User blog:Vel!/IRC log 2014-11-23

A pretty good (if heated) conversation today about BEAF and formality.

[10:34:03] previously you discussed several attempts at extensions in xE^ [10:34:09]  Do most ppl still think legion arrays are LVO? [10:34:12] aarex's, iko's, vell's, holloms (?) [10:34:18]  Bowers' still seems to be under that impression [10:34:23] hm that's odd [10:34:29] they're usually believed to be psi(W_w) [10:34:35]  i know [10:34:49]  but remember, Bowers' didn't start out knowing anything about ordinals [10:34:57]  Who believes that they reach psi(W_w), actually? [10:35:05]  and is only know incorporating that into his work on BEAF [10:35:06] it's the most common belief [10:35:14] hyp cos used it in his analysis [10:35:22] the wiki had that on the fgh's page [10:36:23] unlike with xE^, whatever the linear-array-e is named will be quite hard to logically extend [10:37:04] SbiisExE: in your xE^ article you say that hollom talked about an extension [10:37:10] but the link couldn't take me to it [10:37:55]  you talking about the one where (#^^a)^^b = #^^(a+b) [10:38:15]  @Wojo Who believe it reaches psi(W_w) actually? Me for one [10:38:41] it doesnt reach anything. its undefined [10:38:47]  Thank you Sbiis for more precise answer than cf's [10:38:58]  I believe the LVO result is an artifact of taking the ordinal-arithmetic too literally [10:39:20]  so there is no catching point for SGH and FGH? [10:39:23] --> Flitris (xxxxxxx@gateway/web/freenode/ip.[IP address censored]) has joined ##googology [10:39:24]  Hello Flitris! [10:39:29]  Hey Flit [10:39:31]  professional mathematicians are talking about something ill-defined? [10:39:39]  Sbiis, depends on fundamental sequences [10:39:43] of course vell [10:39:43]  right [10:39:45] <SbiisExE> so what [10:39:46] <Flitris> oh my lotsa people here today [10:39:53] yeah [10:39:58] <SbiisExE> for a chosen set of fundamental sequences, the question becomes meaningful [10:39:59] <SbiisExE> then [10:40:03] <Wojowu> Exactly [10:40:14] <Wojowu> And it has been answered for few such choices [10:40:22] <SbiisExE> which means there is nothing inherently impossible about defining BEAF [10:40:29] <Wojowu> Of course not [10:40:30] <SbiisExE> It is possible in principle [10:40:34] <Wojowu> Of course it is [10:40:43] Wojowu: remember your fgh vs. peano arithmetic question? did you ever get an answer for the f_a > f_b restriction? [10:40:58] <Wojowu> For example: I define all arrays beyons pentational to evaluate to 1. Here, I gave you a definition [10:41:10] <Wojowu> No, I didn't, vell [10:41:40] ok that's just smartass [10:41:48] <Wojowu> So? [10:41:48] cookiefonster: reality is smartass [10:41:52] <Wojowu> It's a definition [10:42:02] it doesn't help the discussion, it's nothing more than smartass [10:42:02] <Wojowu> Maybe a bad one, but this is subjective now [10:42:14] <Wojowu> It's still a valid defintiion [10:42:16] cookiefonster: it does help the discussion, because it proivdes a counterexample to a claim [10:42:31] but it's smartass [10:42:32] <SbiisExE> as I've said before [10:42:36] <Wojowu> Counterexample to what claim? [10:42:44] <SbiisExE> something within Bowers' parameters [10:42:47] <SbiisExE> not just ANY definition [10:42:56] <SbiisExE> why in the heck would I argue for something so trivial [10:42:57] the implicit claim that any definition of BEAF beyond e0 makes intuitive sense [10:43:04] <SbiisExE> you know that's not what I mean [10:43:17] SbiisExE: okay then what are bowers parameters [10:43:25] when we treat everything as formal we get this smartass shit [10:43:31] which is why we can't treat everything as foral [10:43:35] formal * [10:43:45] 9_9 [10:44:26] <SbiisExE> obviously an array should not evaluate to 1 for all values [10:44:33] <Wojowu> Well, I gave one definition, you can think of it as a shit definition [10:44:40] <SbiisExE> unless the prime and base equal 1 of coarse [10:44:44] <Wojowu> But I didn't say we can't have better definitions [10:45:01] <Flitris> or just the base = 1 [10:45:03] SbiisExE: okay then what are bowers parameters [10:45:10] <SbiisExE> Every new structure should contain every previous structure as a substructure [10:45:25] define a structure [10:45:29] <SbiisExE> so if we have an array of tetrational spaces then [10:45:45] that's the kind of thing that goes without saying [10:45:53] cookiefonster: it sure as hell does not [10:45:55] <SbiisExE> {3,3 X 2 } = { 3^^3 & 3 X 3^^3 & 3 X 3^^3 & 3 } [10:45:56] unless you're a smartass of course [10:46:04] X ? [10:46:07] without a precise definition of "structure" there is room for loopholes and smartassery [10:46:08] <SbiisExE> this would obviously have exactly 3^^3 * 3 non-1 entries [10:46:17] the point of formality is to EVADE this smartassery you speak of [10:46:22] <Wojowu> Okay, then at least specify a way of determining if a structure is a substructure of another one [10:46:27] <SbiisExE> It should also be larger than { 3^^3 & 3 } [10:46:38] <SbiisExE> so how could it be 1 under any REASONABLE definition? [10:46:54] "define reasonable definition [10:47:03] <-- Flitris (xxxxxxxx@gateway/web/freenode/ip.