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 ↙↙↙ <[email protected]> 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 > > - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > archives: http://friam.471366.n2.nabble.com/ > FRIAM-COMIC <http://friam.471366.n2.nabble.com/FRIAM-COMIC> > http://friam-comic.blogspot.com/ >
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