<Deedlit> hey Kthulhu koteitan
<Kthulhu> Hey, Deedlit
<Kthulhu> I wanted to ask you a few questions about FOFT
<Kthulhu> is that alright?
<Deedlit> sure!
<Kthulhu> so from what I understood, 0-functionals are ordinals, right?
<Deedlit> yes
<Kthulhu> then how are 1-functionals comparable to oodinals? Are they the same, "bigger", "smaller"?
<Deedlit> they are a different thing than oodinals
<Kthulhu> I know
<Kthulhu> wait
<Kthulhu> one second
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<Kthulhu> so do you mind explaining it to me?
<Kthulhu> I'm afraid I don't really understand your blog post on the subject
<Kthulhu> I tried to do so, but I lack the complete background
<Deedlit> okay
<Deedlit> so first, do you understand FOOT?
<Kthulhu> yes
<Kthulhu> more or less
<Deedlit> so the idea there is that with FOOT we can describe these large ordinals
<Deedlit> and these ordinals allow us to have more expressive power
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<Deedlit> for example, the first big ordinal that FOOT talks about is what we call Ord
<Deedlit> and it is the smallest ordinal such that V and V_Ord satisfy the same sentences of FOST
<Deedlit> and since we can quantify over V_Ord, we can basically quantify over V in a sense
<Deedlit> so for example we can define a truth predicate over V_Ord (which mirrors a truth predicate for V)
<Deedlit> so now we have this stronger language FOST + Ord
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<Deedlit> and now we would like to quantify over this stronger language to obtain an even stronger language
<Deedlit> and so we define Ord_2 to be the smallest ordinal such that V and V_{Ord_2} satisfy the same sentences in FOST + Ord
<Kthulhu> one second
<Deedlit> and so now in the language FOST + Ord_2 we can quantify over V_Ord_2, so we can define a truth predicate for FOST + Ord for example
<Kthulhu> are you familiar with my norminals thing?
<Deedlit> it's been a while
<koteitan> (I am also listening your explanation)
<Kthulhu> well, I think it may be better for the both of us if we simply compare FOFT to some norminal
<Kthulhu> it's a much simpler system
<Kthulhu> so as it stands
<Kthulhu> FOST is equivalent to 1, when treated as a norminal
<Kthulhu> SOST is equivalent to 2,
<Kthulhu> aOST is equivalent to <1>
<Kthulhu> FOOT is equivalent to <0,1>
<Deedlit> looking at it now
<Kthulhu> which is otherwise notated <0<0>1>
<Kthulhu> this is where it's all present: https://sites.google.com/view/norminals/home
<Deedlit> I disagree that it's a much simpler system
<Kthulhu> really? hmm; maybe it's just a matter of familiarity
<Deedlit> it's more vague, but if you interpret it so that it works out like FOOT then it's the same thing really
<Kthulhu> it's really the same thing, but it goes onwards from there
<Kthulhu> so we've agreed that FOOT is equivalent to <0<0>1>
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<Kthulhu> where do you think FOFT would land?
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<Deedlit> to be as strong as FOFT, you would have to extend your notation very far
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<Deedlit> the strength of higher-order functionals is quite impressive
<Kthulhu> still
<Kthulhu> how do you think certain individual functionals compare?
<Deedlit> Func^0_alpha is along the lines of your <alpha>
<Kthulhu> I'd agree
<Kthulhu> given that Func^1_0 is like FOOT
<Kthulhu> from what I've read
<Kthulhu> would Func^1_a be like <a<0>1>, then?
<Deedlit> No, I think Func^1_a(b) would be like <b<0>a>
<Deedlit> then things really get going at Func^2
<Kthulhu> so Func^0_a(b) would be like <a>
<Kthulhu> so Func^1_a(b) would be like <b<0>a>
<Deedlit> I think so
<Kthulhu> well, how does b come into play with the case of Func^0_a(b)?