[IP address censored]) has quit (Quit: Page closed) [10:47:06] <Wojowu> Did I ever say my definition is reasonable in any way? [10:47:38] what wojowu gave is, indeed, a definition [10:47:50] and since we have no idea what "bowers' parameters" are [10:47:58] (sicne he never formally defined shit) [10:47:59] <SbiisExE> A structure is an ordered set of entries obviously [10:48:06] an ordered set of entries? okay [10:48:33] where "entry" is defined as a position in the array [10:48:37] <SbiisExE> we can determine the structure being used by finding the pilot [10:48:48] well, we can at least give parameters with ... common sense [10:48:50] <SbiisExE> {b,p(1)2} 2 is a pilot at position "w" [10:49:04] <Wojowu> So you assume positions are indexed with ordinals? [10:49:05] <SbiisExE> this expands the "w" structure, called a linear array by Bowers' [10:49:17] i define my structure as the following ordered set of entries: the seventh, the sixth, the eleventh, and then the nineteenth [10:49:21] <SbiisExE> then we define it's growth rate by how many entries are produced when prime = p [10:49:34] <SbiisExE> {b,p (1) 2 } = {b,b,...,b} w/p b's [10:49:45] <SbiisExE> therefore w-structure has growth rate p [10:49:56] <SbiisExE> w^2-structure has growth rate p^2 [10:49:58] <SbiisExE> and so on [10:50:17] <SbiisExE> let a comma be the prime delimiter [10:50:30] <SbiisExE> a successor delimiter forms an array of the previous structure [10:50:38] <SbiisExE> and then we have limit delimiters/structures [10:50:53] okay lets look at what we DO agree on about BEAF, namely what it is up to e0 [10:50:56] <SbiisExE> such as dimensional-structures or tetrational-structures [10:51:03] <SbiisExE> not that any of this is new to anyone [10:51:32] can we agre on this definition: http://googology.wikia.com/wiki/Introduction_to_BEAF#As_ordinals_.28advanced.29 [10:52:14] to the best of my knowledge it is perfectly formal, although nobody has yet proven that it terminates [10:52:37] (shouldnt be hard though, considering how simple wojowu's proof for linear arrays is) [10:54:11] <SbiisExE> @Vel What Wojo gave was indeed a definition. Like I said, no one is going to argue against that. [10:54:24] <SbiisExE> but no one is trying to defend that strawman claim [10:54:33] <SbiisExE> When I say BEAF can be defined in principle [10:54:44] <SbiisExE> I mean in the spirit of BEAF as presented by Bowers [10:54:51] i dont feel no spirit [10:54:56] <SbiisExE> not just ANY definition. That's trivial and is of no interest to anyone [10:54:59] can we agre on this definition: http://googology.wikia.com/wiki/Introduction_to_BEAF#As_ordinals_.28advanced.29 [10:55:09] does everyone agree w/ this? [10:55:09] exactly [10:55:10] <SbiisExE> except those who wish to derail the investigation [10:55:42] i'm trying to get us to reach an agreement on the basics of how BEAF is defined [10:55:58] can everyone say whether they agree or disagree w/ the given definition up to e0 please? [10:56:06] sure wh not [10:56:30] (just fixed a minor error) [10:57:20] <SbiisExE> @Vel fundamentally no. It uses an approach which is not inline with the climbing method, and has, once again, more to do with ordinal arithmetic [10:57:37] what the fuck is the climbing method?! [10:57:46] <SbiisExE> glad you asked [10:57:56] who came up with this [10:57:59] <SbiisExE> It's something that Bowers' mentions on his hypernomials page [10:58:10] <Wojowu> "mentions" [10:58:11] <SbiisExE> and references frequently when discussing going beyond tetrational arrays [10:58:12] and it's what sbiis uses in exe [10:58:20] <SbiisExE> right [10:58:24] <SbiisExE> I've said it a few times too [10:58:41] <SbiisExE> but somehow all of this isn't even a blip on the map even though ... we are all talking about BEAF, no? [11:00:18] <XappolBot> BONG BONG BONG BONG BONG BONG BONG BONG BONG BONG BONG [11:00:34] this "climbing method" is meaningless to me http://polytope.net/hedrondude/hypernomial.htm [11:00:48] right off the bat I see X^X^...^X^X "w/ X X's" [11:00:54] <SbiisExE> It's not really all that meaningless [11:01:07] bowers has discussed that in emails with sbiis [11:01:08] <SbiisExE> it's true that if you took a power tower of the form p^^p [11:01:14] <SbiisExE> and you kept multiplying by p [11:01:24] <Wojowu> He doesn't even give a description of climbing method [11:01:28] <SbiisExE> eventually you could observbe the "climb" as Bowers' suggests [11:01:41] we have no idea what "X" means, let alone what "X^X" or "X^^X" means [11:01:46] <Wojowu> He just gives an example and says "this I'll call climbing methid" [11:02:09] and i dont see how my ordinal definition " is not inline with the climbing method" [11:02:36] all i see in bowers' discussion of the climbing method is a shit-ton of ellipses and meaningless symbols [11:03:30] sbiis gave a more extensive interpretation of this climbing method as we know [11:03:33] using a > instead of a , [11:04:33] <SbiisExE> It's not in line for a very simple reason [11:04:43] <SbiisExE> what is w^^(w2) in your system [11:04:58] <SbiisExE> how many entries would {b,p ( w^^(w2) ) 2 } actually produce? [11:05:03] my system only goes up to e0 [11:05:05] <Wojowu> vell asked only if we agree with this up to e0, didn;t he? [11:05:10] <SbiisExE> exactly p^^(p*2) entries or not? [11:05:19] <SbiisExE> true [11:05:24] now that kind of thing is where it gets troublesome [11:05:24] SbiisExE: that's actually impossible because Bowers is wrong about the entry count [11:05:31] <SbiisExE> but I thought his definition was made to go up to legion space? [11:05:37] there's a multitude of interpretations [11:05:41] <SbiisExE> no [11:05:52] <SbiisExE> because as you guys love to point out it depends [11:06:02] <SbiisExE> on the fundamental sequences [11:06:11] the old "prime block of an X structure has as many entries as you get when replacing X's with p's" thing [11:06:33] that kind of thing ... just doesn't work [11:06:36] <SbiisExE> your fundamental sequences do not produce the required entries [11:06:36] it depends on intuition gained from the Wainer hierarchy, and it just doesnt work [11:06:40] and after the svo, first law of googology [11:06:43] <SbiisExE> and so obviously Bowers' would never use them [11:07:24] <SbiisExE> wrong [11:07:31] <SbiisExE> It can be done [11:07:50] <SbiisExE> It