Re: Replacement of Cardinality

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.

That is the opinion of Peano and his disciples. It holds only for potetial infinity, i.e., definable numbers.

 No, it holds for ALL his numbers.

Not for ℵo, i.e., for most it is wrong:
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo

What is the reason for the gap before omega? How large is it? Are these questions a blasphemy?

 Because it is between two different sorts of number.

There is no gap above zero but e real continuum.

 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.

 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.

 For all 1/n, there is a eps that is smaller than it (like 1/(n+1) )

For all 1/n that you can define.

 And for every eps, there is a unit fraction smaller than it

There are infinitely many, namely almost all.

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

But you have a limited number of eps.
Regards, WM