Groups | Search | Server Info | Keyboard shortcuts | Login | Register [http] [https] [nntp] [nntps]

Groups > comp.programming > #11476

Re: Theorem: There are at least ∞^4 integers in 2 dimensions.

Show all headers | View raw

On 18/11/2020 13:50, Richard Heathfield wrote:
> On 17/11/2020 21:36, David Brown wrote:
>> On 17/11/2020 14:10, Richard Heathfield wrote:
>>> On 17/11/2020 06:09, K R wrote:
>>>> Lets look shapes with 3,4,5,6,... angles.
>>>> They have inside ∞ crosses. Because by dividing areas with 2, we get ∞
>>>> points.
>>>> Lets begin with 2 line connected shapes with 3 and 4 angles.
>>>> We can continue to N shapes.
>>>> Now lets start to go around these shapes with connecting shapes.
>>>> Process can continue forever.
>>>> So we get number N*∞*∞*∞ points.
>>>> The biggest integer is ∞, so there are ∞^4 integers.
>>>
>>> No, ∞ is not an integer. There are infinitely many integers, and
>>> therefore there can be no biggest integer. This is easily proved by
>>> reductio. If we assume there is a biggest integer I, we can immediately
>>> disprove the assumption by adding 1 to I to form a new integer J that is
>>> 1 larger than I. Therefore I is not the biggest integer, even if we
>>> (quite improperly) assign ∞ to I.
>>>
>>
>> It's nice to see someone trying to post something other than spam in
>> this group (and nice to hear you are still around, Richard).
>> Unfortunately, you are responding to a spambot, not a person, and the
>> original post made no sense.
>>
>> If you'd like to talk about infinities, however, I'm game - it's always
>> a fun subject (though only vaguely on-topic here).  We could start by
>> proving that ∞⁴ is equal to ∞.
> 
> I just tried to do that, but what I actually proved (in text that I've
> now deleted) is that there are as many fourth powers of natural numbers
> as there are natural numbers, which is (a) obvious, and (b) not quite
> the same thing as proving that ∞⁴ = ∞ - or at least, if it *is* the same
> thing, I don't quite see why.
> 

Well, if we want to prove this kind of thing, it's important to be
precise.  And that means dumping ∞, because that symbol is used to mean
various things (such as increasing integers or reals without a limit, or
when defining limits, or to mean "infinity" without saying /which/
infinity).  And let's start with ∞² rather than ∞⁴.

The cardinality (size) of the set of natural numbers ℕ = {0, 1, 2, …} is
ℵ₀.  (I hope the unicode comes through okay when you read it - that's
aleph 0.)  It is the smallest transfinite cardinality.

ℵ₀² would then be the size of ℕ², the number of pairs of natural
numbers.  We can show that this is the same as ℵ₀ in a variety of ways -
a diagonalization argument is the most common.  (Look up "Hilbert's
Hotel" on Youtube if you like.)  But this is comp.programming, so let's
use a computer-related method.

For any given natural number, you can write it out in binary: 35 would
be 101101, with the digits given as ..., x5, x4, x3, x2, x1, x0.

Split that sequence into two, with a = ..., x5, x3, x1 and b = ..., x4,
x2, x0.  It's quite obvious (and could be shown more rigorously, written
up in LaTeX instead of an email with Unicode) that each number "x"
corresponds to a pair (a, b), and that each pair (a, b) corresponds to a
number "x".  Thus the number of natural numbers "x" must equal the
number of pairs of natural numbers (a, b), i.e., ℵ₀² = ℵ₀.

Extending this argument to ∞⁴ can be done by applying it twice, or by
splitting "x" into four numbers.  (That is a little less mind-bending
than trying to do a diagonalization argument directly in four
dimensions.)  With a little more extension, ℵ₀ ^ n = ℵ₀ for any finite
natural number n (other than 0).

Next challenge - prove that 2 ^ ℵ₀ is greater than ℵ₀ :-)

Back to comp.programming | Previous | Next — Previous in thread | Next in thread | Find similar | Unroll thread

Thread

Theorem: There are at least ∞^4 integers in 2 dimensions. K R <ownerofachattingportal@gmail.com> - 2020-11-16 22:09 -0800
  Re: Theorem: There are at least ∞^4 integers in 2 dimensions. Richard Heathfield <rjh@cpax.org.uk> - 2020-11-17 13:10 +0000
    Re: Theorem: There are at least ∞^4 integers in 2 dimensions. Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-11-17 17:25 +0000
      Re: Theorem: There are at least ∞^4 integers in 2 dimensions. Richard Heathfield <rjh@cpax.org.uk> - 2020-11-18 12:41 +0000
        Re: Theorem: There are at least ∞^4 integers in 2 dimensions. David Brown <david.brown@hesbynett.no> - 2020-11-18 14:58 +0100
          Re: Theorem: There are at least ∞^4 integers in 2 dimensions. Khritjan Robam <hrrobam@gmail.com> - 2020-11-18 06:01 -0800
          Re: Theorem: There are at least ∞^4 integers in 2 dimensions. Richard Heathfield <rjh@cpax.org.uk> - 2020-11-18 14:42 +0000
            Re: Theorem: There are at least ∞^4 integers in 2 dimensions. David Brown <david.brown@hesbynett.no> - 2020-11-18 16:43 +0100
              Re: Theorem: There are at least ∞^4 integers in 2 dimensions. Richard Heathfield <rjh@cpax.org.uk> - 2020-11-18 18:59 +0000
                Re: Theorem: There are at least ∞^4 integers in 2 dimensions. David Brown <david.brown@hesbynett.no> - 2020-11-18 21:39 +0100
    Re: Theorem: There are at least ∞^4 integers in 2 dimensions. David Brown <david.brown@hesbynett.no> - 2020-11-17 22:36 +0100
      Re: Theorem: There are at least ∞^4 integers in 2 dimensions. Richard Heathfield <rjh@cpax.org.uk> - 2020-11-18 12:50 +0000
        Re: Theorem: There are at least ∞^4 integers in 2 dimensions. David Brown <david.brown@hesbynett.no> - 2020-11-18 15:27 +0100
          Re: Theorem: There are at least ∞^4 integers in 2 dimensions. Richard Heathfield <rjh@cpax.org.uk> - 2020-11-18 14:50 +0000
          Re: Theorem: There are at least ∞^4 integers in 2 dimensions. Ben Bacarisse <ben.usenet@bsb.me.uk> - 2020-11-18 15:44 +0000
            Re: Theorem: There are at least ∞^4 integers in 2 dimensions. David Brown <david.brown@hesbynett.no> - 2020-11-18 18:09 +0100
              Re: Theorem: There are at least ∞^4 integers in 2 dimensions. Peter <peterxpercival@hotmail.com> - 2020-11-18 18:00 +0000
                Re: Theorem: There are at least ∞^4 integers in 2 dimensions. David Brown <david.brown@hesbynett.no> - 2020-11-18 21:42 +0100
          Re: Theorem: There are at least ∞^4 integers in 2 dimensions. Peter <peterxpercival@hotmail.com> - 2020-11-18 17:58 +0000
            Re: Theorem: There are at least ∞^4 integers in 2 dimensions. David Brown <david.brown@hesbynett.no> - 2020-11-18 21:46 +0100
    Re: Theorem: There are at least ∞^4 integers in 2 dimensions. Melzzzzz <Melzzzzz@zzzzz.com> - 2020-11-19 03:49 +0000
      Re: Theorem: There are at least ∞^4 integers in 2 dimensions. Khrictian Robam <hrrobam@gmail.com> - 2020-11-19 01:12 -0800

csiph-web