<Kthulhu> if it's equivalent to <a>
<Deedlit> Func^0_a is just an ordinal
<Deedlit> so there is no parameter
<Kthulhu> oh, right
<Deedlit> Func^1_a is a 1-functional, so it takes an ordinal to an ordinal
<Kthulhu> so Func^2_a(b) would be like...
<Kthulhu> <b<0>0<0>a>?
<Deedlit> nah, it would be a lot more powerful than that
<Kthulhu> then tell me
<Deedlit> Func^2_a takes a 1-functional to a 1-functional
<Deedlit> so for any function on ordinals, you get back a function on ordinals
<Kthulhu> hmm
<Kthulhu> (thinking)
<Deedlit> I would have to really analyze how far FOFT^2 goes, but I don't think that you can easily match it with a simple array notation
<Kthulhu> do consider what happens with the stronger norminals
<Kthulhu> even within the first mural, I am quite certain it can be contained within the norminals present there
<Kthulhu> I'll just give you a reminder
<Kthulhu> or to be precise, an explanation
<Kthulhu> SOST predicates over the truth of FOST, correct?
<Deedlit> SOST is stronger than that
<Kthulhu> hold on
<Kthulhu> right
<Kthulhu> my bad
<Kthulhu> I was getting confused between FOST+T and SOST
<Kthulhu> well
<Kthulhu> would saying "SOST predicates over the truth of the variables over which FOST predicates" be more precise?
<Deedlit> SOST quantifies over predicates of sets
<Deedlit> FOST quantifies over sets
<Kthulhu> yes
<Kthulhu> that's (more or less) what I meant
<Kthulhu> so N1 quantifies over sets
<Kthulhu> N2 quantifies over predicates of sets
<Kthulhu> N3 quantifies over predicates of those predicates
<Kthulhu> similarly, N<1> quantifies over these hierarchies of predication, for any ordinal
<Deedlit> I wonder, can you extend higher-order set theory to infinite ordinals?
<Kthulhu> "infinite"?
<Kthulhu> do you mean "transfinite"?
<Deedlit> same thing
<Kthulhu> no, it's not the same thing
<Deedlit> ?
<Kthulhu> infinite is "the greatest", it's ill-defined
<Deedlit> no, infinite just means not finite
<Kthulhu> that's transfinite
<Kthulhu> whatever
<Deedlit> no, we use infinite all the time
<Kthulhu> semantics
<Kthulhu> semantics
<Kthulhu> semantics
<Kthulhu> to answer your question, then
<Kthulhu> you can
<Kthulhu> and I did
<Deedlit> I think I would like outside corroboration on this
<Deedlit> I'll ask Wojowu when he comes around
<Kthulhu> I mean
<Kthulhu> he did too
<Deedlit> no, he didn't
<Kthulhu> higher order set theory with transfinite ordinals is Na, for some transfinite a
<Kthulhu> isn't it?
<Deedlit> but does it make sense?
<Kthulhu> it does
<Deedlit> what does it mean to have an infinite chain of "predicate of predicate of predicate of..."
<Kthulhu> I'll explain
<Kthulhu> let's start with omega
<Kthulhu> in Nw, we can specify, within the language
<Kthulhu> which finite ordinal we use to create the chain
<Kthulhu> for example, we can have an expression in the language that says "predicate of predicate of predicate of..." twelve googol times
<Kthulhu> similarly, we can do the same with larger ordinals
<Kthulhu> did you understand?
<Deedlit> oh, we can certainly do that
<Deedlit> basically you are diagonalizing over N -> Nth order set theory
<Kthulhu> yup!
<Kthulhu> that's what <1> does, only to all norminals
<Kthulhu> I'm starting to feel that FOFT^1 is lower on the scale than we both thought
<Kthulhu> *all ordinals
<Deedlit> FOFT intentionally avoids higher order set theory
<Deedlit> to put it on better philosophical ground
<Kthulhu> well, then I'm not sure why it's so strong
<Kthulhu> shouldn't SOST be stronger than FOFT, if it builds off something weaker?