just depends how you stack things up [11:08:07] <SbiisExE> you could in theory get it to stack up to be something like X-->X-->X-->X [11:08:10] okay well [11:08:22] <SbiisExE> but Bowers' idea is to stack it up in a way to get BEAF like numbers [11:08:37] hear me out here: i linked to my definition because i wanted us to agree on basic notions of what structures, entries, arrays, etc. are [11:08:53] but obviously that's not even going to wrok [11:09:39] bowers has doen nothing tof ormally define structures themselves [11:09:55] i offered a very simple formal definition: structures are just ordinals [11:10:09] but okay. so you dont want to accept that structures are odinrals. [11:10:12] what are they then? [11:10:23] how do you expect us to argue about strcutres if we cant even fucking agree on what they are? [11:11:20] and don't give me this "it's intuitively obvious" shit. we've already disagreed on what structures are meant to be, so it should be obvious as hell that they are NOT infact induitiviley obvious [11:11:22] <SbiisExE> they are ordered sets of entries [11:11:31] <SbiisExE> that's isomorphic to ordinals so ... close enough [11:11:31] what's an entry [11:11:39] <SbiisExE> That's hardly the point of disagreement [11:11:46] in fact it is [11:11:53] <SbiisExE> LORD [11:12:01] <SbiisExE> what's an entry?! It's an argument [11:12:06] <SbiisExE> of a function [11:12:12] <Wojowu> LEt me ask another question: [11:12:13] <SbiisExE> It's the simplest thing to define [11:12:20] <Wojowu> Your entries are ordered, by what? [11:12:30] <Wojowu> What are indices of possible positions? [11:12:34] <SbiisExE> that's why I use "argument" instead of "entry". Because it's the standard term in professional mathematics [11:12:48] <SbiisExE> If you prefer you can also use "parameter" from computer science [11:13:04] <SbiisExE> are you going to tell me these are vague ill-defined terms? [11:13:10] i define an entry as an ordered pair consisting of an ordinal and the value that it maps to. [11:13:25] i'm asking these "dumb" questions because we apparently don't agree that arrays are based on ordinals [11:14:28] <SbiisExE> do you believe in order-types? [11:14:43] <Wojowu> What order types do you mean now? [11:14:46] <SbiisExE> why are you asking the obvious things. You can order them using standard ordinal notations if you like [11:14:54] i define an array as a function mapping from e0 to the positive integers, such that only a finite number of entries are greater than one [11:14:57] <SbiisExE> the usual meaning! [11:15:11] okay so we DO agree on the definition of arrays that i provided? [11:15:15] <SbiisExE> ordinals and other things which can be ordered [11:15:46] <SbiisExE> I agree that the approach of indexing entries by ordinals is handy [11:15:52] so ... nothing beyond e0? [11:15:57] <SbiisExE> but not sufficient to define legion space in and of itself [11:15:57] cookiefonster: just for now [11:16:12] <SbiisExE> you can't even define tetrational arrays with that alone [11:16:15] ok, so that's up to tetrational arrays [11:16:20] SbiisExE: so you don't agree that it works in the long term [11:16:25] then what would work instead? [11:16:26] <SbiisExE> but you can create array notations with it [11:16:29] <SbiisExE> no [11:16:39] <SbiisExE> It works as a basic framework [11:16:47] <SbiisExE> look BEAF works on two levels [11:16:56] <Wojowu> Brb [11:17:05] <SbiisExE> on one level, we have the 5 or so fundamental laws of BEAF [11:17:17] <SbiisExE> That define what to do in the basic situations [11:17:27] <SbiisExE> ie. if there is less than 3 entries [11:17:33] <SbiisExE> if prime=1 [11:17:45] <SbiisExE> If the pilot is at a successor entry or limit entry [11:17:46] <SbiisExE> etc. [11:18:01] <SbiisExE> Then you start defining your spaces using ordinal indexed entries [11:18:24] <SbiisExE> The only thing left to figure out then is, what happens when the pilot is at limit ordinals [11:18:46] <SbiisExE> So I agree that underlying Bowers' theory is ordinals, whether or not he realized it [11:18:59] brb [11:19:08] <SbiisExE> the problem is, if we talk about stuff like tetrational arrays of legion arrays [11:19:26] <SbiisExE> then we have to think about how those "spaces" translate into sizes under Bowers' scheme [11:19:40] and there's the problem: bowers' scheme is vaguely defined [11:19:41] <SbiisExE> this requires another level which is not simply ordinal related [11:19:52] <SbiisExE> It's fundamental sequence related [11:20:00] so you alter the fundamental sequences. there. [11:20:01] <SbiisExE> It's about the choices that are made [11:20:07] <SbiisExE> yes [11:20:12] what's the problem then [11:20:25] <SbiisExE> but clearly you shouldn't alter anything which Bowers' already established [11:20:38] except bowers' didn't establish shit [11:20:51] <SbiisExE> and also, one can see implicit rules based on what Bowers' has already provided [11:20:57] <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 arithmetic [11:21:21] <SbiisExE> can I get everyone to agree that an array of structure A, has p times the number of entries for structure A? [11:21:41] <SbiisExE> (most likely no one is going to even understand what I mean except cf) [11:21:59] <SbiisExE> (and it's a perfectly simple and well-defined concept btw) [11:22:04] i'm trying to figure out what this is supposed to mean? [11:22:08] what's an array of structure A? [11:22:11] <SbiisExE> right [11:22:15] <SbiisExE> so let me break it down [11:22:34] <SbiisExE> let's just say, for the sake of the argument, that some subsystem of BEAF is understood [11:22:42] i get what you mean [11:22:48] <SbiisExE> so let {b,p (A) 2} diagonalize over that system [11:23:10] <SbiisExE> and say