Re: The infintely small number b

On Nov 19, 11:50 pm, Randy Poe <poespam-t...@xxxxxxxxx> wrote:
On Nov 19, 1:38 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:





On Nov 19, 8:12 am, Randy Poe <poespam-t...@xxxxxxxxx> wrote:

On Nov 18, 8:36 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:

On Nov 18, 11:58 pm, William Hughes <wpihug...@xxxxxxxxxxx> wrote:

On Nov 18, 12:28 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:

Regarding sets, interval and measure: Defining an interval as a set of
all real numbers between a given pair of real numbers is dependent on
assumption that the interval is made up by distinct real numbers.

Yes. Two real numbers which aren't the same are distinct.

Such an interval doesn't have distinct and different elements,

Of course it does

unless those
element values are explicitly specified. It is just continuum.

That just means that if x is a real number there is no
next real number. It does not mean x is not distict from
all other real numbers.

When there is no next number then the notation {x:0<x<1} is equivalent
to {x:0<=x<=1}.

No, the notation is not equivalent. 0 is a member of the
second set, but not the first. 0 does not pass the
membership test 0 < x < 1.

It does not pass the membership test because we assume that "strictly
less than" is valid for real numbers.

And it is. If two real numbers x and y are not identical,
then either x<y or y>x. Furthermore, you can determine
which is true by looking at their decimal representations.
If x is not equal to y, then they will differ in at least
one decimal place, and you can use that difference to
say unambiguously whether x<y or y>x.

Why do you find it unfathomable that I can tell whether
3.4198205 is less than or greater than 3.4198206?

Also, it implies that we can't translate an exclusive interval into an
inclusive interval and vice versa. (a,b) <=> [c,d] is not possible.

Every post seems to use an undefined word. The word
for this post is "translate". What do you mean by
"translate"?

Do you mean there is no one-to-one map between
(a,b) and [a,b]? That is false.

This should indicate an issue,

... if it were true, but it isn't. Perhaps you have
some other meaning of "translate" that indicates an
"issue" for you. You haven't articulated it yet.

I hoped the plain english meaning would just work fine, but I should
have used "conversion" or be more explicit. "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). Ofcourse, this involves finding of inclusive bounds
of the exclusive interval (3,4) and this is not possible because there
is no "next" number.

An interval of a physical continuum wouldn't need two representations
if the bounds and extent are same. Only an unnatural interval which is
assumed to be a "string" of points would need the artifical dual
representation of intervals with same bounds and same extent but
different models.

- venkat

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