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bitRAKE
r22 wrote:
Quote: G<S(G$)>G Ref: some good reading... http://www.math.ohiostate.edu/~friedman/manuscripts.html 

11 Mar 2008, 22:33 

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) 

12 Mar 2008, 00:35 

bitRAKE
revolution wrote:


12 Mar 2008, 00:52 

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
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
I need to apologise to victor. For some reason I had misread the submission:
victor wrote: S^S(G)(G) I will try to pay more attention next time. 

13 Mar 2008, 03:04 

revolution
Plue wrote: How big do you want it? 

13 Mar 2008, 03:07 

bitRAKE
$FFFFFFFF valid hex, lol.


13 Mar 2008, 03:31 

victor
bitRAKE wrote: ...Just making it up... revolution wrote: I need to apologise to victor. 

13 Mar 2008, 03:44 

bitRAKE
victor wrote:


13 Mar 2008, 04:45 

victor
revolution wrote:
Quote: A set is uncountable if its cardinal number is larger than that of the natural numbers. Ok. Countability (countable/uncountable) and size (smaller/bigger) are two different concepts. I had the two mixed up. 

14 Mar 2008, 09:04 

edfed
ÿÿÿÿÿÿÿÿÿ
ÿ is the last char in ascii on windows. 

14 Mar 2008, 10:53 

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 N1'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
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. 

14 Mar 2008, 19:34 

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
There are a finite number of K_x which halt?
Quote: Every Turing degree contains exactly Aleph0 (that is, countably, infinitely many) sets. 

14 Mar 2008, 20:29 

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
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
edfed wrote: ÿÿÿÿÿÿÿÿÿ 

17 Mar 2008, 01:24 

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