SbiisExE
but try to figure out when we hit pentational arrays
3:17:56 PMSbiisExE
and this idea, in and of itself, is insufficient
3:18:05 PMSbiisExE
This that clear to you?
3:18:23 PMvell
it is not, in fact
3:18:30 PMWojowu
Well, the idea of just having this v(A) function is not enough for anything
3:18:37 PMWojowu
Unless we define how v works
3:18:37 PMvell
what stops us from defining pentational arrays w/ ordinals?
3:18:38 PMSbiisExE
this idea cant not define pentational arrays
3:18:54 PMWojowu
Why so?
3:19:04 PMWojowu
We can have them as ordinals below zeta_0
3:19:10 PMSbiisExE
because of the disconnect between the number of entries and the ordinal expression you use
3:19:36 PMWojowu
This all collapses to defining fundamental sequences, I'd believe
3:19:46 PMSbiisExE
Maybe so
3:19:54 PMvell
the wikipedia article on the veblen hierarchy has defn's for FS's up to Gamma_0
3:19:55 PMSbiisExE
but no one has successfully done so
3:20:12 PMSbiisExE
all the sequences so far arbitarily SAY what the size of an array is
3:20:18 PMSbiisExE
instead of caring about what it actually is
3:20:22 PMvell
what is the size of an array
3:20:33 PMSbiisExE
I will try one more time
3:20:35 PMWojowu
Number of non-1 entries, I'd say
3:20:43 PMSbiisExE
please, if you understand it, stop asking me the question
3:20:47 PMvell
thanks
3:20:54 PMSbiisExE
Let A be an ordinal
3:21:00 PMSbiisExE
assume it's a limit ordinal
3:21:20 PMSbiisExE
{b,p(A)2} produces A(p) entries = to b
3:21:34 PMWojowu
What is A(p)?
3:21:36 PMSbiisExE
that function A(p) is the size of the structure
3:21:46 PMWojowu
Okay
3:21:51 PMSbiisExE
a function which returns the number of entries
3:22:07 PMWojowu
You lost me
3:22:09 PMSbiisExE
for example {b,p(0,1)2} would be the function A(p) = p^p
3:22:13 PMvell
so you mean |Pi(a)| as defined here http://googology.wikia.com/wiki/Introduction_to_BEAF#As_ordinals_.28advanced.29
3:22:19 PMSbiisExE
right
3:22:22 PMSbiisExE
basically
3:22:26 PMSbiisExE
see what I mean
3:22:26 PMWojowu
Ah, okay, I see
3:22:33 PMSbiisExE
you already know what I mean, you just don't know it
3:22:54 PMvell
whatever yogi berra :P
3:22:56 PMWojowu
But, 0,1 isn't an ordinal
3:23:07 PMSbiisExE
meaningless point
3:23:16 PMvell
w
3:23:18 PMvell
^w
3:23:25 PMWojowu
That's better
3:23:26 PMSbiisExE
Any set which can be ordered can be treated as an ordinal notation
3:23:34 PMWojowu
False
3:23:43 PMSbiisExE
how is that false
3:23:50 PMSbiisExE
sigh
3:23:50 PMWojowu
Set of real nmbers can be ordered :P
3:23:55 PMSbiisExE
you guys love the technicalities
3:24:01 PMWojowu
Of course we do
3:24:07 PMvell
thats why we do math
3:24:19 PMSbiisExE
well anyway, you know perfectly well that has nothing to do with 0,1 as an ordinal notation
3:24:34 PMWojowu
Indeed
3:24:39 PMSbiisExE
okay
3:24:46 PMvell
it should be reasonably easy to map bowers delimiter notation to ordinals, but im too lazy to bother
3:24:54 PMSbiisExE
So how do we define ordinals as opposed to order-types then?
3:24:59 PMSbiisExE
this is a genuine question
3:25:18 PMvell
um, an ordinal is an order type of a well-ordered set? idgi
3:25:25 PMWojowu
You want a formal defintiion?
3:25:28 PMWojowu
vell gave you one
3:25:49 PMSbiisExE
well, I need a better understanding of well-ordered to make sense of that
3:26:00 PMvell
!w well-order
3:26:04 PMvell
!wp well-order
3:26:05 PMXappolBot
http://en.wikipedia.org/wiki/well-order
3:26:18 PMWojowu
In well-ordered set, it's impossible to have infinite descending chain
3:26:31 PMSbiisExE
okay
3:26:33 PMSbiisExE
good enough
3:26:37 PMWojowu
So, for example, integers are not well-ordered, because we have 0>-1>-2>...
