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victor



Joined: 31 Dec 2005
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victor
Another shot: (c$)^(c$) , where c is the cardinality of the continuum. Twisted Evil
Post 07 Mar 2008, 04:49
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revolution
When all else fails, read the source


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revolution
victor wrote:
Another shot: (c$)^(c$) , where c is the cardinality of the continuum. Twisted Evil
Okay, but it is only equivalent to your previous "Beth two", not big enough to enter into the top five
Post 07 Mar 2008, 04:58
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victor



Joined: 31 Dec 2005
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victor
Quote:
...it is only equivalent to your previous "Beth two"...

Beth two = 2^c

c$ = (c!)^(c!)^(c!)^...

Hence, (c$)^(c$) should be much, much bigger than Beth two. Evil or Very Mad
Post 07 Mar 2008, 05:07
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revolution
When all else fails, read the source


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revolution
victor wrote:
Quote:
...it is only equivalent to your previous "Beth two"...

Beth two = 2^c

c$ = (c!)^(c!)^(c!)^...

Hence, (c$)^(c$) should be much, much bigger than Beth two. Evil or Very Mad
I am not an expert here, I made a mistake but I don't think it is any bigger than Beth three.

Let's break it down:

1. c$ = (c!)^(c!)^(c!)^...

2. c! = c

3. c^c = Beth-1^Beth-1 = Beth-2

4. c^c^c = Beth-1^Beth-1^Beth-1 = (Beth-1^Beth-1)^Beth-1 =
Beth-2^Beth-1 = Beth-2

So no matter how many more ^c terms you add you still get C$ = Beth-2

Therefore (c$)^(c$) = (Beth-2)^(Beth-2) = Beth-3

See here
Post 07 Mar 2008, 05:36
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Tomasz Grysztar
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Joined: 16 Jun 2003
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Tomasz Grysztar
revolution wrote:
2. c! = c

Is this defined at all?
Post 07 Mar 2008, 06:05
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edfed



Joined: 20 Feb 2006
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edfed
lim ∑x^x
x->oo

or simply:
x


Last edited by edfed on 07 Mar 2008, 19:40; edited 2 times in total
Post 07 Mar 2008, 06:17
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revolution
When all else fails, read the source


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revolution
Tomasz Grysztar wrote:
revolution wrote:
2. c! = c

Is this defined at all?
1. c! = c*(c-1)!

2. c! = c*(c-1)*(c-2)*...*1,

3. c! = 1*c*(c-1)*(c-2)*...*2, ;1*c=c

4. c! = c*(c-1)*(c-2)*...*2,

5. c! = 2*c*(c-1)*(c-2)*...*3, ;2*c=c

6. c! = c*(c-1)*(c-2)*...*3, etc.

7. c! = c*(c-1),

8. c! = c
Post 07 Mar 2008, 06:18
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revolution
When all else fails, read the source


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revolution
edfed wrote:
lim sigma(x^x)
x->oo

sigma can be considered as the sigma symbol ( 1char) , as i don't have the char tables..
Is this a proposed answer? You don't need any char tables, the answers must all be in 7bit ASCII. And your answer is more than 9 characters.

Take your pick: ∑Σ
Post 07 Mar 2008, 06:24
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Tomasz Grysztar
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Tomasz Grysztar
revolution wrote:
Tomasz Grysztar wrote:
revolution wrote:
2. c! = c

Is this defined at all?
1. c! = c*(c-1)!

2. c! = c*(c-1)*(c-2)*...*1,

3. c! = 1*c*(c-1)*(c-2)*...*2, ;1*c=c

4. c! = c*(c-1)*(c-2)*...*2,

5. c! = 2*c*(c-1)*(c-2)*...*3, ;2*c=c

6. c! = c*(c-1)*(c-2)*...*3, etc.

7. c! = c*(c-1),

8. c! = c

Are you trying to do a transfinite induction here? Then you have to assume axiom of choice and make some total order on c. And still, you also need to define some kind of transfinite multiplication in order to equations above make any sense at all.
Post 07 Mar 2008, 06:42
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revolution
When all else fails, read the source


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revolution
Tomasz Grysztar wrote:
Are you trying to do a transfinite induction here? Then you have to assume axiom of choice and make some total order on c. And still, you also need to define some kind of transfinite multiplication in order to equations above make any sense at all.
Okay, I don't know. I used the idea of things like this as my basis. Basically it suggests that u*v=max(u,v). But I may have applied it wrongly.
Post 07 Mar 2008, 06:49
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Tomasz Grysztar
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Tomasz Grysztar
revolution wrote:
Okay, I don't know. I used the idea of things like this as my basis. Basically it suggests that u*v=max(u,v). But I may have applied it wrongly.

