What do you think of Googology?
>>17016302Not a legitimate field of research. Some of the big numbers in question point to something with real applications but the concept of googology is just "ooooh, number big."
>>17016308>but the concept of googology is just "ooooh, number big."No, not really. Way more often it deals with sequences and ordinals rather than big numbers themselves. And the framework and notations it provides is useful when you need to compare big numbers that arise from other fields like graph theory, computer science, logic and such
>>17016316what are concrete examples of this?
>me when learning that the numerical value of TREE(3) is so large that even trying to prove it’s a finite number would require so many mathematical symbols that a black hole would instantly form from the sheer amount of informationOkay, but what about the largest finite number that can be defined using TREE(TREE(3)) amount of mathematical symbols? Checkmate, googolologists.
>>17016316You have it backwards. The "framework and notations" of googology are borrowed from the fields said big numbers arise from. Ordinals are from set theory. TREE(n) is from graph theory. Rayo's number extends from first-order logic. No "Googologist" ever defined a non-trivial large number with any applications.
>OOOOHH, NUMBER BIG
>>17016350I mean, it is interesting how new notation systems need to be developed to define increasingly large numbers. Eventually you need to start using formal logic systems because there’s no notation that could possibly define a number so large. That’s has legitimate intellectual value that goes beyond “oooo, number big”
>>17016354what value? how big is this value as far as numbers are concerned?
for me it's Cantor's Attic
>>17016319Well, you have large numbers and function growths arising from various fields of math. And if you want to compare them or just estimate their general size, you need tools for that. Of course you can construct proofs for each pair you want to compare, but it's often more useful to use things like fast growing hierarchy(with sufficiently powerful ordinal system). It's not very unlike from using big-O notation for comparing algorithmic complexity, both originally arise from complexity theory anyway.For example TREE sequence far exceeds [math]f_{\Gamma_0}(n)[/math] in FGH which is as far as the standard notation of ordinals goes AFAIK, so you gotta look for alternative, more powerful notations if you want to reason about numbers at this scale, like ordinal collapsing functions.>>17016345>Ordinals are from set theory. TREE(n) is from graph theory. Rayo's number extends from first-order logic.>No "Googologist" ever defined a non-trivial large number with any applications.I literally said that the numbers come from other fields of math. Yes, you can trivially define very large numbers given sufficiently powerful notation, but that's not why they were invented. The actual value comes from analyzing numbers you get from other places in math.
>>17016350This image is terrible.
>>17016374you're not answering the question. i asked for a concrete example and you just gave me more nonsense. congratulations, you're very good at saying nothing
>>17016374>I literally said that the numbers come from other fields of math.That wasn't the point I was making. I was replying to this:>the framework and notations it [googology] providesGoogology literally adopts the framework and notation from other fields. It never provided any of it.
>>17016380>Way more often it deals with sequences and ordinals rather than big numbers themselves. And the framework and notations it provides is useful when you need to compare big numbers that arise from other fields like graph theory, computer science, logic and such>what are concrete examples of this?Well, I gave you a concrete example of such sequence arising from other field (graph theory) and example of tools you'd use to deal with it. If you are expecting me to provide a concrete proof of bounds of tree sequence in terms of FGH with some non standard ordinal system then that's rather beyond my capabilities. It's not like we even have good bounds for that, there are some proofs circling around stackexchange and such but there were some problems found with it eventually.>>17016381Well sure like any area of math it shares terminology and notation from other areas, but things like BEAF, Hyper-E and their extensions, various generalization of veblen function and such did arise from googology.
>>17016354>Eventually you need to start using formal logic systems because there’s no notation that could possibly define a number so largeThat's a very hand-wavy thing to say and could mean a lot of things. I would be rather careful with statements like that.I think it's very important to make a distinction between computable and uncomputable googology. You can define uncomputably large numbers like Rayo's or LNGN using some higher order set theories, but so what. It's not like you can ever reach them or even compare them(if they are based on different theories), as in compute them, it just boils down to "biggest number you can name using X symbols" but actually properly defined. I mean, PTOs and related ordinal analysis of mathematical theories is very interesting at least, but they are not tangible and as a *cough* computer scientist I find more interest in computable numbers. Here you actually provide means to construct these numbers and it comes with unique challenges. Like how you inevitably reach fixed points if you try to naively extend recursive ordinals, and how you can mitigate them and numerate over them just to find more general fixed points. I like to think about computable ordinals in context of FGH as encoding of specific combination of recursions and diagonalizations, allowing for describing growth rate limits of functions or even what kind of functions can be proven to be total within a given theory.