3:26:54 PMWojowu
But natural numbers are well-ordered
3:27:09 PMSbiisExE
So then I can say, any well-ordered set, can represent an ordinal notation
3:27:14 PMvell
"<Wojowu> In well-ordered set, it's impossible to have infinite descending chain" this is dependent on the axiom of dependent choice
3:27:22 PMvell
but we're in zfc i'm assuming :P
3:27:40 PMWojowu
Well, maybe
3:27:55 PMSbiisExE
I could understand the hesitation
3:28:17 PMSbiisExE
But anyway, the point is, that I often refer to any well-ordered set as an ordinal notation
3:28:27 PMWojowu
Uhh
3:28:28 PMSbiisExE
So to me it's clear that (0,1) is an ordinal notation
3:28:37 PMvell
um
3:28:38 PMWojowu
What is ordinal notation for you?
3:28:40 PMSbiisExE
what now
3:28:49 PMSbiisExE
A way to label ordinals?
3:29:03 PMSbiisExE
why, your going to know give me the technical definition of an ordinal notation?
3:29:09 PMvell
"a partial function from finite strings in a finite alphabet to ordinals"
3:29:10 PMWojowu
In that case, well-ordered set doesn't give you ordinal notation
3:29:35 PMWojowu
Because if I say "set of countable ordinals" I give you no way of labelling ordinals
3:29:42 PMSbiisExE
A well-ordered set can always be put into one-to-one correspondence with some subset of the ordinals, no?
3:29:52 PMSbiisExE
god
3:30:09 PMSbiisExE
You guys ever considered that some things should be implied
3:30:17 PMSbiisExE
and that nitpicking wastes time
3:30:30 PMWojowu
But I honestly don't see what you mean
3:30:31 PMSbiisExE
Obviously I was only thinking of stuff that can be bounded by a countable ordinal
3:30:34 PMvell
i honestly dont get what you mean by "I often refer to any well-ordered set as an ordinal notation"
3:30:37 PMWojowu
Okay them
3:30:49 PMWojowu
"set of ordinals below epsilon_0"
3:30:54 PMSbiisExE
I don't see what's confusing
3:31:00 PMWojowu
Does this set, per se, give you a way of labelling ordinals?
3:31:01 PMSbiisExE
ExE uses a set of delimiters
3:31:05 PMSbiisExE
that can be well-ordered
3:31:18 PMSbiisExE
and so #^^^# is just another ordinal notation for gamma(0)
3:32:12 PMWojowu
Okay, maybe we can put this aside for now
3:32:16 PMWojowu
Back to BEAF
3:32:21 PMWojowu
From what I can see so far
3:32:39 PMWojowu
Only thing we have yet to say is how function A(p) looks depending on A
3:33:03 PMSbiisExE
sure
3:33:13 PMWojowu
Wait, there is something else
3:33:14 PMSbiisExE
and on how we might chose to select entries
3:33:24 PMWojowu
A(p) only tells us how many entries there will be
3:33:33 PMWojowu
Not where they will be in new array
3:33:38 PMSbiisExE
ie. on an ordinal and something akin to it's fundamental sequence
3:33:46 PMSbiisExE
sure
3:33:57 PMSbiisExE
but it's easy to set it up in the definition to define this as well
3:34:11 PMWojowu
Yes
3:34:18 PMSbiisExE
For example X&(b,p) = b,b,b,...,b w/p bs
3:34:38 PMWojowu
Now now, we only have to do so for all ordinals we care about
3:34:59 PMSbiisExE
X^2&(b,p) = X&(b,p) (1) X&(b,p) (1) X&(b,p) (1) ... (1) X&(b,p) w/p X&(b,p)s
3:35:21 PMSbiisExE
ie. treat it like a string
3:35:33 PMSbiisExE
this makes the operations very simple and mechanical
3:35:35 PMWojowu
Can you please use ordinals instead of X's?
3:35:56 PMSbiisExE
ordinal positions and multi-dimensional spaces can then be thought of an theoretical interpretations
3:36:00 PMSbiisExE
why?
3:36:05 PMSbiisExE
we are talking about BEAF
3:36:15 PMWojowu
Well, I thought we are trying to define BEAF in terms of ordinals now
3:36:18 PMSbiisExE
so I assume we are trying to define the Xs
3:36:30 PMWojowu
Or do that
3:36:31 PMSbiisExE
Ordinal notation are irrelevant?
3:36:41 PMWojowu
They actually are
3:36:42 PMSbiisExE
ordinal notations are irrelevant?
