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

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

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

On 11/02/2024 07:10 AM, Jim Burns wrote:

>

[...]

>
The delta-epsilonics of course,
or some put it "delta-epsilontics",
with little d and smaller e,

>
or, as others put it,
all arbitrarily.small δ > 0 with
each having small.enough ε > 0 existing
>

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
>
j and k follow true.or.not.first.false.ly from
the line studied by those students being described
as containing all ratios of finites and also
sufficient points more such that,
wherever a function jumps,
there is a point.of.discontinuity.
(intermediate value theorem)
>

I.e., in a manner of speaking,
the infinite transfinite cardinals
don't exist in delta-epsilonics
any more than plain manner-of-speaking "infinity".

>
In a manner of speaking,
when I use my computer,
I am _not_ using
transistors and logic gates and bit packets.
Certainly, I _can_ use it without
any awareness of all that.
>
However, that and those things not.existing
are different.
>
>

The delta-epsilonics is a perfectly suitable approach to defining
infinite limit in theories of infinitesimal analysis.
How about atomism, is there a theory there with
truly abstractly uncuttable objects like the
theoretical atom?