<Kthulhu> as in, FOFT
<Deedlit> SOST is stronger than FOFT
<Kthulhu> oh, right
<Kthulhu> well, then,---
<Kthulhu> N2 would be stronger than anything FOFT related
<Kthulhu> if it was better-defined
<Kthulhu> as SOST is kind of- well -sick
<Deedlit> sure, if you believe in the absolute truth of Nth-order set theory
<Kthulhu> I don't take things for granted
<Kthulhu> if nobody has yet properly defined it, it's ill-defined
<Kthulhu> I'm not asking for a formal proof
<Deedlit> then you can define Rayo_n(m) to be "the smallest number greater than all numbers definable in nth order set theory using at most m symbols"
<Kthulhu> *definition
<Kthulhu> yes, but is "nth order set theory" well defined?
<Kthulhu> that's the crux
<Kthulhu> here's a thought!
<Kthulhu> if you sneeze N_<<1>>^10(10^100) times,
<Kthulhu> this video will eventually be created: https://www.youtube.com/watch?v=cB_AnteqNJ0
<Deedlit> ha
<Kthulhu> but on another similar subject
<Kthulhu> how do you think FOFT and little bigeddon compare?
<Deedlit> Emlightened's language is equivalent to FOFT^1_1
<Kthulhu> hmm
<Kthulhu> it's not much greater than FOOT
<Kthulhu> we've agreed that (if norminals were to use the FOST+T thing instead of SOST), Func^1_a would be equivalent to <0<0>a>
<Kthulhu> and that Func^0_a would be to <a>
<Kthulhu> right?
<Deedlit> assuming <a<0>b> is defined how I am thinking, yes
<Kthulhu> and how do you think it's defined?
<Deedlit> well, I would use the language I used before with FOOT
<Deedlit> that you didn't like all that much
<Kthulhu> describe it in terms of other norminals
<Kthulhu> that's how it's actually defined
<Deedlit> <a+1<0>b> would be the smallest ordinal such that V and V_<a+1<0>b> satisfied the same sentences of V + <a<0>b>
<Kthulhu> yes
<Kthulhu> but I'm talking about <0<0>a>
<Kthulhu> do you understand how <0<0>a+1> is defined?
<Deedlit> and <0<0>a+1> would be the smallest ordinal such that V and V_<0<0>a+1> satisfied the same sentences of V + the function b -> <b<0>a>
<Kthulhu> pretty sure that's right
<Kthulhu> to put it in my words,
<Kthulhu> <0<0>1> is the limit of all <a>
<Kthulhu> <0<0>a> is the limit of all <c<0>b>, where b < a
<Kthulhu> <b+1<0>a> is the limit of all <b<0>a>
<Kthulhu> so yes
<Deedlit> it's not clear to me what that means
<Kthulhu> read in my site, under the section dedicated to explaining what "limit" means
<Deedlit> I have to say, your definitions are quite vague
<Kthulhu> yes, I know they might be slightly vague
<Kthulhu> that's because my knowledge is rusty
<Kthulhu> still
<Kthulhu> one can understand what they mean
<Deedlit> it's somewhat hard to pin down actually
<Deedlit> for example, something like <0<0>1>_1 "which would be the limit of all well-ordered array norminals (similarly to how omega_1 functions)"
<Kthulhu> what's the problem? I could explain
<Deedlit> what exactly are the well-ordered array norminals?
<Kthulhu> oh
<Kthulhu> right, I got mixed up again
<Kthulhu> *facepalm*
<Kthulhu> I meant all reorderings, you know
<Kthulhu> wait one second
<Kthulhu> let me find it
<Kthulhu> there
<Kthulhu> tell me if the new definition is any good
<Deedlit> I still don't see what you mean
<Deedlit> what are "all array norminals"
<Kthulhu> is it any better now?
<Kthulhu> the limit of all norminals derived via surjection from <1>
<Deedlit> as far as I can tell, norminals are not derived via surjection from <1>
<Kthulhu> well
<Kthulhu> reordering?
