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

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  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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#11464 — Theorem: There are at least ∞^4 integers in 2 dimensions.

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.

Kristjan Robam

Ph.: 372 57865442
E-m.: mi23523523 fastmail c o m

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#11466

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.

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

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#11468

Richard Heathfield <rjh@cpax.org.uk> writes:

> 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.

That makes it sound like it is the infinitude itself that leads to the
conclusion, but there are lots of infinite sets with a largest element.

> 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.

This is what leads to the conclusion: the axioms for the > relation.

> Therefore I is not
> the biggest integer, even if we (quite improperly) assign ∞ to I.

(Oh, and hi.  Haven't seen you around for ages.)

-- 
Ben.

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#11471

On 17/11/2020 17:25, Ben Bacarisse wrote:
> Richard Heathfield <rjh@cpax.org.uk> writes:
> 
>> 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.
> 
> That makes it sound like it is the infinitude itself that leads to the
> conclusion, but there are lots of infinite sets with a largest element.

Yes. The set of negative integers, for example; or the set of reals 
between 0 and 1.

>> 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.
> 
> This is what leads to the conclusion: the axioms for the > relation.
> 
>> Therefore I is not
>> the biggest integer, even if we (quite improperly) assign ∞ to I.
> 
> (Oh, and hi.  Haven't seen you around for ages.)

(Yeah, hi. I'm around, but mostly here, on sci.crypt, and on 
rec.puzzles, all three of which are low-traffic - or low-signal, anyway. 
I stopped bothering with clc when it started morphing into a religious 
debating forum. I haven't been back to check whether that trend continued.)

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

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#11473

On 18/11/2020 13:41, Richard Heathfield wrote:
> On 17/11/2020 17:25, Ben Bacarisse wrote:
>> Richard Heathfield <rjh@cpax.org.uk> writes:
>>
>>> 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.
>>
>> That makes it sound like it is the infinitude itself that leads to the
>> conclusion, but there are lots of infinite sets with a largest element.
> 
> Yes. The set of negative integers, for example; or the set of reals
> between 0 and 1.
> 

In fact, there are 2 ^ 2 ^ ∞ infinite sets of real numbers with a
largest element.  That's ℶ₂, which is quite big.

>>> 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.
>>
>> This is what leads to the conclusion: the axioms for the > relation.
>>
>>> Therefore I is not
>>> the biggest integer, even if we (quite improperly) assign ∞ to I.
>>
>> (Oh, and hi.  Haven't seen you around for ages.)
> 
> (Yeah, hi. I'm around, but mostly here, on sci.crypt, and on
> rec.puzzles, all three of which are low-traffic - or low-signal, anyway.
> I stopped bothering with clc when it started morphing into a religious
> debating forum. I haven't been back to check whether that trend continued.)
> 

There's not been much religion on c.l.c. in a while (unless you count
the schism between C and C++ as a religious battle).  But it turns up
occasionally, and there are always some trolls and other less helpful
posters.  (Mind you, last time I told you there had been few religious
posts in c.l.c., and you subscribed again, Rick followed you - by
coincidence, I assume.)

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#11475

Hey people, please ignore this topic and look into 
https://groups.google.com/g/alt.math/c/JN0q36zaW-Q
if you are interested of my idea development.

Khritjan Robam
Ph.: 372 57865442 
E-m.: mi23523523 fastmail c o m

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#11477

On 18/11/2020 13:58, David Brown wrote:
> On 18/11/2020 13:41, Richard Heathfield wrote:
>> On 17/11/2020 17:25, Ben Bacarisse wrote:
>>> Richard Heathfield <rjh@cpax.org.uk> writes:
>>>
>>>> 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.
>>>
>>> That makes it sound like it is the infinitude itself that leads to the
>>> conclusion, but there are lots of infinite sets with a largest element.
>>
>> Yes. The set of negative integers, for example; or the set of reals
>> between 0 and 1.
>>
> 
> In fact, there are 2 ^ 2 ^ ∞ infinite sets of real numbers with a
> largest element.  That's ℶ₂, which is quite big.

Did you remember the set of real numbers between 19.00041 and 23.00043? 
It's easily overlooked. You'd better check your list; if it's not there, 
add it on, and that would make 2 ^ 2 ^ ∞ + 1, which is ever so slightly 
bigger.

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

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#11479

On 18/11/2020 15:42, Richard Heathfield wrote:
> On 18/11/2020 13:58, David Brown wrote:
>> On 18/11/2020 13:41, Richard Heathfield wrote:
>>> On 17/11/2020 17:25, Ben Bacarisse wrote:
>>>> Richard Heathfield <rjh@cpax.org.uk> writes:
>>>>
>>>>> 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.
>>>>
>>>> That makes it sound like it is the infinitude itself that leads to the
>>>> conclusion, but there are lots of infinite sets with a largest element.
>>>
>>> Yes. The set of negative integers, for example; or the set of reals
>>> between 0 and 1.
>>>
>>
>> In fact, there are 2 ^ 2 ^ ∞ infinite sets of real numbers with a
>> largest element.  That's ℶ₂, which is quite big.
> 
> Did you remember the set of real numbers between 19.00041 and 23.00043?

Yes, it's in there.

