# Cantor infinities and zeros **URL:** \ **Category:** In My Humble Opinion\ **Created:** [September 23, 2022, 9:45pm UTC](https://boards.straightdope.com/t/cantor-infinities-and-zeros/972103 "2022-09-23T21:45:29Z")\ **Posts on this page:** 8\ **Page:** 3
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [October 1, 2022, 9:36pm UTC](https://boards.straightdope.com/t/cantor-infinities-and-zeros/972103/41 "2022-10-01T21:36:49Z")
> [@Chronos](#): > > It seems like it should be possible to unambiguously extend the notion of “volume”, such that any specified set can have its volume determined You really do not like the Axiom of Choice 🙂 > [@Dr.Strangelove](#): > > I don’t see why it should be possible to assign a volume to the individual pieces. The Axiom of Choice implies that non-measurable sets exist. ---
**Author:** ![Grrr](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/grrr/32/146_2.png) [@Grrr](https://boards.straightdope.com/u/Grrr)\ **Post date:** [October 1, 2022, 9:39pm UTC](https://boards.straightdope.com/t/cantor-infinities-and-zeros/972103/42 "2022-10-01T21:39:26Z")
I’m not big with numbers (heh), but I just happened to watch a really great documentary on Netflix called: A trip to Infinity. The feeling I got from the documentary is that infinity is a lot like Schrodinger’s Cat. Infinity is all numbers until you observe it. (Loved the infinity hotel paradox) ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [October 1, 2022, 9:47pm UTC](https://boards.straightdope.com/t/cantor-infinities-and-zeros/972103/43 "2022-10-01T21:47:39Z")
> [@Grrr](#): > > Infinity is all numbers until you observe it. Isn’t that a fundamental trick in set theory? One starts with a universe, and expands it to a larger universe having the property you want. So you change things like “all natural numbers”. AIUI this is how to prove the independence of the continuum hypothesis. ---
**Author:** ![xtenkfarpl](https://avatars.discourse-cdn.com/v4/letter/x/c2a13f/32.png) [@xtenkfarpl](https://boards.straightdope.com/u/xtenkfarpl)\ **Post date:** [October 1, 2022, 11:13pm UTC](https://boards.straightdope.com/t/cantor-infinities-and-zeros/972103/44 "2022-10-01T23:13:51Z")
> [@DPRK](#): > > > Nevertheless, I am not sure about hyperreal numbers, but it seems you cannot get too far in geometry without at least considering the possibility of extending your coordinates with things like “dual numbers” Yes, this does seem like an extension just like complex numbers from the reals. Every ‘real’ number could be considered the sum of a real and ‘imaginary’ part where the latter is zero. So maybe we can add an extra ‘dimension’ of infinitesimals which also are not part of the ‘real number line’ but work ‘outside it’? I guess tht was roughly what Robinson was thinking? ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [October 1, 2022, 11:37pm UTC](https://boards.straightdope.com/t/cantor-infinities-and-zeros/972103/45 "2022-10-01T23:37:57Z")
> [@xtenkfarpl](#): > > I guess tht was roughly what Robinson was thinking? The closest analogy to the complex numbers would be the dual numbers cited above, where you insert an epsilon such that \epsilon^2=0. That leads to a two-dimensional number system, similar to the complex numbers. It’s sufficient for calculus purposes (I think). Robinson though worked with a full-fledged hyperreal system, with not just a single infinitesimal value, but an infinite number of ever-smaller ones, as well as an infinite tower of infinities (which are just 1 \over \epsilon, 1 \over \epsilon^2, etc.). So it’s a bit more general than the dual numbers. I wonder if there’s any utility to “triple numbers”, “quad numbers”, etc., where \epsilon^N=0, but \epsilon^{N-1}\>0, and \epsilon\>\epsilon^2\>\epsilon^3, .... The universe seems to only require some finite number of derivatives for its operation. Perhaps it’s enough to only store a few layers of infinitesimals. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [October 2, 2022, 12:44am UTC](https://boards.straightdope.com/t/cantor-infinities-and-zeros/972103/46 "2022-10-02T00:44:40Z")
> [@Chronos](#): > > But what’s the volume of each of the pieces? You have put your finger on it precisely. The pieces that appear in B-T are non-measurable. The existence of such sets requires AC. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [October 2, 2022, 1:43am UTC](https://boards.straightdope.com/t/cantor-infinities-and-zeros/972103/47 "2022-10-02T01:43:58Z")
Yes, precisely. I’d rather take measurability of sets as an axiom than take the Axiom of Choice. ---
**Author:** ![ricksummon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ricksummon/32/3333_2.png) [@ricksummon](https://boards.straightdope.com/u/ricksummon)\ **Post date:** [November 14, 2022, 4:23pm UTC](https://boards.straightdope.com/t/cantor-infinities-and-zeros/972103/48 "2022-11-14T16:23:04Z")
What; no love for the generalized continuum hypothesis? 🙂 Aleph-1 all the way! [Previous page](https://boards.straightdope.com/t/cantor-infinities-and-zeros/972103.md?page=2)