William Hughes <wpihug...@xxxxxxxxxxx> wrote:





On Nov 22, 12:01 am, Venkat Reddy <vred...@xxxxxxxxx> wrote:

On Nov 22, 7:39 am, mike3 <mike4...@xxxxxxxxx> wrote:

On Nov 21, 6:18 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:

On Nov 22, 2:39 am, mike3 <mike4...@xxxxxxxxx> wrote:

On Nov 19, 5:45 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:

<snip>

"Translation" means
finding an alternate and equivalent value or an expression for a given
one. In this case, given (a,b) find [c,d] which is equivalent to
(a,b). For example, given (3,4) find values for c and such that [c,d]
represents (3,4).

OK, so what criteria must it satisfy such that it "represents" (3, 4)?
What does "represent" mean? I could have a whole different idea of
what that means than you do. Does it mean, have two points c and d
such that the set of all x where c <= x <= d is the same set as (3, 4)
(ie. [c, d] = (3, 4))? If so, then such c and d do not exist on the
reals, otherwise they would be the endpoints of (3, 4) and that would
contradict our idea that (3, 4) is an open interval. How is it a big
issue for you that some sets do not have endpoints in them?

So, atleast we agree that there is no closed interval (say C) on the
Real line which is equivalent to a given open interval (say D) on the
Real line. However since both intervals represent line segments on a
line, it suggests that there must be another parallel number system
which contains D. What is that number system? My latest thread
attempts to answer this.

- venkat

There IS a number system that contains D. It is R, the set of real
numbers. It also contains the interval C as well.

A few lines up, you have agreed that the endpoints of D do not exist
in R.

Well, yes, but only because the statment "the endpoints of D do not
exist"
implies "the endpoints of D do not exist in R". D does
not have endpoints, it does not have a largest or smallest
element (It has bounds, but these bounds are not part of D).
There is no contradiction in saying the R contains D but
R does not contain the nonexistent endpoints of D.
You may not be able to imagine a finite segment that
has bounds but not endpoints. Others do
not have such a limited imagination.

-William Hughes

That's right.

So, you imagination allows you to imagine a finite segment without
endpoints? This contradicts the "finiteness" of the line segment, or
those endpoints live in some other imaginary space, and still giving
the finite extent to the line segment which lives in this world?

- venkat



.

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