## WelcomeEdit

Hi, welcome to Googology Wiki! Thanks for your edit to the **Talk:Goodstein sequence** page.

Please leave a message on my talk page if I can help with anything! -- FB100Z (Talk) 03:39, September 20, 2012

## Strength of & operatorEdit

Hey, Deedlit. You probably thought that \(f_{\Gamma_0}(n) \approx \{n,n / 2\}\) from the reason that:

\(f_\omega(n) \approx \{n,n,n\}\) and \(f_{\varphi(\omega,0)}(n) \approx \{X,X,X\} \&\ n\).

\(f_{\omega+1}(n) \approx \{n,n,1,2\}\) and \(f_{\varphi(\omega+1,0)}(n)\) therefore should be \(\{X,X,1,2\} \&\ n\).

That seems reasonable, but we can look at that by the other way: \(f_\varphi(\alpha,0)(n)\) is the:

\(\alpha+1\)-th hyper-operator applied to X's. (When \(\alpha\) is a finite number)

\(\alpha\)-th hyper-operator applied to X's. (When \(\alpha\) is a transfinite ordinal)

So f_{\(f_\varphi(n,0)\)}(n) corresponds to \(\{X,X,n+1\} \&\ n\) and \(f_\varphi(\omega,0)\)

Next look at \(\varphi(\omega+1,0)\). It is equal to \(f_\varphi(\omega,\varphi(\omega,\cdots,\varphi(\omega,0))...))\) (with \(\omega\text{ }\varphi\)'s (in the more compact way, it is the first ordinal such that \(\alpha\) = \(\varphi(1,\alpha)\)). Hence, it behaves in the same way as, say \(\varphi(2,0) = \alpha = \varphi(1,\alpha)\). We can pretty confident that \(\varphi(\omega+1,0)\) corresponds to \(\{X,X,X+1\} = \{X,p,X+1\} = \{X,\{X,p-1,X\},X+1\}\) structure.

When the one structure represents the hyper-operator of the other, X times, we reach expandal arrays, which is surely corresponds to \(\varphi(1,0,0) = \Gamma_0\).

More generally, \(\varphi\) function with n entries behaves at the same as linear arrays with n-1 entries. Thus, \(\{X,X (1) 2\} \&\ n\} \approx f_{\vartheta(\Omega^\omega)}(n)\). Chris Bird received the same results, and after more considerations you can see that limit ordinal of the non-legion arrays is the \(\vartheta(\Omega^\Omega)\), the Large Veblen Ordinal.

To be clear, I just present the selection of some comparisons:

\(X \uparrow\uparrow X \&\ n \approx f_{\varepsilon_0}(n)\)

\(X \uparrow\uparrow\uparrow X \&\ n \approx f_{\zeta_0}(n)\)

\(X \uparrow\uparrow\uparrow\uparrow X \&\ n \approx f_{\eta_0}(n)\)

\(\{X,X,X\} \&\ n \approx f_{\varphi(\omega,0)}(n)\)

\(\{X,X,X+1\} \&\ n \approx f_{\varphi(\omega+1,0)}(n)\)

\(\{X,X,2X\} \&\ n \approx f_{\varphi(\omega 2,0)}(n)\)

\(\{X,X,X^2\} \&\ n \approx f_{\varphi(\omega^2,0)}(n)\)

\(\{X,X,X^X\} \&\ n \approx f_{\varphi(\omega^\omega,0)}(n)\)

\(\{X,X,X \uparrow\uparrow X\} \&\ n \approx f_{\varphi(\varepsilon_0,0)}(n)\)

\(\{X,X,\{X,X,X\}\} \&\ n \approx f_{\varphi(\varphi(\omega,0),0)}(n)\)

\(\{X,X,1,2\} \&\ n \approx f_{\varphi(1,0,0)}(n)\)

\(\{X,X,2,2\} \&\ n \approx f_{\varphi(1,1,0)}(n)\)

\(\{X,X,3,2\} \&\ n \approx f_{\varphi(1,2,0)}(n)\)