<Kthulhu> something?
<Kthulhu> I'm trying to find the right word
<Kthulhu> you know the definition of omega_1, right?
<Deedlit> yes
<Kthulhu> well
<Deedlit> but I'm not sure how that helps here
<Deedlit> the next cardinal after <1> would not be all that much larger
<Kthulhu> yes, certainly
<Kthulhu> the limit of all norminals derived via some well-defined reordering of entries within the array, perhaps?
<Deedlit> I don't think that works
<Deedlit> if we take the set of arrays of the form <a1<A1>a2<A2>...<An>a(n+1)>, where the Ai are themselves arrays
<Deedlit> reordering entries would stay within the set
<Kthulhu> fine
<Kthulhu> I'll just erase that part
<Kthulhu> I mean, considering it isn't crucial for anything further than itself
<Kthulhu> there
<Kthulhu> done
<Deedlit> it doesn't look like you have any definition of AO[n] though, other than anything in AO[2] is bigger than anything in AO[1]
<Kthulhu> I do, that being:
<Kthulhu> AO[n] is bigger than anything in AO[m], when n > m
<Kthulhu> I do not know, they are the only cost way way of resurrection, I do not know, it is my name and I alea, but if we think your name is Weslie your child, older age I can say that I guess, and my growing tree of growth, I think to me as Lara Lala speaks Adult, my name is Maria, whatever I am 3 is simply buried I can not do planting the seeds, it is big in size, I will say what I want, mega oom where - white hair, but it shows tree
<Kthulhu> oh, whoops
<Kthulhu> wrong IRC
<Deedlit> that's not a definition
<Deedlit> AO[n] could be anything
<Deedlit> like, AO[n] = {n}
<Kthulhu> AO[n] isn't an individual norminal
<Kthulhu> read
<Kthulhu> the class-ordinary of AO[n] is, however
<Kthulhu> and that is defined as the smallest norminal in AO[n] that is bigger than anything in AO[m], when n > m
<Deedlit> but you haven't defined what AO[n] is
<Deedlit> just being bigger than the last one isn't even close to a unique description
<Kthulhu> it's a category of norminals
<Kthulhu> I don't see what else I have to add
<Deedlit> saying what it is would be a good start
<Deedlit> let's say I tell you this
<Deedlit> A_1 is a set of numbers
<Deedlit> A_2 is a set of numbers, all of which are bigger than the numbers in A_1
<Deedlit> okay, no I have defined A_1 and A_2
<Deedlit> *now
<Kthulhu> I already said what it is, on the site, I just don't remember to quote the whole thing in one go
<Deedlit> I've read your site, it's not there
<Kthulhu> "Introducing AO[2], a new type of norminal variable identical to AO[1], only that we specify two things: One, all AO[2] are larger than all AO[1]; and two, they are within a different category than AO[1]. This, inevitably, allows us to create languages so useful, hence functions so powerful, and hence numbers so large that the mind cannot even begin to conceive of them."
<Kthulhu> oh whoops
<Kthulhu> meant to quote less
<Kthulhu> "Introducing AO[2], a new type of norminal variable identical to AO[1], only that we specify two things: One, all AO[2] are larger than all AO[1]; and two, they are within a different category than AO[1]."
<Kthulhu> here I state what they are
<Kthulhu> "Note that these aren't individual norminals, but rather categories of norminals."
<Deedlit> that's just like the A_1 and A_2 I defined above
<Kthulhu> right
<Deedlit> can you tell me what numbers are in my A_1?
<Kthulhu> which is already there, great
<Kthulhu> hmm
<Kthulhu> A_1 is AO[1], right?
<Deedlit> nope
<Kthulhu> then what is it?
