Groups | Search | Server Info | Login | Register
| Subject | Re: Cardinalities of sets |
|---|---|
| Newsgroups | sci.math, sci.physics.relativity, sci.electronics.design |
| References | (18 earlier) <9bGdnZo4hL-aDQX0nZ2dnZfqnPqdnZ2d@giganews.com> <10na9a8$kc8a$1@dont-email.me> <10nabhd$k7kl$1@dont-email.me> <10nci9d$7c6h$2@gwaiyur.mb-net.net> <z6acnXqIZ5dpRAT0nZ2dnZfqnPidnZ2d@giganews.com> |
| From | Ross Finlayson <ross.a.finlayson@gmail.com> |
| Date | 2026-02-21 08:30 -0800 |
| Message-ID | <IWednWDVRZ8SQAT0nZ2dnZfqnPadnZ2d@giganews.com> (permalink) |
Cross-posted to 3 groups.
On 02/21/2026 08:14 AM, Ross Finlayson wrote:
> On 02/21/2026 07:16 AM, Thomas 'PointedEars' Lahn wrote:
>> [X-Post & F'up2 sci.math]
>>
>> Jeroen Belleman wrote:
>>> On 2/20/26 19:31, Bill Sloman wrote:
>>>> I know enough to know that the infinite number of integers is a smaller
>>>> number than the infinite number of rational numbers,
>>
>> "One of the great challenges in this world is knowing enough about
>> a subject to think you're right, but not enough about the subject
>> to know you're wrong."
>>
>> --Neil deGrasse Tyson, astrophysicist and science communicator
>> (in his MasterClass promotion video:
>> <https://www.youtube.com/watch?v=io6QdGcoWMU>)
>>
>> (SCNR)
>>
>>>> but I don't get excited about it.
>>>
>>> I don't think that is correct. Both the sets of natural and rational
>>> numbers are aleph-0 in size,
>>
>> More precisely, their _cardinality_ is ℵ₀ (strictly: _alef_-0).
>>
>>> because it's possible to create a one-to-one mapping of every rational
>>> number to every integer.
>>
>> Otherwise correct (as purportedly proven by Georg Cantor at the end of
>> the
>> 19th/beginning of the 20th century): |ℤ| = |ℚ| = ℵ₀.
>>
>> The misconception that this would not be so can arise from the assumption
>> that ℚ = ℤ × ℤ. But actually, ℚ ⊊ ℤ × ℤ since e.g. 2/2 = 1/1 and
>> ℤ/0 ∉ ℚ as ℚ := {p/q : p, q ∈ ℤ, q > 0}.
>>
>> But then |ℚ| < |ℤ × ℤ|; and while |ℤ| < |ℤ × ℤ|, |ℤ| < |ℚ| does NOT
>> follow,
>> and is fact false: |ℤ| = |ℚ| < |ℤ × ℤ|.
>>
>> ISTM that Bill Sloman's statement would be true when comparing the
>> cardinalities of ℤ (or ℚ) and ℝ, the set of _real_ numbers, instead.
>> ℤ (and
>> ℚ) is/are countable (countably infinite), while ℝ is uncountable
>> (uncountably infinite, as also purportedly proven by Cantor). |ℤ| =
>> |ℚ| =
>> ℵ₀; |ℝ| = 2^ℵ₀, and (ISTM uncontroversial that) ℵ₀ < 2^ℵ₀, so then
>> |ℤ| = |ℚ| < |ℝ|.
>>
>> <https://en.wikipedia.org/wiki/Aleph_number>
>>
>
>
> Cardinality is rather _less precise_ than other matters
> of size relation like, for example, asymptotic density.
>
> Or, "half of the integers are even".
>
> Cardinality establishes a transitive inequality among sets,
> where "cardinals" themselves as equivalence classes of sets
> having any transitive bijective relation, are, besides zero,
> rather too large to be sets in ordinary set theories like ZF(C).
>
> Cardinality is rather specific to sets, and, set theory
> rather _describes_ numbers than _is_ numbers,
> that though "descriptive set theory" is a great account
> of formalization in mathematics.
>
> Emil duBois-Reymond discovered various arguments for
> the uncountability of reals, later Cantor wrote them
> in set theory.
>
>
> About the Continuum Hypothesis of G. Cantor, there's
> that Goedel showed it consistent one way and von Neumann
> another, then P. Cohen added an axiom to make it
> independent instead of inconsistent, set theory.
>
>
>
As one might imagine, that's a bit messy, since then
thusly one may derive contradictions in set theory
itself, and not even talking about how to derive
contradictions in set theory about description of
other theories of one relation, like ordinals for
order theory or about class/set distinction, or
about theories of other objects like those of
geometry or number theory, as modeled in
ordinary set theory.
Back to sci.math | Previous | Next — Previous in thread | Next in thread | Find similar
Cardinalities of sets (was: energy and mass) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-02-21 16:16 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-21 08:11 -0800
Cardinalities of sets wm <wolfgang.mueckenheim@tha.de> - 2026-02-22 19:04 +0100
Re: Cardinalities of sets Alan Mackenzie <acm@muc.de> - 2026-02-22 20:32 +0000
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-22 22:41 +0100
Re: Cardinalities of sets Alan Mackenzie <acm@muc.de> - 2026-02-22 22:08 +0000
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-22 21:07 -0800
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-23 13:32 +0100
Re: Cardinalities of sets Alan Mackenzie <acm@muc.de> - 2026-02-23 13:45 +0000
Re: Cardinalities of sets wm <wolfgang.mueckenheim@tha.de> - 2026-02-23 16:51 +0100
Re: Cardinalities of sets Alan Mackenzie <acm@muc.de> - 2026-02-23 17:16 +0000
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-23 18:27 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-23 09:38 -0800
Re: Cardinalities of sets Alan Mackenzie <acm@muc.de> - 2026-02-23 21:10 +0000
Re: Cardinalities of sets wm <wolfgang.mueckenheim@tha.de> - 2026-02-24 19:33 +0100
Re: Cardinalities of sets Alan Mackenzie <acm@muc.de> - 2026-02-25 12:52 +0000
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-25 21:28 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-26 08:59 -0800
Re: Cardinalities of sets wm <wolfgang.mueckenheim@tha.de> - 2026-02-24 18:40 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-21 08:14 -0800
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-21 08:30 -0800
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-21 08:51 -0800
Cardinalities of sets wm <wolfgang.mueckenheim@tha.de> - 2026-02-22 18:55 +0100
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-02-22 22:52 +0100
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-23 13:24 +0100
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-02-23 23:41 +0100
Re: Cardinalities of sets wm <wolfgang.mueckenheim@tha.de> - 2026-02-24 18:46 +0100
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-02-25 18:10 +0100
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-25 21:32 +0100
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-02-26 00:36 +0100
Re: Cardinalities of sets "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2026-02-26 01:10 -0800
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-26 16:36 +0100
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-02-26 22:46 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-26 18:08 -0800
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-27 16:52 +0100
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-02-28 00:39 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-27 20:03 -0800
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-02-27 20:12 -0800
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-01 08:12 -0800
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-04 10:15 -0800
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-02-28 19:29 +0100
Re: Cardinalities of sets Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-03-17 15:57 +0100
Re: Cardinalities of sets WM <wolfgang.mueckenheim@tha.de> - 2026-03-17 18:11 +0100
Re: Cardinalities of sets Ross Finlayson <ross.a.finlayson@gmail.com> - 2026-03-17 14:54 -0700
csiph-web