Well, this multiplication takes just two arguments - and is associative, so you can extend it to any finite number of arguments. But that's about all. Making infinite products is not such an easy task.
Post 07 Mar 2008, 06:54
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revolution
When all else fails, read the source


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revolution
I can find nothing to support a different answer than c!=c. How about this, c^c = 2^c = Beth-2, we are given this, and c! < c^c, and by GCH there are no cardinals strictly between c and 2^c, therefore, with c! less than c^c it can only take the value c.
Post 07 Mar 2008, 08:44
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Tomasz Grysztar
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Tomasz Grysztar
revolution wrote:
I can find nothing to support a different answer than c!=c. How about this, c^c = 2^c = Beth-2, we are given this, and c! < c^c, and by GCH there are no cardinals strictly between c and 2^c, therefore, with c! less than c^c it can only take the value c.

The hasn't be to be answer at all. The value of c! doesn't even exist until we define the transfinite meaning of "!". And we may have different definitions leading to different results. How do you know that c!<c^c when you haven't even defined the c! yet?
Post 07 Mar 2008, 08:58
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revolution
When all else fails, read the source


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revolution
Tomasz Grysztar wrote:
The value of c! doesn't even exist until we define the transfinite meaning of "!".
Okay, I get your point. But how to solve it? It would seem logical as an extension of the typical factorial where n! < n^n if n>1. Is it not allowed to take that deduction and apply it to c?

Does the following hold?

c^c = 2^c

c^(c-1) < c^c and therefore,

c^(c-1) < 2^c

Perhaps the "c-1" thing is also not defined/allowed? Anybody know?
Post 07 Mar 2008, 09:05
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Tomasz Grysztar
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Tomasz Grysztar
revolution wrote:
Perhaps the "c-1" thing is also not defined/allowed? Anybody know?

Even for ordinals this wouldn't work - there are ordinals that don't have a preceding element, they are called the limit ordinals.
Transfinite induction doesn't work with just a +1 steps, as they cannot get you to the points after the omega.
Post 07 Mar 2008, 09:11
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revolution
When all else fails, read the source


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revolution
So does that mean I have to disqualify the entry "(c$)^(c$)" because it is not well defined?
Post 07 Mar 2008, 09:13
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Tomasz Grysztar
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Tomasz Grysztar
Yes, unless we are given some acceptable definition of $ operator for cardinals.

Still, as I already wrote above, I don't think that allowing both ordinals and cardinals is a good idea.
Post 07 Mar 2008, 09:19
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revolution
When all else fails, read the source


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revolution
Tomasz Grysztar wrote:
Still, as I already wrote above, I don't think that allowing both ordinals and cardinals is a good idea.
You are currently the head of the non-Aleph list, but there are higher numbers than the ones currently given.

Code:
non-Aleph:
5. 9$$$$$$$$       bitRAKE
4.   9($^9$$$)       Tomasz Grysztar
3.   G($^G$$$)       Tomasz Grysztar (belatedly accepted)
2.      BB(BB(9))       Tomasz Grysztar
1.   (BB^9)(9)       Tomasz Grysztar (belatedly accepted)

Purely mathematical:
5.      Beth-Z$^Z       bitRAKE
4.   Aleph-9$$       Tomasz Grysztar
3.   Aleph-F$$       MHajduk
2.   Aleph-Z$$       MHajduk
1.   Beth-Z$$$       bitRAKE    
Post 07 Mar 2008, 09:25
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Tomasz Grysztar
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Tomasz Grysztar
To take further the discussion of problems I mentioned: let's now take the ring of 2-adic integers, Z_2. Since there's no natural order there, we have to make up some order, otherwise we wouldn't be able to tell whether some number is bigger compared to the other one.
I propose this very simple and intuitive order:
For 2-adic integers a=[a_0 a_1 a_2 ...] and b=[b_0 b_1 b_2 ...] we've got a<=b iff there exists i such that a_j=b_j for j<i and a_i<=b_i.
So 0=[000...]<1=[1000...], 2=[01000...]<3=[11000...], but 1=[1000...]>2=[01000...]

Now what biggest number can we get here? Well, it's -1=[11111...].
It's the maximal element in this order, and thus there exist no element greater that it. Wink
Post 07 Mar 2008, 10:33
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revolution
When all else fails, read the source


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revolution
Okay, you're describing a bit-reflected version of standard integer binary. But even a non-bit-reflected version still has -1 as the "greatest" element.

Can you point out if I have made a mistake in this puzzle?

Specifically have I made a comparison error in the two lists? Maybe comparing things that can't be compared? Or allowing something that makes no sense?

Or should I define an extra parameter (a rule) that further submissions must meet?
Post 07 Mar 2008, 10:43
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