Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

On 11/04/2024 03:09 PM, Jim Burns wrote:

On 11/4/2024 2:39 PM, Ross Finlayson wrote:

On 11/04/2024 03:52 AM, Jim Burns wrote:

On 11/2/2024 6:01 PM, Ross Finlayson wrote:

On 11/02/2024 12:37 PM, Jim Burns wrote:

On 11/2/2024 2:02 PM, Ross Finlayson wrote:

>

The delta-epsilonics of course,
or some put it "delta-epsilontics",
with little d and smaller e,
of often for induction arbitrary m and larger n,
is well-known to all students of calculus.
"The infinitesimal analysis", ....

>
The delta.epsilonics well.known to students of calculus
is not infinitesimal analysis.
For δ > 0 and ε > 0
there are _finite_ j and k such that
δ > ⅟j > 0 and ε > ⅟k > 0

>

Sure it is,
the delta-epsilonics is well known,

>
For δ > 0
there is finite j such that δ > ⅟j > 0
>
⎛ Assume otherwise.
⎜ Assume, for δ > 0, that
⎜ no ⅟j exists: δ > ⅟j > 0
⎜ δ is a lower bound of ⅟ℕ
⎜ ⅟ℕ = {⅟i: i ∈ ℕ⁺ ∧ i finite}
⎜
⎜ Let β be the greatest lower bound of ⅟ℕ
⎜ β ≥ δ > 0
⎜ 2⋅β > β > ½⋅β
⎜
⎜ ⅟ℕ ᵉᵃᶜʰ≥ β > ½⋅β
⎜ ⅟ℕ ᵉᵃᶜʰ≥ ½⋅β
⎜ ½⋅β is a lower bound of ⅟ℕ  [1]
⎜
⎜ If 2⋅β is a lower.bound of ⅟ℕ
⎜ then 2.β > β is a greater.than.β lower bound of ⅟ℕ
⎜ which is not a thing.
⎜ 2⋅β is not a lower bound of ⅟ℕ
⎜
⎜ 2⋅β is not a lower bound of ⅟ℕ
⎜ exists ⅟k ∈ ⅟ℕ: ⅟k > 2⋅β
⎜ exists ¼⋅⅟k ∈ ⅟ℕ: ¼⋅⅟k > ½⋅β
⎜ ½⋅β is not a lower bound of ⅟v ℕ
⎜
⎜ However,
⎜ [1]  ½⋅β is a lower bound of ⅟ℕ
⎝ Contradiction.
>
Therefore,
For δ > 0
there is finite j such that δ > ⅟j > 0
>

What I'm saying is that since antiquity,
it is known,
that there are at least two models of continuity,
and you may call it Archimedean and Democritan,
about the field of rationals versus atomism,
and that infinitesimal analysis includes both.

>
Calculus class.
Complete ordered field.
Delta.epsilonics.
No infinitesimals.
>

So, infinitesimal analysis includes delta-epsilonics,
if not the other way around.

>
Infinitesimal analysis without infinitesimals.
So creative!
>
>
>

Actually, in infinitesimal analysis, there's
a reasoning that there _are_ infinitesimals
in the geometric series, that the limit, to exist,
_is_ the sum, so, in infinitesimal analysis,
it's merely a simple definition of limit that
thusly "attains to" the limit if never reaching
it, that in infinitesimal analysis, "reaches".
In that sense it's merely a polite conceit.
That is to say, and I hope it's clear when mentioned,
there's a reasoning that the limit must be the infinite
limit and must actually as a matter of "fact" _reach_
the limit to exist - because, for example, then in
the integral calculus, "the sum of sums" must make
for not letting a case for "sum of zeros".
I.e., it's certainly agreeable dealing with any
derived quantity that it "is" itself, ...,
less so that it isn't. Then of course if not
one would needfully declare that all the declarations
are cumulative and the notation representing a
manner-of-speaking, this "integral" and "differential",
about "infinity" and, ..., "zero", it's certainly so
that some have the infinite limit _is_ the sum,
and not merely "no different".
So, in infinitesimal analysis, it's a sort of
constructive development, delta-epsilonics and
the limit, that is to say that constructivist
means avoiding proof-by-contradiction, with
regards to "there is ..." and "there isn't not ...".
It's a sort of not constructivist development,
while it's agreeably constructive.
I.e., in infinitesimal analysis or "where quantities,
even non-zero infinitesimals exist", then it's the
same sort of development, yet for finitists it
never completes while for the infinitary it's perfect.
Then, some maintain that to be correct it's perfect,
and that the limit _is_ the sum, because, otherwise
there's a reductio where it's also _less_ the sum.
Yet, people are told it's acceptable to make do,
and told that it suffices. Others simply have it not so.
Hey thanks though, it's spirited the craigh,
yet you'll see again it's not quite entailed
from the wider milieu.