> It's easily overlooked. You'd better check your list; if it's not there,
> add it on, and that would make 2 ^ 2 ^ ∞ + 1, which is ever so slightly
> bigger.
> 

The weird thing about infinities is the difference between cardinals
such as ℵ₀ and ordinals such as 𝜔 :  2 ^ 2 ^ ℵ₀ + 1 is the same size as
2 ^ 2 ^ ℵ₀ + 1, while  2 ^ 2 ^ 𝜔 + 1 is different from (and comes just
after) 2 ^ 2 ^ 𝜔.

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#11485

On 18/11/2020 15:43, David Brown wrote:

<snip>

> The weird thing about infinities is the difference between cardinals
> such as ℵ₀ and ordinals such as 𝜔 :  2 ^ 2 ^ ℵ₀ + 1 is the same size as
> 2 ^ 2 ^ ℵ₀ + 1, while  2 ^ 2 ^ 𝜔 + 1 is different from (and comes just
> after) 2 ^ 2 ^ 𝜔.

The weird thing about Richards is that you can't always tell whether or 
not they're pulling your leg.

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

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#11486

On 18/11/2020 19:59, Richard Heathfield wrote:
> On 18/11/2020 15:43, David Brown wrote:
> 
> <snip>
> 
>> The weird thing about infinities is the difference between cardinals
>> such as ℵ₀ and ordinals such as 𝜔 :  2 ^ 2 ^ ℵ₀ + 1 is the same size as
>> 2 ^ 2 ^ ℵ₀ + 1, while  2 ^ 2 ^ 𝜔 + 1 is different from (and comes just
>> after) 2 ^ 2 ^ 𝜔.
> 
> The weird thing about Richards is that you can't always tell whether or
> not they're pulling your leg.
> 

Maybe /you/ can't tell, but /I/ can :-)

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#11469

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 ∞.

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#11472

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.

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

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#11476

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 :-)

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#11478

On 18/11/2020 14:27, David Brown wrote:

<snip>

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

No problem.

Like you, I adopted a computer-based approach.

Here's the source code, followed by the output:

me@here:~/alldata/dev/aleph$ cat aleph.c
#include <stdio.h>
#include <stdlib.h>

int main(void)
{
   unsigned long aleph = strtoul("ℵ₀", NULL, 10);
   unsigned long topowerofaleph = strtoul("2 ^ ℵ₀", NULL, 10);
   if(topowerofaleph > aleph)
   {
     printf("QED\n");
   }
   else
   {
     printf("The statement is false.\n");
   }
   return 0;
}

me@here:~/alldata/dev/aleph$ ./aleph
QED

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

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#11480

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.

-- 
Ben.

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#11482

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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#11484

David Brown wrote:
> 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.
> 
I suspect that the OP is taking about neither, though he doesn't know it.


-- 
When, once, reference was made to a statesman almost universally
recognized as one of the villains of this century, in order to
induce him to a negative judgment, he replied: "My situation is
so different from his, that it is not for me to pass judgment".
Ernst Specker on Paul Bernays

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#11487

On 18/11/2020 19:00, Peter wrote:
> David Brown wrote:
>> On 18/11/2020 16:44, Ben Bacarisse wrote:

>>> ∞ 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.
>>
> I suspect that the OP is taking about neither, though he doesn't know it.
> 

I'm not yet sure if the OP is a very sophisticated robot, or a very
unsophisticated human.  Either way, it seems more concerned with
spamming some website, or service, or something, than making sense.  But
we can still use the original post as an excuse to talk about weird
infinities and to memorise the Unicode sequences for some fun maths symbols.

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#11483

David Brown wrote:
> [...]
> 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).  

I must protest.  Expressions such as "lim_{x -> infty} f(x)", when 
replaced by their definitions, make no reference to infinity.

As for the number of lines and what-have-you in various geometric 
figures, I would refer interested parties to

	Hilbert & Cohn-Vossen /Geometry and the imagination/

first.  Hilbert's 15-th problem sought a rigorous account of the topics 
described there, and those interested parties (are there any?) may like 
to pursue the Schubert calculus.


-- 
When, once, reference was made to a statesman almost universally
recognized as one of the villains of this century, in order to
induce him to a negative judgment, he replied: "My situation is
so different from his, that it is not for me to pass judgment".
Ernst Specker on Paul Bernays

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#11488

On 18/11/2020 18:58, Peter wrote:
> David Brown wrote:
>> [...]
>> 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).  
> 
> I must protest.  Expressions such as "lim_{x -> infty} f(x)", when
> replaced by their definitions, make no reference to infinity.
> 

Indeed.  But you still typically use the symbol ∞ when writing them.

> As for the number of lines and what-have-you in various geometric
> figures, I would refer interested parties to
> 
>     Hilbert & Cohn-Vossen /Geometry and the imagination/
> 
> first.  Hilbert's 15-th problem sought a rigorous account of the topics
> described there, and those interested parties (are there any?) may like
> to pursue the Schubert calculus.
> 

I'm afraid I have no idea what Shubert calculus is - I guess a trip to
Wikipedia or Wolfram Alpha is in order...  (And while I am aware of
Hilbert's list of problems, and know of a few, I don't have the full
list in my head - and I know that for several of them I can't begin to
comprehend the question, never mind the answer.)

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