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

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On 18/11/2020 16:44, Ben Bacarisse wrote:
> David Brown <david.brown@hesbynett.no> writes:
> 
>> 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.
> 
> ∞ is ambiguous. It could mean ℵ₀ or it could mean 𝜔, the smallest
> infinite ordinal.  𝜔⁴ is just 𝜔𝜔𝜔𝜔 ≠ 𝜔.
> 

Yes - I mentioned that ambiguity in another post.  My assumption here is
that we were talking about cardinals, rather than ordinals.

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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

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