Liste des Groupes | Revenir à s logic |
Le 02/08/2024 à 01:53, Richard Damon a écrit :But you have the wrong definition of eps.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.
ℵo is not a "Natural Number" or a number in Peano.Not for ℵo, i.e., for most it is wrong: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.
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
Because the real continuum is a single type of number.There is no gap above zero but e real continuum.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.
Good, Then when talking about omega, gaps between numbers isn't a problem either, we get the sets of 0*Omega + n, as the Natural Numbers, then 1*Omega + n as the first set of transfinite numbers, nd those having a gap of Omega shouldn't be a problem.>All gaps of size 1 do not bother me..
There is a gap between 1 and 2, but that doesn't bother you.
Nope, we can find any one of them we want.There are infinitely many by the definition of accumulation point. You cannot find them. Therefore they are dark.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?
And that is your problem, the set of eps doesn't HAVE a minimum, because it is unbounded.>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.
You just get your order of conditions reversed.
For *ALL* 1/n, PERIOD, since I can define any of them.>For all 1/n that you can define.
For all 1/n, there is a eps that is smaller than it (like 1/(n+1) )
No, FOR ALL.>There are infinitely many, namely almost all.
And for every eps, there is a unit fraction smaller than it
Nope, why do you say I have a limited number of eps?So we have an unlimited number of Unit fractions, and no smallest one.But you have a limited number of eps.
Regards, WM
Les messages affichés proviennent d'usenet.