Talk: Fish number 7

\(Rayo^n(n)\)

What if, \(f_0(n)\) is equal to \(Rayo(n)\)?

\(f_1(n)\) is equal to \(Rayo^n(n)\)! Aarex^Tiao 16:23, January 1, 2014 (UTC)

Yes. This extension was already considered, and I'm pretty sure you considered it too. It is just shifted FGH. LittlePeng9 (talk) 17:42, January 1, 2014 (UTC)

BOOOR-ing. FB100Z • talk • contribs 22:59, January 1, 2014 (UTC)

Defining notation showing its power

It certainly must grow much faster than Rayo's function, as \(R_\alpha(n)\) is not a shifted FGH which would mean than \(R_\alpha(n) = f_{\beta+\alpha}(n)\) for some \(f_\beta \approx Rayo(n)\). But even Rayo's function is still an uncharted territory, we have to define more explicit notation which shows its power. Ikosarakt1 (talk ^ contribs) 09:43, May 28, 2014 (UTC)

In first undefinable ordinal, it is written that "The growth rate of Rayo's function is conjectured to be related to this ordinal." If this conjecture, \(Rayo(n) \approx f_{\omega_1^\text{DEF}}(n)\) is true, then we might be able to approximate as \(F_7 \approx f_{\omega_{\zeta_0}^\text{DEF}}^{63}(10^{100})\). As it is just a conjecture, I did not write it in the main text. Kyodaisuu (talk) 09:56, May 28, 2014 (UTC)

First undefinable ordinal, by the way, must be the first ordinal which cannot be defined by any formal system. Although I don't understand first order logic, I guess it is the formal system, and n-order logic too.

Well, every notation has its own first undefinable ordinal. We can say that, for example, with addition, exponentiation, 1 and \(\omega\) the first undefinable ordinal is \(\varepsilon_0\). If we define binary Veblen's function, then it would be \(\Gamma_0\), etc. The better term for the ordinal which I meant is the "first formally undefinable ordinal." Ikosarakt1 (talk ^ contribs) 10:04, May 28, 2014 (UTC)

I agree. As the "first undefinable ordinal" can vary with the formal system that is used, the notation of DEF would be better to be changed to something which represents the name of the formal system. As it is in the language of first-order logic, the symbol which represents the first-order logic would be more preferable. Something like \(\omega_1^\text{FOL}\). Kyodaisuu (talk) 10:11, May 28, 2014 (UTC)

Fish number 7 function indeed does grow much faster than Rayo function. I removed the mentioned claim in first undefinable ordinal article, because Rayo's function probably outgrows it a lot. There is also no ordinal undefinable in any formal system, as there are formal systems allowing infinitely long sentences. I also don't think that \(\omega_{\zeta_0}^\text{DEF}\) is a right notation for ordinal you think about, because for me it looks like "\(\zeta_0\)'th undefinable ordinal, which is just \(\omega_1^\text{DEF}+\zeta_0\). LittlePeng9 (talk) 13:46, May 28, 2014 (UTC)

Hmm... so anyway there seems to be no way to approximate Rayo(n) with FGH right now. We must switch to Rayo hierarchy. Kyodaisuu (talk) 14:35, May 28, 2014 (UTC)

"There is also no ordinal undefinable in any formal system, as there are formal systems allowing infinitely long sentences." - infinitely, but countably. If it turns out that for any sequence of formal systems there would be always a formal system which diagonalizes through it, I think \(\omega_1\) is actually the first formally undefinable ordinal for which I'm looking. Ikosarakt1 (talk ^ contribs) 15:34, May 28, 2014 (UTC)

There is uncountable many infinite formal systems. LittlePeng9 (talk) 15:43, May 28, 2014 (UTC)

Was the page of first undefinable ordinal deleted? Kyodaisuu (talk) 16:06, May 28, 2014 (UTC)

I see that Ikosarakt1 deleted it. And now only the Japanese version remains... Kyodaisuu (talk) 16:18, May 28, 2014 (UTC)

Yes, we agreed (on the chat) to delete the article, because its name makes people think of the truly undefinable ordinals, and not undefinable in some formal system, which leads to pretty serious paradoxes. LittlePeng9 (talk) 16:29, May 28, 2014 (UTC)

I see. Kyodaisuu (talk) 16:34, May 28, 2014 (UTC)

Also it was made up by members of this wiki and it's not in any sources. you're.so.pretty! 20:41, May 28, 2014 (UTC)

Is it better than FGH'?