3:36:45 PMWojowu
I believe
3:36:50 PMSbiisExE
why do we have to keep returning to the same points
3:36:57 PMWojowu
?
3:37:01 PMSbiisExE
why do they matter?
3:37:16 PMWojowu
Because they give us a way to write ordinals down
3:37:22 PMSbiisExE
(I just got through discussing how countable well-ordered sets can be used as ordinal notations)
3:37:27 PMWojowu
For example, w^w is an ordinal notation for some ordinal
3:37:28 PMSbiisExE
of coarse
3:37:37 PMSbiisExE
but using #^# or w^w or X^X shouldn't matter
3:37:44 PMSbiisExE
so why bring it up?
3:37:58 PMWojowu
Because there isn't always an obvious isomorphism
3:38:08 PMWojowu
What should be zeta_0 in X's?
3:38:26 PMWojowu
If we use ordinals, we should stick to ordinals
3:38:33 PMSbiisExE
also using something more esoteric like sigma(0,sigma(0,1))
3:38:36 PMWojowu
And their standard notations
3:38:43 PMSbiisExE
w/e
3:38:57 PMSbiisExE
if ordinals are something that transcend notation then what you are saying is pedantic
3:39:06 PMSbiisExE
we are sticking to ordinals regardless of what notation is used
3:39:22 PMSbiisExE
what you mean is you want to stick to the standard notation
3:39:29 PMSbiisExE
I don't see why that matters so much
3:39:44 PMWojowu
Actually, ordinals transcend all notations, but it's not the point
3:40:03 PMSbiisExE
knowing the definition of gamma(0), SVO, LVO, BHO, etc. isn't going to help us define BEAF structures. It's not going to tell us what X^^^X is for example
3:40:09 PMSbiisExE
k
3:40:11 PMWojowu
True
3:40:14 PMSbiisExE
what's the point then
3:40:21 PMWojowu
But can you prove that X structures are well-ordered?
3:40:27 PMvell
hey guys, quick distraction: https://github.com/EnterpriseQualityCoding/FizzBuzzEnterpriseEdition
3:40:30 PMWojowu
How do you order them, first of all?
3:40:34 PMSbiisExE
can you prove w is well-ordered?
3:40:44 PMSbiisExE
0,1,2,3,4,5,... etc.
3:41:01 PMWojowu
If we define w in set-theoretic terms, yes I can
3:41:14 PMSbiisExE
(btw going to get knocked off in 6 min. Library closing)
3:41:30 PMvell
mind if i post this entire conversation publicly?
3:41:31 PMSbiisExE
k
3:41:36 PMWojowu
Not at all
3:41:37 PMSbiisExE
i guess
3:41:39 PMSbiisExE
where though
3:41:44 PMvell
wiki probably
3:41:48 PMSbiisExE
k
3:41:53 PMSbiisExE
is it that important?
3:41:58 PMWojowu
I guess vell wants to keep it for future generations
3:42:05 PMSbiisExE
k
3:42:13 PMSbiisExE
as long as it isn't some sort of witch hunt
3:43:27 PMSbiisExE
*isn't turned into one I mean
3:43:56 PMvell
i just thing that some of the points we made are quite representative of the general conflict over beaf
3:44:12 PMvell
we were getting to the heart of the matter in this conversation
3:44:33 PMSbiisExE
ic
3:44:36 PMSbiisExE
that's fine then
3:44:58 PMSbiisExE
I just don't want this to be used as some sort of quote mine for smear campaigns
3:45:28 PMvell
i dont mean to shame or mock anyone, i just want to showcase this discussion because it was good
3:45:29 PMSbiisExE
taking stuff out of context can easily be twisted to an agenda
3:45:40 PMSbiisExE
okay that's fine then
3:46:07 PMWojowu
Like this:
3:46:07 PMWojowu
<SbiisExE> @Vel fundamentally [I disagree with your definition]. It uses an approach which is not inline with the climbing method, and has, once again, more to do with ordinal arithmeti
3:46:11 PMSbiisExE
anyway, getting logged off in 38 seconds
3:46:17 PMWojowu
Lol, so precise
3:46:23 PMSbiisExE
right
3:46:23 PMWojowu
Cya later
3:46:30 PMSbiisExE
but that's not my full view
3:46:39 PMSbiisExE
just a watered down version for the purposes of discussion
3:46:43 PMSbiisExE
my point exactly
3:46:54 PMWojowu
I think 38 secs passed
3:49:33 PMvell
bye sbiis