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Author
r22

Joined: 27 Dec 2004
Posts: 805
r22
Quote:
S^{G}G(G)

Are you compositing S() to the order of a Gth dimensional polytope?
I call NaN on {G}G

x<y>z notation is interesting

x<2>z == x*z
== x + ... + x (z x's)

x<3>z == x^z
== x * ... * x (z x's)

x<4>z == x^^z
== x^...^x (z x's)

x<5>z == x^^^^z ??bitRAKE ascii notation??
== (x^^z)^^..^^(x^^z) (z (x^^z)'s)
If this is the case then...

G<S(G\$)>G
11 Mar 2008, 19:02
bitRAKE

Joined: 21 Jul 2003
Posts: 2915
Location: [RSP+8*5]
bitRAKE
r22 wrote:
Quote:
S^{G}G(G)

Are you compositing S() to the order of a Gth dimensional polytope?
That was the intent - applying {G} to the ^n() notation dreamt up previously. Whereas, S^^^^G(G) = S^{4}G(G).

Quote:
G<S(G\$)>G
...well done. Wouldn't: S(G\$)<G>G be larger? Still much smaller than the composition of S() though.

http://www.math.ohio-state.edu/~friedman/manuscripts.html

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11 Mar 2008, 22:33
revolution
When all else fails, read the source

Joined: 24 Aug 2004
Posts: 17279
revolution
Tetration and x<y>z things are good for generating small numbers. But they can't compete with the really big numbers.
bitRAKE wrote:
S^{G}G(G)
A G-dimensional hyper cube of size G of S() sounds interesting. Where did you find the {} notation?
12 Mar 2008, 00:35
bitRAKE

Joined: 21 Jul 2003
Posts: 2915
Location: [RSP+8*5]
bitRAKE
revolution wrote:
bitRAKE wrote:
S^{G}G(G)
A G-dimensional hyper cube of size G of S() sounds interesting. Where did you find the {} notation?
Just making it up - not that it isn't used somewhere. I've never seen the F^n() notation either. New directions are always fun.

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12 Mar 2008, 00:52
Tomasz Grysztar
Assembly Artist

Joined: 16 Jun 2003
Posts: 7724
Location: Kraków, Poland
Tomasz Grysztar
bitRAKE wrote:
I've never seen the F^n() notation either.

http://en.wikipedia.org/wiki/Iterated_function
This is a widely used notation - have you ever wondered why the inverse function is noted as f^(-1)?

BTW, I just found this article. Might be useful for our transfinite part, if only could grab a copy.
12 Mar 2008, 08:18
Plue

Joined: 15 Dec 2005
Posts: 151
Plue
Quote:
But it doesn't matter really, because, so far, no one has come even close to putting a REALLY big number.

How big do you want it?

(Hex:) F^FFFFFF!
12 Mar 2008, 19:32
revolution
When all else fails, read the source

Joined: 24 Aug 2004
Posts: 17279
revolution
I need to apologise to victor. For some reason I had misread the submission:
victor wrote:
S^S(G)(G)
And of course this is the current leader in the list.

I will try to pay more attention next time.
13 Mar 2008, 03:04
revolution
When all else fails, read the source

Joined: 24 Aug 2004
Posts: 17279
revolution
Plue wrote:
How big do you want it?

(Hex:) F^FFFFFF!
I want it really big. You'll need to make it larger than that if you want to make the top 5.
13 Mar 2008, 03:07
bitRAKE

Joined: 21 Jul 2003
Posts: 2915
Location: [RSP+8*5]
bitRAKE
\$FFFFFFFF valid hex, lol.

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13 Mar 2008, 03:31
victor

Joined: 31 Dec 2005
Posts: 126
Location: Utopia
victor
bitRAKE wrote:
...Just making it up...
bitRAKE is a funny guy! Maybe I should also start making up my own functions, e.g., R(n) is defined to be the total number of sperms ever produced by revolution throughout his life ...

revolution wrote:
I need to apologise to victor.
Never mind.
13 Mar 2008, 03:44
bitRAKE

Joined: 21 Jul 2003
Posts: 2915
Location: [RSP+8*5]
bitRAKE
victor wrote:
bitRAKE wrote:
...Just making it up...
bitRAKE is a funny guy! Maybe I should also start making up my own functions
I hope I'm not the first to break it to you, but it is all made up. No, really - it is! Even that "1+1=2" thing - totally made up. Have you ever seen a "1"? Which one? ..and it wasn't like the other "1" at all. So, it was more like "1+1=11" than "1+1=2". Only guys with money want you to believe "1+1=2" - just imagine if you really knew "1+1=11"!

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13 Mar 2008, 04:45
victor

Joined: 31 Dec 2005
Posts: 126
Location: Utopia
victor
revolution wrote:
victor wrote:
Epsilon nought, E0, is a countable ordinal. So are Ew, EEw, and EEEw. Correct?

w_1, w with subscript 1, is the smallest uncountable ordinal. w_w is the limit of the w_n for natural numbers n.