<Deedlit> I've told you
<Kthulhu> right
<Kthulhu> I see
<Deedlit> each A_n consists of numbers bigger than the previous numbers
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<Kthulhu> you see, with AO[n], we start with the class-ordinary of the category, then begin to build new norminals off it using the system already explained
<Kthulhu> a new class-ordinary is then defined, AO[n+1], as greater than anything in the previous category
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<Kthulhu> and the smallest one
<Kthulhu> because n will always be unique,
<Kthulhu> each class is unique
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<Kthulhu> and every unique norminal has to be in one unique class
<Kthulhu> that's pretty much it
<Deedlit> how is AO[1] defined exactly?
<Kthulhu> oh
<Kthulhu> as I said
<Kthulhu> it's the category assigned to 1
<Deedlit> and how do you define categories
<Kthulhu> ...a set?
<Deedlit> ...that can be anything
<Kthulhu> categories are classes of norminals, each defined by containing norminals smaller than each lower category
<Deedlit> you can do anything with that
<Deedlit> AO[n] could contain just the nth smallest norminal
<Kthulhu> I honestly don't see what's the problem here, or why we're stalling so much on this detail
<Kthulhu> let me try again
<Kthulhu> categories are classes of norminals, each defined by containing norminals smaller than each lower category, starting with their corresponding class-ordinary
<Kthulhu> *,
<Kthulhu> is that enough?
<Deedlit> nope
<Deedlit> you can fit just about anything into that definition
<Kthulhu> how so?
<Deedlit> just make the next category larger than the previous
<Deedlit> but each category can be whatever you want it to be
<Kthulhu> ...how would you do it, then?
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<Deedlit> I'm not sure that you can do what you seem to be trying to do
<Deedlit> it seems like you are trying to make AO[2] surpass all "array notations", or something like that
<Kthulhu> not exactly, but it's close to that
<Deedlit> but "array notation" is too vague and general to do that, I think
<Kthulhu> yes, which is why we have the "limit of" thing
<Deedlit> how does "limit of" solve things?
<Kthulhu> if we take some array notation, then "limit of" it, we get the next diagonalisation, right?
<Deedlit> well, for a specific array notation indexed by n, you can diagonalize over n, yes
<Kthulhu> so we have "limit of" to diagonalise,
<Kthulhu> and the simple successor to expand further
<Kthulhu> given those two things,
<Kthulhu> we can express "any array notation"
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<Kthulhu> the next category simply diagonalises over all that
<Deedlit> I think you are confusing things here
<Deedlit> if we have an ordinal, taking limits and successors will take us up through the class of ordinals
<Deedlit> for an array notation, you have to specify exactly what you are doing
<Kthulhu> well
<Kthulhu> I'm starting to get slightly tired of this now
<Deedlit> yeah
<Kthulhu> I'll think about it
<Kthulhu> it's entirely possible that all along,
<Kthulhu> actually, hold on
<Kthulhu> if we specify some variable,
<Kthulhu> that being the variable in AO[a]
<Kthulhu> , what's the problem of just using it?
<Deedlit> just using it for what?
<Kthulhu> to diagonalise over anything with a lower value to that variable
<Deedlit> I'm not sure what you mean
<Deedlit> but you don't want to have arbitrary diagonalization
<Kthulhu> I want to buy ONE yoga
<Kthulhu> oh, shit
<Deedlit> or arbitrary array notations
<Deedlit> lol
<Kthulhu> wrong IRC again
<Kthulhu> lol
<Kthulhu> what's the problem with arbitrary diagonalisation? I'd like to hear the specific reason
<Deedlit> well, it should be easy to define array notations A_n such that "the largest number in A_n using a googol symbols" is at least n
<Deedlit> in which case, if you can access all of those array notations, "the largest number definable in a googol symbols" becomes infinite
<Kthulhu> to be completely honest,
<Kthulhu> I was dodging this fact the whole time
<Kthulhu> I am ashamed of myself
<Deedlit> no need to be
<Kthulhu> oh well, XD
<Kthulhu> guess it's just a matter of definition higher and higher array notations...
