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It boggles my mind that a number (an uncomputable number, granted) like BB(748) can be "independent of ZFC". It feels like a category error or something.
I wonder if the visible universe is large enough to write down the exact value of BB(6).
It boggles my mind that a number (an uncomputable number, granted) like BB(748) can be "independent of ZFC". It feels like a category error or something.
It boggles my mind that a number (an uncomputable number, granted) like BB(748) can be "independent of ZFC". It feels like a category error or something.
The number itself is not independent of ZFC. (Every integer can be expressed in ZFC.) What's independent of ZFC is the process of computing BB(748).
It boggles my mind that a number (an uncomputable number, granted) like BB(748) can be "independent of ZFC". It feels like a category error or something.
The category error is in thinking that BB(748) is in fact, a number. It's merely a mathematical concept.
I wonder if the visible universe is large enough to write down the exact value of BB(6).
The mass-energy includes ordinary matter, dark matter, and dark energy. Current estimates suggest the observable universe contains roughly 10^53 kg of mass-energy equivalent.
Plugging these into S ≤ 2πER/ℏc gives someting on the order of 10^120 bits of maximum information content.
S ≤ 2πER/ℏc
S ≤ (2 × 3.141593 × 3.036e+71 × 4.399e+26)/(1.055e-34 × 299792458)
S ≤ 2.654135e+124
S ≤ 10^120
So, no.
I thought it was a typo. First time I encounter tetration.
I wonder if the visible universe is large enough to write down the exact value of BB(6).
It's not.
Look at 3 sub 10 = which is (10^(10^10)). So that is 10 to the power of 10 billion. In regular decimal notation, that is a "1" with 10 billion "0"s following it. It takes 10 gigabytes of ram to represent the number in decimal notation, naively.
The number of atoms in the universe is only 10^80, or 1,000...000 (80 zeroes). 10-million sub 10 is so huge, how much ram to represent it.
This example is from https://www.statisticshowto.com/tetration-function-simple-de...
It boggles my mind that a number (an uncomputable number, granted) like BB(748) can be "independent of ZFC". It feels like a category error or something.
The category error is in thinking that BB(748) is in fact, a number. It's merely a mathematical concept.
If you are refering to the idea that nothing that can't exist in the real universe "really exists", then the "Busy Beaver" portion of that idea is extraneous, as 100% of integers can't exist in the real universe, and therefore, 100% of integers are equally just "mathematical concepts". That one of them is identified by BB(748) isn't a particularly important aspect. But certainly, a very specific number is identified by that designation, though nothing in this universe is going to know what it is in any meaningful sense.
Earlier quoted context omitted.
The number itself is not independent of ZFC. (Every integer can be expressed in ZFC.) What's independent of ZFC is the process of computing BB(748).
Sure, if someone just gives you the number, ZFC can represent it. But ZFC cannot prove that the value is correct, so how do you know you have the right number? Use a stronger proof system? Go a bit bigger and same issue.
Most of these 'uncomputable' problems are uncomputable in the sense of the halting problem: you can write down an algorithm that should compute them, but it might never halt. That's the sense in which BB(x) is uncomputable: you won't know if you're done ever, because you can't distinguish a machine that never halts from one that just hasn't halted yet (since it has an infinite number of states, you can't just wait for a loop).
So presumably the independence of a number from ZFC is like that also: you can't prove it's the value of BB(745) because you won't know if you've proved it; the only way to prove it is essentially to run those Turing machines until they stop and you'll never know if you're done.
I'm guessing that for the very small Turing machines there is not enough structure possible to encode whatever infinitely complex states end up being impossible to deduce halting from, so they end up being Collatz-like and then you can go prove things about them using math. As you add states the possible iteration steps go wild and eventually do stuff that is beyond ZFC to analyze.
So the finite value 745 isn't really where the infinity/uncomputability comes from-it comes from the infinite tape that can produce arbitrarily complex functions. (I wonder if over a certain number of states it becomes possible to encoding a larger Turing machine in the tape somehow, causing a sort of divergence to infinite complexity?)