\(\{X,X,X,2\} \&\ n \approx f_{\varphi(1,\omega,0)}(n)\)

\(\{X,X,\{X,X,X\},2\} \&\ n \approx f_{\varphi(1,\varphi(\omega,0),0)}(n)\)

\(\{X,X,\{X,X,1,2\},2\} \&\ n \approx f_{\varphi(1,\varphi(1,0,0),0)}(n)\)

\(\{X,X,1,3\} \&\ n \approx f_{\varphi(2,0,0)}(n)\)

\(\{X,X,1,4\} \&\ n \approx f_{\varphi(3,0,0)}(n)\)

\(\{X,X,1,X\} \&\ n \approx f_{\varphi(\omega,0,0)}(n)\)

\(\{X,X,1,\{X,X,1,2\}\} \&\ n \approx f_{\varphi(\varphi(1,0,0),0,0)}(n)\)

\(\{X,X,1,1,2\} \&\ n \approx f_{\varphi(1,0,0,0)}(n)\)

\(\{X,X,1,1,1,2\} \&\ n \approx f_{\varphi(1,0,0,0,0)}(n)\)

\(\{X,X (1) 2\} \&\ n \approx f_{\vartheta(\Omega^\omega)}(n)\)

\(\{X,X,2 (1) 2\} \&\ n \approx f_{\vartheta(\Omega^\omega+1)}(n)\)

\(\{X,X,3 (1) 2\} \&\ n \approx f_{\vartheta(\Omega^\omega+2)}(n)\)

\(\{X,X,X (1) 2\} \&\ n \approx f_{\vartheta(\Omega^\omega+\omega)}(n)\)

\(\{X,X,1,2 (1) 2\} \&\ n \approx f_{\vartheta((\Omega^\omega)2)}(n)\)

\(\{X,X,1,3 (1) 2\} \&\ n \approx f_{\vartheta((\Omega^\omega)3)}(n)\)

\(\{X,X,1,X (1) 2\} \&\ n \approx f_{\vartheta((\Omega^\omega)\omega)}(n)\)

\(\{X,X,1,1,2 (1) 2\} \&\ n \approx f_{\vartheta((\Omega^\omega)\vartheta(\Omega))}(n)\)

\(\{X,X,1,1,1,2 (1) 2\} \&\ n \approx f_{\vartheta((\Omega^\omega)\vartheta(\Omega^2))}(n)\)

\(\{X,X (1) 3\} \&\ n \approx f_{\vartheta((\Omega^\omega)\vartheta(\Omega^\omega))}(n)\)

\(\{X,X (1) X\} \&\ n \approx f_{\vartheta(\Omega^{\omega+1})}(n)\)

Notice that all these comparisons looks very similar with the regular arrays, where \(\{n,n (1) n\}\) represented by \(X+1\) structure. In general, if we have A structure (in BEAF), then \(A \&\ n \approx f_{\vartheta(\Omega^\alpha)}(n)\), where \(\alpha\) is the ordinal that associated with A. So, continuing onwards:

\(\{X,X (1) X,X\} \&\ n \approx f_{\vartheta(\Omega^{\omega+2})}(n)\)

\(\{X,X (1) X,X,X\} \&\ n \approx f_{\vartheta(\Omega^{\omega+3})}(n)\)

\(\{X,X (1)(1) 2\} \&\ n \approx f_{\vartheta(\Omega^{\omega 2})}(n)\)

\(\{X,X (1)(1)(1) 2\} \&\ n \approx f_{\vartheta(\Omega^{\omega 3})}(n)\)

\(\{X,X (2) 2\} \&\ n \approx f_{\vartheta(\Omega^{\omega^2})}(n)\)

\(\{X,X (3) 2\} \&\ n \approx f_{\vartheta(\Omega^{\omega^3})}(n)\)

\(\{X,X (0,1) 2\} \&\ n \approx f_{\vartheta(\Omega^{\omega^\omega})}(n)\)

\(\{X,X (1,1) 2\} \&\ n \approx f_{\vartheta(\Omega^{\omega^{\omega+1}})}(n)\)