<Kthulhu> because it is
<Deedlit> yeah, that seems like what one ultimately winds up doing in this quest for higher numbers
<Kthulhu> which was my point yesterday
<Kthulhu> it "kills" googology
<Kthulhu> no longer whimsical and fun, just diagonals over diagonals
<Deedlit> well, I actually find that interesting
<Deedlit> since it matters a great deal HOW you do it
<Kthulhu> not all too much,
<Kthulhu> anyone can expand an already existing array notation
<Deedlit> well yes
<Deedlit> if you take the view of "I can just take your number and beat it"
<Kthulhu> no, even without that
<Deedlit> no, I disagree
<Kthulhu> there aren't that many unique ways one can create an array notation
<Kthulhu> there's like
<Kthulhu> 1
<Deedlit> I think you just aren't familar with the stronger notations out there
<Deedlit> take ordinal collapsing functions for instance
<Kthulhu> I am, and I've seen one thing
<Deedlit> there isn't like 1
<Kthulhu> when you have something like that
<Kthulhu> it's really just the concise way of saying something similar
<Kthulhu> take TAN, for example
<Kthulhu> take BEAF
<Kthulhu> they're all, in the end, the same
<Deedlit> I disagree
<Kthulhu> different tastes of the same treat
<Kthulhu> well, I guess it's a matter of how you see things
<Deedlit> I guess so
<Deedlit> I don't see how you could say TAN is like BEAF
<Kthulhu> if you can find points of comparison, where the diagonalisation meets,
<Kthulhu> you found the similarity
<Deedlit> every number is representable as 1+1+1+...+1
<Deedlit> so every number is similar there
<Deedlit> but finding a powerful way to describe a lot of those +1's is a different sort of thing
<Deedlit> same with diagonalization
<Deedlit> TAN is way more powerful than BEAF, for instance
<Kthulhu> ...is it?
<Kthulhu> it might be, for all I know
<Kthulhu> I don't remember where I left off
<Deedlit> it is
<Kthulhu> at which point does it become more powerful?
<Deedlit> even at the first level of TAN, which only goes up to the Bachmann-Howard ordinal, goes beyond where BEAF is well-defined
<Kthulhu> hold up
<Kthulhu> are we talking about the same TAN?
<Deedlit> maybe not
<Deedlit> I was talking about Taranovsky's notation, but you were probably talking about something else
<Kthulhu> http://googology.wikia.com/wiki/User_blog:KthulhuHimself/Terminal_array_notation.
<Kthulhu> lel
<Kthulhu> that's what I was thinking about
<Deedlit> oh okay, lol
<Deedlit> but my point, is it's not just diagonals over diagonals
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<Deedlit> if your method is to diagonalize in some repetitive way, then you can be easily beaten by someone who loops over what you are doing
<Kthulhu> that's just the thing
<Kthulhu> for every method you create
<Kthulhu> there can be devised a method to beat it
<Kthulhu> using it
<Deedlit> but that's the point
<Deedlit> numbers are endless
<Kthulhu> that's the whole concept behind norminals
<Kthulhu> here, look
<Kthulhu> you have Func^a_b, right?
<Deedlit> okay
<Kthulhu> we can put that in an array
<Kthulhu> <a,b>
<Kthulhu> right?
<Kthulhu> actually, let's notate it (a,b)
<Kthulhu> just for clarity
<Deedlit> I know where you are going
<Kthulhu> we can then define (0,0,1) as the limit of all (a,b), by letting it access a and b
<Kthulhu> of course
<Kthulhu> it's obvious
<Kthulhu> I should just repair the part about categories and all that
<Deedlit> it sort of seems like a naive extension though
<Deedlit> because with Func^a_b we used more powerful methods then what you would get with a normal array
<Kthulhu> I know
<Kthulhu> come to think of it, everything I've done until now was futile
<Kthulhu> it all kind of is
<Deedlit> if you are looking for an endpoint, then yes, it is futile
<Kthulhu> no
<Kthulhu> that's not it
<Kthulhu> whatever
<Kthulhu> I'll come back sometime later with an answer
<Deedlit> okay