it creates { A(p)&b } [11:23:28] <SbiisExE> where A(p) is a function of p. A(p) is the number of entries that result [11:23:58] <SbiisExE> somehow it also codifies the entries in the structure that get non-1 values, and which don't (btw this can also be defined in sound ways at least up to e0) [11:24:18] <SbiisExE> Now create a new delimiter, called (A+1) [11:24:24] <SbiisExE> can we agree that [11:25:00] <SbiisExE> {b,p(A+1)2} = { A(p)&b (A) A(p)&b (A) ... (A) A(p)&b } w/p copies of A(p)&b [11:25:10] <SbiisExE> and that this would have exactly p*A(p) [11:25:16] <SbiisExE> entries [11:25:39] essentially you are taking a structure A and asking about A*w [11:25:47] <SbiisExE> right [11:26:02] <SbiisExE> so the 2 would now be in position A*w [11:26:08] <SbiisExE> Is that not clear? [11:26:11] with any extension of the Wainer hierarchy, yes that is true [11:26:17] <SbiisExE> w/e [11:26:23] <SbiisExE> were talking about BEAF [11:26:33] <SbiisExE> but I understand that means you agree [11:26:50] because the ordinal definition of BEAF so far uses the wainer hierarchy [11:27:10] <SbiisExE> that's part of the parameters spelled out by Bowers' writings [11:27:11] will be afk, gotta rake leaves [11:27:23] <SbiisExE> so simply defining everything = 1 is not in line with this [11:27:44] <SbiisExE> But why can't this extend to ANY ordinal? [11:28:22] clarify [11:28:34] <SbiisExE> If we agree to subsystem up to ordinal A [11:28:45] <SbiisExE> then we already have a definition up to ordinal A*w [11:28:46] <SbiisExE> no? [11:29:12] <SbiisExE> (well technically you need A(p)&b first which is something distinct from the subsystem up to A) [11:29:23] <SbiisExE> (assume we have already chosen such a diagonalization) [11:29:47] sure [11:29:52] <SbiisExE> okay [11:29:59] <SbiisExE> so there is at least one thing we agree one [11:30:01] <SbiisExE> on [11:30:24] <SbiisExE> Of coarse this doesn't really get us very far [11:30:35] <SbiisExE> we can only iterate this up to A*w^w [11:30:55] <SbiisExE> but that should also be pretty clear since Bowers' shows us how dimensional arrays work [11:31:13] <SbiisExE> which means we can have dimensional arrays of A-blocks, as opposed to just entries [11:31:51] sure [11:31:56] <SbiisExE> You say Bowers' didn't establish shit [11:32:00] <SbiisExE> but that's not true [11:32:11] <SbiisExE> that's why we can more or less agree up to e0 [11:32:32] <SbiisExE> and there is enough to go on to get to e0*w^w based on what we agreed on already [11:32:38] <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 arithmetic [11:32:43] <SbiisExE> that's why we can more or less agree up to e0 [11:32:47] <SbiisExE> So it follows that if A is known, then subsystem A*A is also known [11:33:05] <SbiisExE> I meant past e0 [11:33:20] <SbiisExE> up to e0 it should be fine if your just using the wainer hierarchy [11:33:50] <SbiisExE> Next we can nest structure A to get to A^X [11:34:02] <SbiisExE> is there something equivalent to A^X in your notation? [11:34:31] <SbiisExE> like, can I take any epsilon number and have {{w,w,2},w} ? [11:35:49] my definition does not define BEAF over ordinals [11:35:55] <SbiisExE> ? [11:36:01] it does not allow ordinal entries [11:36:15] if by that you mean e0^w, then no, my definition does not cover that [11:36:33] <SbiisExE> well there's the problem then [11:36:48] um. it's delibereately designed to only go op to e0 [11:36:48] <SbiisExE> undoubtably you use w^(e0*w) instead [11:37:16] <SbiisExE> I distinctly remember you originally defined it to go to legion space [11:37:27] <SbiisExE> which you believed was LVO according to your notation [11:37:35] <SbiisExE> have you altered it since then? [11:37:37] that was a conjecture according to the work of iko/wyth/etc. [11:37:47] later i just gave up on it after finding a critical error [11:38:09] i decided that it's just not worth my time. bowers can formalize his own god damn notation [11:38:23] <SbiisExE> sure [11:38:34] <SbiisExE> or other interested parties can pick up where he left off [11:39:06] <SbiisExE> It's not as if your "doing his work for him" [11:39:32] <SbiisExE> If someone defines something useful out of BEAF then they deserve credit for helping to formalize the idea [11:39:47] i realized that there was really no point in what i was doing [11:39:52] <SbiisExE> should we just disregard all of Chris Bird's work because ... he's just working with Bowers' arrays [11:39:55] <SbiisExE> of coarse not [11:40:05] <SbiisExE> he is rightly recognized for his contribution [11:40:15] <SbiisExE> and Bowers' was definitely helped out by his work [11:40:27] if bowers didn't bother to create a formal notation, then the emperor wears no clothes and i'm not patching them up for him [11:40:32] <SbiisExE> none the less Bowers' still get's credit for the initial idea [11:41:12] <SbiisExE> What he has is a sloppy idea [11:41:15] <SbiisExE> not no idea at all [11:41:17] <SbiisExE> that's all [11:41:28] <SbiisExE> But you basically want to say he's got nothing [11:42:20] <SbiisExE> that he should be disregarded and ignored [11:42:26] <SbiisExE> but I don't understand that [11:42:36] <SbiisExE> I much prefer Bowers' as a founding father over Joyce [11:42:39] what i mean to say is that the only reason i was trying to formalize his notation [11:42:45] <SbiisExE> at least his vision has far more soaring [11:42:56] was that BEAF has been put on a pedestal [11:42:59] <SbiisExE> and ... actually a lot better defined, despite it's considerable limitations [11:43:12] <SbiisExE> yes [11:43:15] and from trying to fix it, i realized...why is it even being put on a pedestal at all [11:43:18] <SbiisExE> this is undeniably true [11:43:27] <SbiisExE> should we dethrone