\(\{X,X (0,2) 2\} \&\ n \approx f_{\vartheta(\Omega^{\omega^{\omega 2}})}(n)\)

\(\{X,X (0,0,1) 2\} \&\ n \approx f_{\vartheta(\Omega^{\omega^{\omega^2}})}(n)\)

\(\{X,X (0,0,0,1) 2\} \&\ n \approx f_{\vartheta(\Omega^{\omega^{\omega^3}})}(n)\)

\(\{X,X ((1) 1) 2\} \&\ n \approx f_{\vartheta(\Omega^{\omega^{\omega^\omega}})}(n)\)

\(\{X,X ((0,1) 1) 2\} \&\ n \approx f_{\vartheta(\Omega^{\omega^{\omega^{\omega^\omega}}})}(n)\)

\(X \uparrow\uparrow X \&\ n \&\ n \approx f_{\vartheta(\Omega^{\vartheta(1)})}(n)\)

\(X \uparrow\uparrow\uparrow X \&\ n \&\ n \approx f_{\vartheta(\Omega^{\vartheta(2)})}(n)\)

\(X \uparrow\uparrow\uparrow\uparrow X \&\ n \&\ n \approx f_{\vartheta(\Omega^{\vartheta(3)})}(n)\)

\(\{X,X,X\} \&\ n \&\ n \approx f_{\vartheta(\Omega^{\vartheta(\omega)})}(n)\)

\(\{X,X (1) 2\} \&\ n \&\ n \approx f_{\vartheta(\Omega^{\vartheta(\Omega^\omega)})}(n)\)

\(\{X,X (2) 2\} \&\ n \&\ n \approx f_{\vartheta(\Omega^{\vartheta(\Omega^{\omega^2})})}(n)\)

\(\{X,X (3) 2\} \&\ n \&\ n \approx f_{\vartheta(\Omega^{\vartheta(\Omega^{\omega^3})})}(n)\)

\(\{X,X (0,1) 2\} \&\ n \&\ n \approx f_{\vartheta(\Omega^{\vartheta(\Omega^{\omega^\omega})})}(n)\)

\(\{X,X ((1) 1) 2\} \&\ n \&\ n \approx f_{\vartheta(\Omega^{\vartheta(\Omega^{\omega^{\omega^\omega}})})}(n)\)

\(\{X,X ((0,1) 1) 2\} \&\ n \&\ n \approx f_{\vartheta(\Omega^{\vartheta(\Omega^{\omega^{\omega^{\omega^\omega}}})})}(n)\)

\(X \uparrow\uparrow X \&\ n \&\ n \&\ n \approx f_{\vartheta(\Omega^{\vartheta(\Omega^{\vartheta(1)})})}(n)\)

\(X \uparrow\uparrow X \&\ n \&\ n \&\ n \&\ n \approx f_{\vartheta(\Omega^{\vartheta(\Omega^{\vartheta(\Omega^{\vartheta(1)})})})}(n)\)

All that limits to the Large Veblen Ordinal, \(\vartheta(\Omega^\Omega)\). Ikosarakt1 (talk ^ contribs) 20:28, March 24, 2013 (UTC)

If you are right, what ordinal may be limit of BEAF? If we reach LVO without even && operator, I wonder how big meameamealokkapoowa oompa may really be. LittlePeng9 (talk) 21:36, March 24, 2013 (UTC)

Yes, I have recently come to the same conclusion concerning \(\{X,X,1,2\}\) and \(\Gamma_0\). I haven't examined linear arrays and above yet, but your conclusions seem reasonable. Of course, we need a rigorous definition for Bowers' arrays to be sure.

