Alan Turing posited the state-machine in 1936 and we, programmers, are still living in his world. Think of a state-machine as a flowchart; we'll define the "order" of the machine with how many circles are in it. Esther/Ling Fu and Sarah Pan have a succint summary (pdf); lately youtuber "Up and Atom" has a video.
Alonzo Church proved that Turing Machines (TM) are equivalent to algorithms. So "language" means "possible input": the set of strings that the machine accepts
. Programmers know the "machine language" as the possible instructions the running CPU can understand; in turn your program is in a language, which all those layers of dot.NET hopefully interpret as making sense, down to the machine-language. Your enduser sends more strings to that running program; these are in the language you've told your enduser will work on your program.
Turing's machine comes with its own mathematic, rather metamathematic, quirks. Fu and Pan start with the "Acceptance Problem". Say there's a "language" which includes every TM as accepts an argument ω. Yay recursion! Turns out, that language is "undecidable". That seems... obvious; an infinite set of infinite possibilities. A subset of this language is those with all TM as halt on ω. There's the Halting Problem: that's undecidable too. Hence why any decent interpreter forces call-stacks upon our recursive functions. Which adds more circles to your flowchart.
I'm not here to prove the above. I'm cutting a lot out because (1) I'm rushing to the good bit and (2) you should read Fu and Pan.
So: not all flowcharts halt, and you can't tell if the flowchart halts ahead of time. What about the flowcharts that take too long before halting? That's the pathologic case of the "Busy Beaver". A halting flowchart of order 1 halts at the first step OBVIOUSLY. A two-circle 'chart, through the two-circle "maze", maxes stepcount at BB(2): six. They call that stepcount "state shift". BB(3)=21.
BB(4) and BB(5) proved... harder. It turned out that finding BB(n) was uncomputable if not undecidable: there's no equation or algorithm for "public ulong BB(int n) {...return??;}".
I wonder beyond a certain n, pathologic arbitrary-shift can be pseudocoded for inclusion as circles in the chart. Thus making that BB(n) infinite, if countably so. Looks like ZFC.
At the time of writing, Fu and Pan and everyone else knew BB(5)≥47176870, since someone wrote a function for that, but nobody knew if that was the highest-shift function for 5-order. Just last year, it was proven. This was shown, also, through a Turing Machine: by the "interactive theorem prover" Coq, which since everybody hates laughter is now in v.9 called "Rocq" (what would Alan think??).
BB(6) is at least 10^^15. Nobody ain't solving that until some major breakthrough in metamathematics is made, not even by "quantum".
The Goldbach Conjecture, per "code golf addict", has a 31-state halting TM since reduced to 27. The solution of BB(27) is, then, the limit to how long to wait before the Goldbach Conjecture will grind either to a solution or be nonhalting. The solution - if it exists - will be the proof that Goldbach was wrong. Riemann, meanwhile, over 2016-20 was sitting at BB(744).