We can write down how long we should expect to wait to encounter every book in the Library of Babel with a pretty high probability, and it will be nothing peanuts compared to even some of these small large finite numbers.
The point is that this tree theorem stuff is a different combinatorial problem than the monkeys-on-typewriters problem.
Doesn’t that obviously follow from the definition? I am curious to hear about these applications, though!
I think what you’re saying is probably right, but the formulation might invite some misunderstandings. For one, of course, no particular integer is uncomputable; even for BB(745), there exists a finite program that outputs it, which is just ‘print BB(745)’. What is true is that the proposition that says something like ‘BB(745) is equal to (some ungodly large number)’ is independent of the axioms of ZFC. Also, what one considers ‘inaccessible to mathematics’ is to some extent a matter of… well maybe not taste, exactly, but certainly debate. The TM used to yield BB(745) encodes some problem undecidable in ZFC, but there are stronger axiom systems, involving e.g. large cardinal axioms, where that TM can be proved to run forever. So what is ‘inaccessible to mathematics’ will depend on what sort of axiom systems you’re willing to trust.
Quite possible, and the elaboration is appreciated. I should emphasize that it’s the BB function itself that’s incomputable, not any particular number. You can write a finite program to print the first n of them, but there’s no finite program that will compute all of them. Otherwise one could solve the halting problem by simply running any program BB(n) steps and seeing if it’s halted yet or not.
I may have been a bit poetic there. Still, ZFC encompasses what most people think of “normal” mathematics. I think it is quite remarkable that there is a finite boundary out there at which point our axioms are insufficient. It would be one thing if we were talking about hierarchies of infinity or something. But BB(745) is an otherwise perfectly ordinary integer, one that you could simply count to if you had enough time, and yet impossible to land on with the axioms you started with.
The weirder possibility is that ZFC actually is inconsistent, but it only gets that way for extremely long proofs which we have no hope of discovering.
But we can always include another axiom.
That’s true. Another striking demonstration of that comes via Chaitin’s halting probability \Omega: for any given \Omega, ZFC can determine only finitely many initial bits, meaning that there is some index beyond which the values of all further bits are undecidable propositions. Moreover, these aren’t necessarily long initial strings—in some cases, that index may even be zero!
So there are quite short sequences of ones and zeros, and yet, after a certain point, ZFC just shrugs it’s shoulders.
This is quite similar to my thoughts on the objection to infinity “not existing”. If there aren’t “really” an infinite number of natural numbers then what is the alternative? They somehow stop somewhere? Where is that point and why that point? Etc.
It’s not just enough to deny infinity, the denier needs to provide a workable alternative.
It quickly gets ridiculous weird if you try to suss something out along those lines.
It comes down to whether one can properly talk about “an” infinity– a singular object like a set that can be called an infinity. So to speak, can a TARDIS be infinitely large on the inside yet be bundled up into a single blue box? My uneducated guess is that yes you can but one has to avoid certain questions that might pop up as a result.
You can talk about anything you want, including infinite sets—if you do not, you are not going to get very far in set theory let alone algebra, number theory, etc.
The reason is that any mathematical statement or derivation [that you are going to deal with] has finite length. In fact, you can construct and check formal proofs (re. the Continuum Hypothesis, if you want something infinite) using a computer—play with it and you will see what I mean!
It has been a serious topic. We get into the world of finitism.
I’m not sure “deny” is the right term. It’s not like an infinity is anything that’s been experimentally observed. Maybe it’s better to say that if a mathematician doesn’t want to consider infinities, then they need to work with alternatives.
From that link:
" While all natural numbers are accepted as existing, the set of all natural numbers is not considered to exist as a mathematical object."
So: “You know those things over there. There’s a lot of them. Just don’t call them a set. Call them a herd.” The Abbott and Costello sketch practically writes itself.
One can explore various forms of finitism if one wants, but that does not mean there is any mathematical problem with using infinite sets. Unless I am missing something?
That’s pretty much what was done with sets and classes, though. Self-referential sets causing paradoxes? No problem. We’ll just call those a class. And classes aren’t allowed to talk about classes, just sets. Sometimes the naming actually matters.
Speaking of which, and apropos of the thread, it’s apparently been proven that all Chaitlin’s constants (there’s one for every programming language) are normal in every base. I don’t know if it’s reassuring that there’s something that we can know about those unknowable numbers, or disturbing that the only specific numbers known to be absolutely normal are the unknowable ones.
Please explain to an idiot. Pi is a ratio, yes? If it comes to zero, isn’t it done?
First, to quote Wikipedia:
So yes, it is a ratio (the ratio of a circle’s circumference to its diameter), but it’s not a rational number—that is, not a ratio of integers.
The decimal representation of pi starts out
As you can see, some of those digits are zeros. There’s no reason a decimal number can’t have digits that are zeros, followed by more digits that aren’t. (For example, it happens in the decimal representation of the fraction 1/11 = 0.09090909…)
In the part I copied, you don’t see “00” (two zeros in a row), but if I went on to show more digits, you would. It might take a while for you to see “000” (three zeros in a row), but it’s there eventually. The OP’s question was whether there is a limit to the number of zeros in a row in the decimal expansion of pi, if you consider all of the infinite string of digits that come after the decimal point.
It is counterintuitive, but easy to understand in the broad strokes: it’s (algorithmically) random. If we had any means whatsoever of predicting the digits (with a program that was shorter than the number of digits you hope to predict), it would be somewhat less than perfectly random.
If a Chaitin’s constant wasn’t absolutely normal, then given some string of digits (in whatever base), then you’d be able to predict the next one with better than even odds. Like if there was a maximum number of zeroes you could have in a row, then as soon as you had a string that long you could predict the next number as “anything but zero”. But that’s not possible due to its construction.
Unfortunately, none of this helps the case of pi, since there are short programs that can generate as many digits as you want.
I don’t think that holds up. A number could be non-normal without it being known just precisely how it’s non-normal.
Iteratively of course, as in “pi is the limit of repeating this program forever”. Otherwise pi wouldn’t be transcendent.