TREE(3) will have to be moved down the chain, to around the Small Veblen ordinal. (We don't know the exact ordinal for TREE(3) but I guess we will assume that it is not much larger than the Small Veblen ordinal.) Deedlit11 (**talk**) 22:01, March 24, 2013 (UTC)

@LittlePeng9: Chris Bird analyzed Bowers' notation and concluded that it ends well below the Bachmann-Howard ordinal, and I think below \(\vartheta (\Omega^{\Omega^{\Omega}})\). Deedlit11 (**talk**) 00:05, March 25, 2013 (UTC)

Some measurements for legion arrays:

If {n,n/2} is at the level of the LVO, then {n,n/3} is at the level of LVO*2, {n,n/4} is at the level of {n,n/4} is at the level of LVO*3, and {n,n/1+F} is at the level of LVO*alpha, where we take alpha to be the ordinal associated to F. So {n,n / 1 / 2} will be at the level of LVO^2, {n,n / 1 / 1 / 2} will be at the level of LVO^3, {n,n (/1) 2} will be at the level of LVO^omega, and {n,n (/F) 2} will be at the level of LVO^(omega^alpha). Thus n && n && n will be at the level of LVO^LVO, and {n,n // 2} will be at the level of epsilon_{LVO+1}. We go through the same hierarchy up with //'2 so we get that {n, n /// 2} is at the level of epsilon_{LVO+2}. In general, {L, X^F} = {n,n (F)/ 2} will be at the level of epsilon_{LVO + alpha}. It follows that {L, L} = epsilon_{epsilon_{LVO+1}}, and probably {L, L, X} = phi(2, LVO+1). It's not clear to me how higher arrays are defined in terms of L - it seems like very bad notation to me - but it seems reasonable that {L, L, ..., L, X} = phi(n, LVO+1). So L2 space is at the level of Gamma_{LVO+1} = phi(1, 0, LVO+1). L3 space will be at level phi(1, 0, LVO+2), LF space will be at level phi(1, 0, LVO + alpha). LLF space will perhaps be at level phi(1, 1, LVO + alpha) (but again, I would need a precise definition of what LF is), and perhaps (L^F)G will be at phi(1, alpha, LVO + beta). So (L^L)F would be at level phi(2, 0, LVO + alpha).

Alternatively, perhaps {L, L, ..., L} is at level phi (1, 0, ..., 0, LVO+1}, with the same number of terms in each. So L2-space will be the second ordinal fixed by the Schutte Klammersymbolen, i.e. \(\theta(\Omega^{\Omega},1)\). In general, LF space will be the alphath ordinal fixed by the Schutte Kalmmersymbolen, i.e. \(\theta(\Omega^{\Omega}, \alpha)\). So perhaps LLF space will be \(\theta(\Omega^{\Omega}+1, \alpha)\), and in general (L^F)G space will be \(\theta(\Omega^{\Omega}+\alpha, \beta)\), and thus the closure space (I guess we are calling this L^L) will be at the level of \(\theta (\Omega^{\Omega} + \Omega, 0)\). So we are very far from \(\theta(\Omega^{\Omega^{\Omega}}, 0)\). Deedlit11 (**talk**) 01:54, March 25, 2013 (UTC)

But in BEAF there are array spaces such absurdal as L & L. Also, is there any reading about Klammersymbolen? I understand general concept, but nothing more LittlePeng9 (talk) 06:33, March 25, 2013 (UTC)

- L & L? My analysis went well past the "&" operator.
- I couldn't find much on the web on the Klammersymbolen, except for:

- http://www.cs.man.ac.uk/~hsimmons/ORDINAL-NOTATIONS/FromBelow.pdf
- http://www.cs.swan.ac.uk/~csetzer/articles/ordsyscor010124.ps

- Both of those papers generalize the Klammersymbolen so they are harder to read than need be.
- But, the notation is not that complicated; it's just an extension of the Extended Veblen notation into the transfinite.
- To do that, we need to explicitly name the place values of the variables. For example, instead of phi(8, 0, 0, 0, 3, 0, 5, 6, 4), represent it as (8 @ 8, 3 @ 4, 5 @ 2, 6 @ 1, 4 @ 0). We can represent all Extended Veblen notations this way, but we can also say (1 @ omega, c @ 0), which is defined as the cth ordinal that satisfies (x @ n, 0 @ 0) = x for all n < omega.
- More generally, define (a_1 @ b_1, a_2 @ b_2, ..., a_n @ b_n, c @ 0) as the cth ordinal that satisfies the system of equalities (a_1 @ b_1, a_2 @ b_2, ..., a_{n-1} @ b_{n-1}, d @ b_n, x @ e) = x for all d < a_n, e < b_n. Deedlit11 (
**talk**) 03:50, March 26, 2013 (UTC)