him? [11:43:29] why should i care about this half-broken notation [11:43:41] <SbiisExE> by simply saying "it's all ill-defined" and amounts to nothing [11:43:44] SbiisExE: i said nothing about discounting ALL of bowers work, including his illions [11:43:49] <SbiisExE> doesn't seem very googological in spirit [11:44:09] if his notation offered something new or genuinely interesting, then i would be more inclined to try to fix it [11:44:20] <SbiisExE> let's just disregard any work with potential. Then we don't ever have to try and outdo it, in either percision or scale [11:44:39] except i would argue that BEAF has no potential [11:44:56] <SbiisExE> it does thogh [11:45:15] <SbiisExE> it does though. He independently discovered a lot of things and put an interesting twist on it [11:45:26] such...as...? [11:45:39] <SbiisExE> @Vel BEAF has no potential. That's nonsense [11:45:43] <SbiisExE> of coarse it has potential [11:45:54] <Wojowu> Back [11:45:58] <SbiisExE> It's got as much potential as trying to get SGH to catch up to FGH [11:46:00] <SbiisExE> at least [11:46:27] SGH, HH, and FGH are extremely simple and well-defined ways to generate large numbers on virtually unlimited scales [11:46:30] <Wojowu> I don't get the comparison [11:47:03] <SbiisExE> and so is BEAF [11:47:04] BEAF is a clunky, half-broken system that is at most as powerful as the above three [11:47:14] <SbiisExE> it's no more or less ill-defined than SGH, FGH, or HH [11:47:24] fuck no [11:47:29] we can't even agree on the definition of a structure [11:47:31] <SbiisExE> fuck yes [11:47:37] <SbiisExE> What's FGH? 3 rules [11:47:43] but we can agree on the definition of an ordinal [11:47:45] <SbiisExE> would you say it's well defined? [11:47:50] <Wojowu> I'd say so [11:48:00] <SbiisExE> so why is there a problem applying that to BEAF? [11:48:11] <Wojowu> we can't even agree on the definition of a structure [11:48:19] <SbiisExE> FGH, SGH, HH just need fundamental sequences to run [11:48:23] because we can't agree on the basic definitions of the fundamentals of BEAF because bowers didnt even bother to define those [11:48:26] <SbiisExE> and BEAF just needs structures defined [11:48:32] <SbiisExE> It's perfectly well defined [11:48:35] <Wojowu> Then define the structures for god's sake [11:48:37] and by "funcamentals" i mean not FS's but things like entries and shit [11:48:41] <SbiisExE> The only problem is in the 2nd layer [11:49:10] <SbiisExE> Chris Bird did just fine taking array notation very far [11:49:24] i haven't really bothered to look over his work because it's massive [11:49:29] <Wojowu> Same [11:49:32] <SbiisExE> of coarse he NEVER talks about array-sizes, or entry counts. That's the 2nd layer [11:49:46] <Wojowu> Bird took a way around [11:49:49] and i'm skeptical considering that he started the entire fad of going "is comparable to f_a(n)" which is a massive red flag for me [11:49:55] <SbiisExE> But the first layer of BEAF is just as well defined as FGH,SGH, or HH [11:49:56] <Wojowu> So he didn;t have to fiddle with all the Bowers' crap [11:50:37] <SbiisExE> you say we can't agree on the definition of a structure [11:50:41] <SbiisExE> but we already did [11:50:46] <SbiisExE> An ordered set of entries [11:50:51] <Wojowu> First layer, possibly [11:50:51] <SbiisExE> that's no problem [11:50:54] <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 [11:50:59] <Wojowu> But the problem with BEAF is the second layer [11:51:04] <Wojowu> Which FGH doesn't have [11:51:04] <SbiisExE> so? [11:51:13] <SbiisExE> how does argue with what I said [11:51:25] we do not agree on my definition up to e0 [11:51:42] <Wojowu> How does climbing method magically now agree with vell's definition? [11:51:48] <SbiisExE> Let me be more explicit. Your continuation past e0, which you originally posted, does not follow Bowers' guidelines that structures should only be named according to the growth rate of entries [11:51:51] <SbiisExE> for that particular space [11:52:00] <SbiisExE> very specific complaint [11:52:10] <Wojowu> Growth rate of entry? [11:52:12] whats the growth rate of an entry, help [11:52:15] <SbiisExE> It's not a blanket complaint to your entire approach to BEAF, so stop painting it as such [11:52:30] <SbiisExE> sigh [11:52:32] <SbiisExE> I'm sorry [11:52:39] <SbiisExE> I can't talk to you guys about this [11:52:47] <SbiisExE> There just isn't the terminology needed at this point [11:52:48] <Wojowu> Because...? [11:53:02] and that's a result of the fact that bowers' writings are vague [11:53:05] <SbiisExE> Bowers' didn't create one [11:53:15] <SbiisExE> I mean he has terminology but it's used loosely [11:53:28] <SbiisExE> I don't think it's that hard to make sense of [11:53:35] we have these differing interpretations and we don't agree on things because bowers did a crap job in defining notation [11:53:51] this i consider evidence for the view that BEAF does not deserve the respect it has [11:53:53] <SbiisExE> but if you guys need me to explain every term "entry" "structure" "array" "space" "size" etc. [11:54:10] <SbiisExE> then this is just going to be a boring explication of elementary terms [11:54:27] <Wojowu> Well [11:54:27] <SbiisExE> We agree [11:54:33] <SbiisExE> we just don't realize it [11:54:34] <Wojowu> If you want BEAF well-defined [11:54:43] <Wojowu> You would have to define all the terms [11:55:00] <SbiisExE> If you understood what I meant you wouldn't be acting like I'm talking nonsense [11:55:14] <SbiisExE> the basic ideas are pretty simple. But the consequences are hard to work out [11:55:21] " you wouldn't be acting like I'm talking nonsense" thanks for implying that we're being dishonest [11:55:41] <SbiisExE> which is why Bowers' never really understood