FGH' has all rules same as FGH, but \(f_0\) is Rayo's function. Is \(R_2>^*f_{\omega_1^\text{CK}}\) using FGH'? 80.98.179.160 18:35, December 24, 2017 (UTC)

If one can write a fundamental sequence of \(\omega_1^\text{CK}\) using FOST, which is equivalent to writing busy beaver function in FOST, implementing \(R_1(n)\) to that expression may produce the strength of the proposed function and therefore \(R_2>^*f_{\omega_1^\text{CK}}\). I am not quite sure if this kind of discussion is quite meaningful, though. 🐟 Fish fish fish ... 🐠 19:31, December 24, 2017 (UTC)

Is Fish number 7 naive extension of Rayo's number?

(this is in regard to recent reversion on Rayo's number page, I thought it'll be better to discuss this here)

I don't think this is a naive extension. The idea of using functional oracle "f(a)=b" is, in my opinion, an idea which is not just a trivial recursion applied to Rayo function, but a genuine extension of the language which makes it significantly stronger.

Please share your opinions. LittlePeng9 (talk) 15:53, July 1, 2015 (UTC)

I agree, it isn't a naive extension for me Fluoroantimonic Acid (talk) 16:01, July 1, 2015 (UTC)

i'm going by vell's word Cookiefonster (talk) 16:17, July 1, 2015 (UTC)

It is naive relative to BIG FOOT, but it certainly inspired LittlePeng9 to develop the BIG FOOT. I feel that BIG FOOT is too sophisticated to be a measure of naiveness. 🐟 Fish fish fish ... 🐠 16:39, July 1, 2015 (UTC)

From our article on naive extension, to qualify as a naive extension something needs to add no new insight. By this definition, I don't consider the Fish number 7 a naive extension. -- vel! 07:07, July 3, 2015 (UTC)

How strong is \(R_2\) function?

It seems that \(R_2\) approximately works as "Rayo's function with an oracle \(R_1\) function". But how strong is it? Can we use some ordinal to express it? {hyp/^,cos} (talk) 13:01, August 1, 2018 (UTC)

I guess the growth rate of \(R_2\) is \(\alpha\times2\) when \(R_1\) has growth rate \(\alpha\), like how the growt rate of oracle BB is \(\omega^CK_1\times2\).Rpakr (talk) 13:05, August 1, 2018 (UTC)

It is more like ω^CK_2, rather than ω^CK_1 * 2. As we do not have ordinal expression for Rayo function, we cannot use ordinal expression for R2. I remember that once ω^def or something like this was posted on Gwiki, but the article was deleted because it has no source of mathematical soundness. 🐟 Fish fish fish ... 🐠 15:47, August 1, 2018 (UTC)

"It is more like ω^CK_2" Why? Rpakr (talk) 22:10, August 1, 2018 (UTC)

R2 is similar to second order Turing machine because both of them give oracle to the system (Turing machine or FOST) and measure the strength of the system. 🐟 Fish fish fish ... 🐠 05:57, August 2, 2018 (UTC)

And how does that relate to the growth rates of oracle turing machine BB being ω^CK_2? I disagree with that. Rpakr (talk) 22:21, August 2, 2018 (UTC)

And how do you define the oracle BB and estimate its growth rate as \(\omega_1^{\textrm{CK}} \times 2\)? -- P-adic p-adic p-adic... 01:07, August 3, 2018 (UTC)

I define oracle BB like shown in the BB function article (A special "ask" state, in which the TM counts the 1's on the tape (assume it had n 1's), "see" if TM #n halts, and goes to "yes" state if it halts, and to "no" state if not.) I estimated its growth rate as \(\omega_1^{CK}\times2\) because it can calculate the BB function and any recursive extention of that (like f(n)=BB^{BB^n(n)}(n)) but not the limit of these which has growth rate \(\omega_1^{CK}\times2\).Rpakr (talk) 19:08, August 3, 2018 (UTC)

So you mean if a function \(f\) has growth rate \(\geq \alpha\), then a function greater than any recursive extensions of \(f\) has growth rate \(\geq \alpha \times 2\), right? How do you estimate it? -- P-adic p-adic p-adic... 22:14, August 3, 2018 (UTC)

I was probably wrong. Maybe it is \(\alpha+\omega^{CK}_1\) instead of \(\alpha\times2\) Rpakr (talk) 22:30, August 3, 2018 (UTC)