## Array ofEdit

We found out that the & is the array of. And i am sorry for the confusion that we caused in BEAF. ^{\(a\)\(l\)\(t\)} 04:19, April 14, 2013 (UTC)

But you never said is array of... ^{\(a\)\(l\)\(t\)} 04:24, April 14, 2013 (UTC)

- Sorry, I thought you knew. Deedlit11 (
**talk**) 06:00, April 14, 2013 (UTC)

## SDA forums? Edit

Just curious, are you the same person as "Deedlit" on the Speed Demos Archive forums? --Ixfd64 (talk) 19:51, April 16, 2013 (UTC)

- Yep, that's me. I speedran FF12 a few years back. I go by Deedlit in a bunch of places. —Preceding unsigned comment added by Deedlit11 (
**talk**• contribs)

- If I not miscalculated, even if you turn all 0's and 1's in the Internet to elementary particles and crumple them in one ball, this ball will be so small that you will barely able to see it with the naked eye! Internet contains roughly 4.2 zettabytes = 33.6 zettabits (that is, 0's and 1's). Human body contains about 64 octillion elementary particles, and then \({6.4 \times 10^{28} \over 3.36 \times 10^{22}} \approx 1904762\). In other words, this ball would be roughly 1904762 times smaller than body of average human, and we're speaking about entire Internet! Ikosarakt1 (talk ^ contribs) 20:42, April 16, 2013 (UTC)

- I guess it's not surprising that the internet would be about a drop of water (1/18 of a gram in atoms, definitely visible), since the internet has to be stored, and we aren't anywhere near 3D atomic level of storage. But of course this stuff is increasing exponentially - it won't take all that long before we surpass a human in size. (well, it should be within our lifetimes.) Deedlit11 (
**talk**) 20:56, April 16, 2013 (UTC) - It has been calculated that if we take all electrons moving due to global Internet transfer, mass-energy we are giving them for the flow is equivalent to 50 grams! Nicely explained here. LittlePeng9 (talk) 05:23, April 17, 2013 (UTC)

- I guess it's not surprising that the internet would be about a drop of water (1/18 of a gram in atoms, definitely visible), since the internet has to be stored, and we aren't anywhere near 3D atomic level of storage. But of course this stuff is increasing exponentially - it won't take all that long before we surpass a human in size. (well, it should be within our lifetimes.) Deedlit11 (

## Template:wedgeEdit

Vote now! ^{\(a\)\(l\)\(t\)} 12:09, April 26, 2013 (UTC)

## Loader.c/Calculus of Constructions Edit

Few days ago me together with Ikosaract had a short conversation about Loader's function and CoC. He asked me if I could make a user blog post about this calculus, so everyone would understand it better, but I think you would be a lot better person to ask for this. So I'm asking you - would you be able to write a blog post (or maybe a whole wiki article?) about CoC and how Loader's program incorporates it? Thanks in advance. LittlePeng9 (talk) 19:53, April 8, 2014 (UTC)

- You might want to check out the Coq proof assistant to familiarize yourself with the system. I tried my hand at figuring out CoC several times, and I still don't fully understand it. you're.so.pretty! 20:02, April 8, 2014 (UTC)
- Actually, I don't understand the Calculus of Constructions that well either. I could try writing something on it, but I would have to base it on what I can glean from the internet. Perhaps it would be best to start the simple typed lambda calculus, then work our way up to System T, System F, then the CoC. Deedlit11 (
**talk**) 20:37, April 9, 2014 (UTC)

- Actually, I don't understand the Calculus of Constructions that well either. I could try writing something on it, but I would have to base it on what I can glean from the internet. Perhaps it would be best to start the simple typed lambda calculus, then work our way up to System T, System F, then the CoC. Deedlit11 (