[FRIAM] actual vs potential ∞

[Frank Wimberly]

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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