the beast he unleashed when he tried to imagine legion space [11:55:47] <Wojowu> I'm just pointing out what I think is required for formal definition [11:55:57] <SbiisExE> I not saying your being dishonest [11:56:06] <SbiisExE> I think your bafflement is genuine [11:56:16] <SbiisExE> but for whatever reason the language being used here is a barrier [11:56:18] fair enough [11:56:58] <SbiisExE> I think Bowers' gave enough of a framework to generate a formal definition [11:57:07] <SbiisExE> at least for variants of what we might call BEAF [11:57:31] <SbiisExE> the only difficultly is in gauging the "sizes" (theres that technical term again) of the structures generated [11:57:53] <SbiisExE> then this is just going to be a boring explication of elementary terms [11:58:00] <SbiisExE> right [11:58:06] without these elementary terms, we get into discussions like the above [11:58:12] that's WHY we define them [11:58:24] <SbiisExE> I'd rather avoid having to explain these terms. I thought they'd be well understood to anyone who has spent time trying to understand BEAF [11:58:33] <SbiisExE> You shouldn't have to ask me what an entry is, at least [11:58:49] <SbiisExE> And a structure, while vaguely discussed in Bowers' own writing [11:59:01] <SbiisExE> clearly refers to an ordered set of entries [11:59:31] <SbiisExE> as bowers says there are linear structures, planar structures, dimensional structures, structures that need tetration to define, etc. [11:59:59] <SbiisExE> What all these and more have in common is that the entries can always be put in order, in one-to-one correspondence with some subset of the countable ordinals [12:00:16] <SbiisExE> So what's so unclear about the term "structure" then [12:00:18] <XappolBot> BONG BONG BONG BONG BONG BONG BONG BONG BONG BONG BONG BONG [12:00:35] <SbiisExE> who cares what he calls it, tetrational-structure, X^^X-structure [12:00:52] <SbiisExE> The underlying idea is still an ordered-set of entries of order-type e0 [12:00:58] <SbiisExE> Is that ill-defined? [12:01:46] did bowers ever explicitly specify this this? [12:02:03] <SbiisExE> That is one genuine problem [12:02:13] that is the genuine problem with the entirety of beaf [12:02:20] <SbiisExE> Bowers' entire work is written in an informal style [12:03:01] <SbiisExE> But (1) that doesn't mean he is talking nonsense. Just that he is not the best communicator (2) It doesn't mean his idea is inherently wrong [12:03:22] <SbiisExE> But I really don't understand why you object to such a simple definition [12:03:36] <SbiisExE> to me it seems very clearly in the spirit of what Bowers' was writing about [12:03:41] its communicated in a vague enough way that there is "room for interpretation" [12:03:47] which is okay in art, and bad in mathematics [12:03:50] <SbiisExE> He obviously thinks of structures as "spaces" made up of entries [12:04:29] <SbiisExE> and the easiest way to make sense of them is to order the entries using ordinals rather than his more esoteric idea of literally listing the entries out in multiple dimensional space and beyond [12:05:21] <SbiisExE> He also clearly thinks that there is a structure/space corresponding to every dimension we could define using the powers of w [12:05:44] <SbiisExE> But I don't think you actually disagree with me [12:05:53] <SbiisExE> We're just saying the same thing in different ways [12:06:40] we both intuitively understand how BEAF is "supposed" to work [12:06:58] and i'm sure we both agree that the three main rules of beaf are clearly specified enough [12:07:20] ...given the foundations of how arrays are defined [12:07:22] <SbiisExE> okay [12:07:40] <SbiisExE> I dislike his definition of dimensional arrays btw [12:07:53] <SbiisExE> It's ridiculous that he needs that many words to define it [12:08:10] <SbiisExE> but that is because he didn't even bother to have a writable notation past planar arrays [12:08:16] and your stance on "arrays are functions mapping ordinals to positive integers" is not 100% ckear to me but i think it's agreement? [12:08:18] <SbiisExE> because the entries would be "in another dimension" [12:08:52] <SbiisExE> Chris Bird's approach is much more practical. In line everything, treat the "space" as a manner of speaking, and use delimiters to denote the same concept [12:08:56] <Wojowu> (I think we need it so that all but finitely many ordinals get mapped to 1) [12:09:10] Wojowu: i left that out to save typing [12:09:31] <SbiisExE> well I wouldn't agree with that [12:10:01] <SbiisExE> It's a weird function which takes number-ordinal pairs and maps it to a number [12:10:17] i think you misunderstood me [12:10:20] <SbiisExE> and of coarse it's not like a regular function, because it can take as many or as few arguments as you like [12:10:21] <Wojowu> Why should it take pairs? [12:10:32] <SbiisExE> The position and value of each entry [12:10:38] <SbiisExE> is an ordinal-number pair [12:10:43] an array itself -- separate from the "v(A)" function -- is itself a function mapping ordinals to positive integers [12:10:48] w/ the restrictio wojwo mentioned [12:10:51] <SbiisExE> take a few of these pairs and they define an array to solve [12:11:03] <Wojowu> No, an array would be a function from ordinals to numbers [12:11:03] <SbiisExE> I guess [12:11:05] beaf is a function that maps arrays to positive integers [12:11:13] <SbiisExE> but it's not the way I usually think about it [12:11:20] <Wojowu> So that, on input (position) we get output (value at the position) [12:11:32] <SbiisExE> you mean like {3,3,3} --> w^2*3+w*3+3, right? [12:11:36] no [12:11:59] {3,3,3} is the following function: f(0) = 3, f(1) = 3, f(2) = 3, f(3) = 1, f(4) = 1, .... f(w) = 1, ... [12:12:14] <SbiisExE> what? [12:12:23] <SbiisExE> oh [12:12:38] <SbiisExE> I don't see why your defining such a function as BEAF [12:12:50] this is the array itself, separate from evaluation [12:12:54] <SbiisExE> sure, that may be part of how BEAF works [12:13:04] <SbiisExE> but I'm talking about {3,3,3} mapping to a value [12:13:06] tritri is what you get when you plug this "f" into the evalution function "v" [12:13:08] <SbiisExE> of 3^^^3 [12:13:19] <SbiisExE> k [12:13:24] <SbiisExE> fine [12:13:26] <Wojowu> vell described a formal way to talk about arrays [12:13:31] <SbiisExE> k [12:13:39] do you have a better definition you suggest? you seem skeptical [12:13:49] <SbiisExE> like I said, we agree on the fundamentals. [12:14:00] <SbiisExE> You guys are mainly objecting to the terminology [12:14:35] <SbiisExE> But there is nothing wrong in what I said either [12:14:49] <SbiisExE> BEAF takes a set of ordinal-number pairs and returns a number [12:14:57] <SbiisExE> It's just a way of looking at it [12:15:24] <Wojowu> "set of ordinal-number pairs" is isomorphic to "function from ordinals to numbers" [12:15:32] <SbiisExE> so there is no problem [12:15:35] and your stance on "arrays are functions mapping ordinals to positive integers" is not 100% ckear to me but i think it's agreement? [12:15:38] so the answer to this is yes [12:15:41] <SbiisExE> how can I be "wrong" then and you "right" [12:15:41] gotcha [12:16:05] <Wojowu> Except that what you said allows things like (w,1) and (w,2) at the same time, but it;s technicality [12:16:15] <SbiisExE> right [12:16:20] <SbiisExE> I realized that of coarse [12:16:29] so from my definition, we can easily define what entries and structures are [12:16:46] <SbiisExE> okay [12:16:53] entry in array A is a pair (a,n) such that A(a) = n [12:16:55] <SbiisExE> except that you can't as I said earlier [12:17:15] <Wojowu> ? [12:17:17] structure is just an ordinal in the domain of A [12:17:21] <SbiisExE> This only formalizes the first layer of Bowers' idea [12:17:36] <SbiisExE> as long as we don't care how big the arrays get [12:17:39] <SbiisExE> then it's no problem [12:17:47] <SbiisExE> but try to figure out when we hit pentational arrays [12:17:55] <SbiisExE> and this idea, in and of itself, is insufficient [12:18:03] <SbiisExE> This that clear to you? [12:18:22] it is not, in fact [12:18:29] <Wojowu> Well, the idea of just having this v(A) function is not enough for anything [12:18:35] <Wojowu> Unless we define how v works [12:18:36] what stops us from defining pentational arrays w/ ordinals? [12:18:38] <SbiisExE> this idea cant not define pentational arrays [12:18:53] <Wojowu> Why so? [12:19:03] <Wojowu> We can have them as ordinals below zeta_0 [12:19:09] <SbiisExE> because of the disconnect between the number of entries and the ordinal expression you use [12:19:35] <Wojowu> This all collapses to defining fundamental sequences, I'd believe [12:19:45] <SbiisExE> Maybe so [12:19:53] the wikipedia article on the veblen hierarchy has defn's for FS's up to Gamma_0 [12:19:54] <SbiisExE> but no one has successfully done so [12:20:09] <SbiisExE> all the sequences so far arbitarily SAY what the size of an array is [12:20:18] <SbiisExE> instead of caring about what it actually is [12:20:21] what is the size of an array [12:20:32] <SbiisExE> I will try one more time [12:20:35] <Wojowu> Number of non-1 entries, I'd say [12:20:42] <SbiisExE> please, if you understand it, stop asking me the question [12:20:46] thanks [12:20:54] <SbiisExE> Let A be an ordinal [12:20:59] <SbiisExE> assume it's a limit ordinal [12:21:19] <SbiisExE> {b,p(A)2} produces A(p) entries = to b [12:21:33] <Wojowu> What is A(p)? [12:21:36] <SbiisExE> that function A(p) is the size of the structure [12:21:45] <Wojowu> Okay [12:21:50] <SbiisExE> a function which returns the number of entries [12:22:07] <Wojowu> You lost me [12:22:09] <SbiisExE> for example {b,p(0,1)2} would be the function A(p) = p^p [12:22:13] so you mean |Pi(a)| as defined here http://googology.wikia.com/wiki/Introduction_to_BEAF#As_ordinals_.28advanced.29 [12:22:19] <SbiisExE> right [12:22:21] <SbiisExE> basically [12:22:25] <SbiisExE> see what I mean [12:22:25] <Wojowu> Ah, okay, I see [12:22:33] <SbiisExE> you already know what I mean, you just don't know it [12:22:53] whatever yogi berra :P [12:22:55] <Wojowu> But, 0,1 isn't an ordinal [12:23:06] <SbiisExE> meaningless point [12:23:15] w [12:23:17] ^w [12:23:24] <Wojowu> That's better [12:23:25] <SbiisExE> Any set which can be ordered can be treated as an ordinal notation [12:23:33] <Wojowu> False [12:23:42] <SbiisExE> how is that false [12:23:49] <SbiisExE> sigh [12:23:50] <Wojowu> Set of real nmbers can be ordered :P [12:23:54] <SbiisExE> you guys love the technicalities [12:24:00] <Wojowu> Of course we do [12:24:06] thats why we do math [12:24:18] <SbiisExE> well anyway, you know perfectly well that has nothing to do with 0,1 as an ordinal notation [12:24:32] <Wojowu> Indeed [12:24:38] <SbiisExE> okay [12:24:44] it should be reasonably easy to map bowers delimiter notation to ordinals, but im too lazy to bother [12:24:53] <SbiisExE> So how do we define ordinals as opposed to order-types then? [12:24:58] <SbiisExE> this is a genuine question [12:25:17] um, an ordinal is an order type of a well-ordered set? idgi [12:25:24] <Wojowu> You want a formal defintiion? [12:25:28] <Wojowu> vell gave you one [12:25:47] <SbiisExE> well, I need a better understanding of well-ordered to make sense of that [12:25:58] !w well-order [12:26:03] !wp well-order [12:26:04] <XappolBot> http://en.wikipedia.org/wiki/well-order [12:26:17] <Wojowu> In well-ordered set, it's impossible to have infinite descending chain [12:26:29] <SbiisExE> okay [12:26:31] <SbiisExE> good