Re: Replacement of Cardinality

Le 02/08/2024 à 17:42, joes a écrit :

Am Fri, 02 Aug 2024 15:09:18 +0000 schrieb WM:

Le 02/08/2024 à 01:53, Richard Damon a écrit :

On 8/1/24 8:27 AM, WM wrote:

 

And thus there is no "smallest" unit fraction, as for any eps, there
are unit fractions smaller,

Your eps cannot be chosen small enough.

All unit fractions are larger than zero, so an epsilon can be chosen

Set theory says:
∀eps > 0:  NUF(eps) ≥ ℵ₀.
Who is right? You or set theory?
By the way, set theory is wrong for all x occupied by unit fractions, at least ℵ₀*2^ℵ₀ points x > 0.

There is a gap between 1 and 2, but that doesn't bother you.

All gaps of size 1 do not bother me..
 

It is the definition of definable numbers. Study the accumulation
point.
Define (separate by an eps from 0) all unit fractions. Fail.

So, which Unit fraction doesn't have an eps that seperates it from 0?

There are infinitely many by the definition of accumulation point. You
cannot find them. Therefore they are dark.

That is not the definition. The "infinitely many" are not the same ones
for every epsilon.

Strawman! Nobody claimed that. The claim is simply infimitely many. Not infinitely many the same.

You just get your order of conditions reversed.

I get it the only corect way. Every eps that you can chose belongs to a
set of chosen eps. This set has a minimum - at every time. It is finite.
Quantifiers therefore can be reversed.

The set of reals is infinite and does not have a minimum.

The set of chosen reals has a minimum at every time you choose.

 

So we have an unlimited number of Unit fractions, and no smallest one.

But you have a limited number of eps.

WTF?

You can choose as many as you like. You will never have chosen infinitely many eps. Therefore your claim that for every 1/n there is an eps separating it, is wrong.
Of course for every 1/n there is a smaller real number. But you cannot choose it. Most are dark.
Regards, WM