[FRIAM] actual vs potential ∞

[Frank Wimberly]

I might modify this slightly to

For any r in R, however large, there exists x in R, and epsilon > 0 in R
such that  1/x > r for x < epsilon.

I'm not sure that makes a difference but it may make it clearer.

On Mon, Aug 3, 2020 at 11:14 AM Frank Wimberly <wimberly3 at gmail.com> wrote:

> My opinion.  1/0 is undefined.  Depending on the context you can define it
> in a way that's useful in that context.
>
> To say that   lim(1/x) as x ->0 = infinity means precisely:
>
> For any r in R, however large, there exists an x in R such that  1/x > r.
>
> Frank
>
> ---
> Frank C. Wimberly
> 140 Calle Ojo Feliz,
> Santa Fe, NM 87505
>
> 505 670-9918
> Santa Fe, NM
>
> On Mon, Aug 3, 2020, 11:03 AM uǝlƃ ↙↙↙ <gepropella at gmail.com> wrote:
>
>>
>> I know I've posted this before. I don't remember it getting any traction
>> with y'all. But it's relevant to my struggles with beliefs in potential vs
>> actual infinity:
>>
>>   Belief in the Sinularity is Fideistic
>>   https://link.springer.com/chapter/10.1007%2F978-3-642-32560-1_19
>>
>> Not unrelated, I've often been a fan of trying identify *where* an
>> argument goes wrong. And because this post mentions not only 1/0, but
>> Isabelle, Coq [⛧], Idris, and Agda, I figured it might be a good follow-up
>> to our modeling discussion on Friday, including my predisposition against
>> upper ontologies.
>>
>>   1/0 = 0
>>   https://www.hillelwayne.com/post/divide-by-zero/
>>
>> Here's the (really uninformative!) SMMRY L7:
>>
>> https://smmry.com/https://www.hillelwayne.com/post/divide-by-zero/#&SM_LENGTH=7
>> > Since 1 0, there is no multiplicative inverse of 0⁻. Okay, now we can
>> talk about division in the reals.
>> >
>> > So what's -1 * π? How do you sum up something times? While it would be
>> nice if division didn't have any "Oddness" to it, we can't guarantee that
>> without kneecapping mathematics.
>> >
>> > We'll define division as follows: IF b = 0 THEN a/b = 1 ELSE a/b = a *
>> b⁻.
>> >
>> > Doing so is mathematically consistent, because under this definition of
>> division you can't take 1/0 = 1 and prove something false.
>> >
>> > The problem is in step: our division theorem is only valid for c 0, so
>> you can't go from 1/0 * 0 to 1 * 0/0. The "Denominator is nonzero" clause
>> prevents us from taking our definition and reaching this contradiction.
>> >
>> > Under this definition of division step in the counterargument above is
>> now valid: we can say that 1/0 * 0 = 1 * 0/0. However, in step we say that
>> 0/0 = 1.
>> >
>> > Ab = cb => a = c but with division by zero we have 1 * 0 = 2 * 0 => 1 =
>> 2.
>>
>>
>>
>> [⛧] I decided awhile back to focus on Coq because it seems to have
>> libraries of theorems for a large body of standard math. But still NOT
>> having explored it much, yet learning some meta-stuff surrounding the
>> domain(s), I'm really leaning toward Isabelle. I suppose, in the end, I
>> won't learn to use any of it, except to pretend like I know what I'm
>> talking about down at the pub.
>>
>> --
>> ↙↙↙ uǝlƃ
>>
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>

-- 
Frank Wimberly
140 Calle Ojo Feliz
Santa Fe, NM 87505
505 670-9918
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