So, under the "Transfinites" section, Beth w_w should be much, much bigger than Beth EEEw. Correct?
I'm not sure here, but I think that you are confusing countable/uncountable with smaller/larger.

Quote:
A set is uncountable if its cardinal number is larger than that of the natural numbers.
Refer to this.

Ok. Countability (countable/uncountable) and size (smaller/bigger) are two different concepts. I had the two mixed up.
14 Mar 2008, 09:04
edfed

Joined: 20 Feb 2006
Posts: 4237
Location: 2018
edfed
ÿÿÿÿÿÿÿÿÿ

ÿ is the last char in ascii on windows.
14 Mar 2008, 10:53
r22

Joined: 27 Dec 2004
Posts: 805
r22
Kleene’s hierarchy: - > Turing Degrees -> hypercomputer -> halting problem
http://en.wikipedia.org/wiki/Post%27s_theorem
http://en.wikipedia.org/wiki/Oracle_Turing_machine
http://en.wikipedia.org/wiki/Hypercomputer
http://en.wikipedia.org/wiki/Halting_problem

So the largest possible number in 9 ASCII characters is ...
Code:
`S_ZM(K_i) or ZM^S(K_i) in think the latter syntax is more obvious    `

The number of (S)teps for a (Z)eno (M)achine to solve the set (K_i) of all solvable programs for the Halting Problem
Quote:
K := { (i, x) | program i will eventually halt if run with input x}.

Rationale:
Regular Turing machines can't solve the halting problem, but hyper computers are theoretically able to do so. K is the set of program & input pairs of turning degree. Have a Turing machine of order N can always solve order N-1's indeterminable problems. Since a Zeno Machine is capable of solving problems of a higher or equal complexity to even Busy Beaver it stands that feeding it K_i (all solvable Turing level programs) it can output a number larger than any other. I think even the count of K (|K|) would be be one of the largest calculable numbers.
14 Mar 2008, 16:28
bitRAKE

Joined: 21 Jul 2003
Posts: 2915
Location: [RSP+8*5]
bitRAKE
The number sounds akin to my CS(), Community Beaver - which could also be defined as: "The number of (S)teps" {} "to solve the set" {} "of all solvable programs for the Halting Problem".

Invoking a hypothetical computational model (Zeno Machine) is pure genius!
Quote:
Unfortunately, though, the naive approach of Zeno machines to infinite calculations leads to problems. For example, unlike with a Turing machine, the output of a Zeno machine after it finishes its computation (i.e., after one time unit) need not be in a defined state; this can happen if the machine continues to alternatively write different outputs, for example. Other complications include undefined internal states of the machine, writeheads that "run away to infinity" and other such things.
How can you say the number is defined?

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14 Mar 2008, 19:34
r22

Joined: 27 Dec 2004
Posts: 805
r22
Quote:
How can you say the number is defined?

I figured since the input consists of all the Solvable programs, then the output would be constant.

But even if the ZM's output is undefined the function is counting the steps to reach the output; not the solutions K_x.

Last edited by r22 on 15 Mar 2008, 01:17; edited 1 time in total
14 Mar 2008, 20:11
bitRAKE

Joined: 21 Jul 2003
Posts: 2915
Location: [RSP+8*5]
bitRAKE
There are a finite number of K_x which halt?
Quote:
Every Turing degree contains exactly Aleph-0 (that is, countably, infinitely many) sets.

_________________
14 Mar 2008, 20:29
r22

Joined: 27 Dec 2004
Posts: 805
r22
K_i is the Solvable program.
K_x would be the input of the program, and the output of the ZM

So there can be infinitely many K_x's.
But by just using the {K_i} set I constrain the input to one instance of a solvable program.

But then, yes, I still have a set of all solvable programs that seems infinite by simple induction.

My ZM will need to be upgraded with an Oracle that tells it to stop after it's won this time consuming and entertaining contest.

If there was a way to measure the complexity of a Turing machine of degree ~G\$\$, then that value would win.
15 Mar 2008, 01:41
bitRAKE

Joined: 21 Jul 2003
Posts: 2915
Location: [RSP+8*5]
bitRAKE
If there was a limiting factor (I wasn't sure) then I was going to attack the definition of a ZM step - which hypothetically need not be greater than one.

_________________
15 Mar 2008, 02:11
revolution
When all else fails, read the source

Joined: 24 Aug 2004
Posts: 17279
revolution
edfed wrote:
ÿÿÿÿÿÿÿÿÿ

ÿ is the last char in ascii on windows.
You are still not following the rules, 0x20-0x7E are the only valid characters, it's stated in the first post.
17 Mar 2008, 01:24
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