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Le 02/08/2024 à 01:53, Richard Damon a écrit :All unit fractions are larger than zero, so an epsilon can be chosenOn 8/1/24 8:27 AM, WM wrote:And thus there is no "smallest" unit fraction, as for any eps, thereYour eps cannot be chosen small enough.
are unit fractions smaller,
That is not the definition. The "infinitely many" are not the same onesNot for ℵo, i.e., for most it is wrong:That is the opinion of Peano and his disciples. It holds only forNo, it holds for ALL his numbers.
potetial infinity, i.e., definable numbers.
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
There is no gap above zero but e real continuum.What is the reason for the gap before omega? How large is it? AreBecause it is between two different sorts of number.
these questions a blasphemy?
>There is a gap between 1 and 2, but that doesn't bother you.All gaps of size 1 do not bother me..
There are infinitely many by the definition of accumulation point. YouIt is the definition of definable numbers. Study the accumulationSo, which Unit fraction doesn't have an eps that seperates it from 0?
point.
Define (separate by an eps from 0) all unit fractions. Fail.
cannot find them. Therefore they are dark.
The set of reals is infinite and does not have a minimum.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.
WTF?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 itThere 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.
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