enough [12:26:36] <Wojowu> So, for example, integers are not well-ordered, because we have 0>-1>-2>... [12:26:51] <Wojowu> But natural numbers are well-ordered [12:27:08] <SbiisExE> So then I can say, any well-ordered set, can represent an ordinal notation [12:27:13] "<Wojowu> In well-ordered set, it's impossible to have infinite descending chain" this is dependent on the axiom of dependent choice [12:27:20] but we're in zfc i'm assuming :P [12:27:38] <Wojowu> Well, maybe [12:27:54] <SbiisExE> I could understand the hesitation [12:28:15] <SbiisExE> But anyway, the point is, that I often refer to any well-ordered set as an ordinal notation [12:28:26] <Wojowu> Uhh [12:28:27] <SbiisExE> So to me it's clear that (0,1) is an ordinal notation [12:28:35] um [12:28:38] <Wojowu> What is ordinal notation for you? [12:28:40] <SbiisExE> what now [12:28:48] <SbiisExE> A way to label ordinals? [12:29:02] <SbiisExE> why, your going to know give me the technical definition of an ordinal notation? [12:29:07] "a partial function from finite strings in a finite alphabet to ordinals" [12:29:08] <Wojowu> In that case, well-ordered set doesn't give you ordinal notation [12:29:34] <Wojowu> Because if I say "set of countable ordinals" I give you no way of labelling ordinals [12:29:42] <SbiisExE> A well-ordered set can always be put into one-to-one correspondence with some subset of the ordinals, no? [12:29:51] <SbiisExE> god [12:30:09] <SbiisExE> You guys ever considered that some things should be implied [12:30:15] <SbiisExE> and that nitpicking wastes time [12:30:29] <Wojowu> But I honestly don't see what you mean [12:30:31] <SbiisExE> Obviously I was only thinking of stuff that can be bounded by a countable ordinal [12:30:34] i honestly dont get what you mean by "I often refer to any well-ordered set as an ordinal notation" [12:30:37] <Wojowu> Okay them [12:30:48] <Wojowu> "set of ordinals below epsilon_0" [12:30:53] <SbiisExE> I don't see what's confusing [12:30:59] <Wojowu> Does this set, per se, give you a way of labelling ordinals? [12:31:00] <SbiisExE> ExE uses a set of delimiters [12:31:05] <SbiisExE> that can be well-ordered [12:31:17] <SbiisExE> and so #^^^# is just another ordinal notation for gamma(0) [12:32:11] <Wojowu> Okay, maybe we can put this aside for now [12:32:15] <Wojowu> Back to BEAF [12:32:20] <Wojowu> From what I can see so far [12:32:38] <Wojowu> Only thing we have yet to say is how function A(p) looks depending on A [12:33:02] <SbiisExE> sure [12:33:12] <Wojowu> Wait, there is something else [12:33:13] <SbiisExE> and on how we might chose to select entries [12:33:24] <Wojowu> A(p) only tells us how many entries there will be [12:33:32] <Wojowu> Not where they will be in new array [12:33:37] <SbiisExE> ie. on an ordinal and something akin to it's fundamental sequence [12:33:46] <SbiisExE> sure [12:33:56] <SbiisExE> but it's easy to set it up in the definition to define this as well [12:34:10] <Wojowu> Yes [12:34:17] <SbiisExE> For example X&(b,p) = b,b,b,...,b w/p bs [12:34:38] <Wojowu> Now now, we only have to do so for all ordinals we care about [12:34:59] <SbiisExE> 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 [12:35:20] <SbiisExE> ie. treat it like a string [12:35:33] <SbiisExE> this makes the operations very simple and mechanical [12:35:34] <Wojowu> Can you please use ordinals instead of X's? [12:35:54] <SbiisExE> ordinal positions and multi-dimensional spaces can then be thought of an theoretical interpretations [12:36:00] <SbiisExE> why? [12:36:04] <SbiisExE> we are talking about BEAF [12:36:14] <Wojowu> Well, I thought we are trying to define BEAF in terms of ordinals now [12:36:17] <SbiisExE> so I assume we are trying to define the Xs [12:36:29] <Wojowu> Or do that [12:36:30] <SbiisExE> Ordinal notation are irrelevant? [12:36:39] <Wojowu> They actually are [12:36:42] <SbiisExE> ordinal notations are irrelevant? [12:36:45] <Wojowu> I believe [12:36:49] <SbiisExE> why do we have to keep returning to the same points [12:36:56] <Wojowu> ? [12:37:00] <SbiisExE> why do they matter? [12:37:16] <Wojowu> Because they give us a way to write ordinals down [12:37:22] <SbiisExE> (I just got through discussing how countable well-ordered sets can be used as ordinal notations) [12:37:27] <Wojowu> For example, w^w is an ordinal notation for some ordinal [12:37:27] <SbiisExE> of coarse [12:37:36] <SbiisExE> but using #^# or w^w or X^X shouldn't matter [12:37:43] <SbiisExE> so why bring it up? [12:37:57] <Wojowu> Because there isn't always an obvious isomorphism [12:38:07] <Wojowu> What should be zeta_0 in X's? [12:38:26] <Wojowu> If we use ordinals, we should stick to ordinals [12:38:32] <SbiisExE> also using something more esoteric like sigma(0,sigma(0,1)) [12:38:35] <Wojowu> And their standard notations [12:38:43] <SbiisExE> w/e [12:38:57] <SbiisExE> if ordinals are something that transcend notation then what you are saying is pedantic [12:39:06] <SbiisExE> we are sticking to ordinals regardless of what notation is used [12:39:22] <SbiisExE> what you mean is you want to stick to the standard notation [12:39:29] <SbiisExE> I don't see why that matters so much [12:39:43] <Wojowu> Actually, ordinals transcend all notations, but it's not the point [12:40:02] <SbiisExE> 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 [12:40:08] <SbiisExE> k [12:40:11] <Wojowu> True [12:40:14] <SbiisExE> what's the point then [12:40:20] <Wojowu> But can you prove that X structures are well-ordered? [12:40:30] <Wojowu> How do you order them, first of all? [12:40:33] <SbiisExE> can you prove w is well-ordered? [12:40:43] <SbiisExE> 0,1,2,3,4,5,... etc. [12:41:00] <Wojowu> If we define w in set-theoretic terms, yes I can