User talk:Vel!/BEAF

There's a lot of talk going around of BEAF beyond \(\varepsilon_0\) being ill-defined. Has anyone yet found an error in this definition? you're.so.pretty! 15:41, June 2, 2014 (UTC)

All I can give you is my opinion based on my own personal work on BEAF. There appears to be a non-ambiguous interpretation of BEAF up to e0, and I even formalized it at one point in my personal writings. This was achieved some time around 2007-2008 before the inception of my site. Much effort was put out since then to figure out how to go further. The result has been no end of confusion. I believe that in principle it is possible to create a system for legion space. The reason is because eventually a block of entries has to reach any desired size. If you keep multiplying by the prime eventually you can get any number of entries in a block. However there are technical difficulties to formalizing this concept. Some of these technical difficulties can in principle be overcome, though the solutions may not be particularly elegant. However it is not entirely clear that there may not be some "hitch" down the road. Bowers' simply assumed that the array structures existed, that they made sense, and that there was a unique non-ambiguous way to interpret them. In short, there is no way to know whether or not it's ill-defined until someone finds definite proof that something is definitely unworkable or someone demonstrates that an explicit definition of BEAF can be formulated.

We could debate whether or not it's ill-defined, but let me suggest a better way to settle the debate. If it's well-defined, then someone needs to create an explicit definition of it to demonstrate this. If it's ill-defined then someone needs to find a definite reason why Bowers' idea is ambiguous or problematic. Short of this we are merely speculating.

Sbiis Saibian (talk) 16:31, June 2, 2014 (UTC)


 * I would say that there generally isn't a "definite reason" why what someone says is ambiguous, other than the fact that you read/hear what they write/say, and you can tell that they are missing details so that you can't pinpoint specifically what they mean. After reading Bowers' description of legion arrays, it is clear to me that he hasn't precisely defined the notation - no definite reason beyond that.


 * Of course, someone could define a notation that seems like it could be what Bowers really meant by array spaces and legion spaces. But, I'm willing to bet that there are multiple ways to due this, due to the very ambiguity that we are trying to eliminate. So I don't think one can come up with a notation and say, "This is how fast Bowers' notation grows". Deedlit11 (talk) 07:42, June 5, 2014 (UTC)

Definition
"An array is a function A : W -> W\{0} where the number of outputs greater than one is finite." What does that mean, actually? And what does it output? Wait nvm. So it can handle limit ordinals now? King2218 (talk) 23:44, June 2, 2014 (UTC)


 * Old news, man. Get with the times. you're.so.pretty! 01:30, June 3, 2014 (UTC)


 * What I'm saying is that you can probably figure it out from the page. you're.so.pretty! 01:50, June 3, 2014 (UTC)
 * Nvm, I already figured that out before you posted :P King2218 (talk) 02:06, June 3, 2014 (UTC)

Stewardess Problem
It must be at the end. (try doing some analysis to see for yourself) King2218 (talk) 18:19, June 9, 2014 (UTC)


 * Can you be more clear? you're.so.pretty! 01:42, June 10, 2014 (UTC)
 * Okay. See how $$\{\omega,\omega,\omega\}$$ in your definition is the limit of $$\{\{\omega,1,\omega\},\{\omega,2,\omega\},\{\omega,3,\omega\},...\}$$. $$\{\omega,1,\omega\}$$ is equal to 1 so no problem there. $$\{\omega,2,\omega\}$$ is the limit of $$\{\{\omega,2,1\},\{\omega,2,2\},\{\omega,2,3\},...\}$$, $$\{\omega\uparrow2, \omega\uparrow\uparrow2,\omega\uparrow\uparrow\uparrow2,...\}$$, or basically $$\{\omega\cdot\omega,\omega\uparrow\omega,\omega\uparrow\uparrow\omega,...\}$$. The limit of those is $$\omega\uparrow^\omega\omega$$ or $$\omega\{\omega\}\omega$$ which makes it inconsistent with Bowers's definition. King2218 (talk) 11:53, June 10, 2014 (UTC)

Larger structures
While dealing with normal arrays like $$\{\omega,\omega,1,2 (1) \omega (2) 5\}$$ might not be problem, you didn't define the rules for something like $$\{\omega,5,2\}$$ & $$\{\omega,1,1,2\}$$ & $$\omega$$.

I don't believe that the power of sub-legion BEAF is $$\psi(\Omega_\omega)$$. The variant of BEAF, considered by Hyp cos, uses arrays with more than $$\omega$$ entries (if we let X = $$\omega$$) and yet he used structures $$X_n$$, which is the core of the power and nether Bowers nor you mention it.

Also, why the article doesn't explain rules for past-legion spaces at all? How $$\{\omega,\omega+1 / 2\}$$ & n must be solved for instance? I know that it must be somehow translated into your ternary & operator, but it isn't included into definition. Ikosarakt1 (talk ^ contribs) 07:27, June 10, 2014 (UTC)

Analysis between \(\{\omega,\omega (1) 2\}\) and \(\{\omega,\omega,2 (1) 2\}\)
Let $$\{\omega,\omega (1) 2\} = \psi(\Omega^{\Omega^\omega})$$

Then, if indeed $$\{\omega,\omega+n (1) 2\} = \{\{\omega,\omega+n-1 (1) 2\},\{\omega,\omega+n-1 (1) 2\},\cdots,\{\omega,\omega+n-1 (1) 2\}\}$$ (with n+1 $$\{\omega,\omega+n-1 (1) 2\}$$'s):

Then $$\{\omega,\omega*2 (1) 2\}$$ would be equivalent to $$\psi(\Omega^{\Omega^\omega}*2)$$ and nowhere close to $$\psi(\Omega^{\Omega^{\omega*2}})$$. Hyp cos proposed to represent intermediate ordinals using entries at transfinite positions, but by our definition that's disallowed. By that we can come to $$\{\omega,\omega,2 (1) 2\}$$ is merely $$\psi(\Omega^{\Omega^\omega+1})$$ and nowhere LVO. Ikosarakt1 (talk ^ contribs) 07:47, June 10, 2014 (UTC)

MathJax
Can you allow to replace MathJax to LaTeX? Ikosarakt1 (talk ^ contribs) 07:47, June 10, 2014 (UTC)