On Uncomputable Numbers 

Yup. The probability of finding non-computable random "real number" is 1. Belief in Emperor's New Clothes could be called a kind of experiencing, though?

Yeah, or apparently any random number? (like pi but with 3 random digits replaced or removed, like the Dr. Evans said?) The question of whether these numbers are interesting or not was satisfying haha Maybe they're not! (though I do find Chaitin's constant to be interesting)

@@DylanNCF ... The difference is that the probability that a random program halts is not a random number. It is a perfectly defined unique number. We don't know its value but that's a different matter. "Pi with 3 random digits removed" is not a well-defined number. There are many different numbers that match that definition.

adb012 You rock!

Numberphile mention Chaitin's number in the video below if someone get interested ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-5TkIe60y2GI.html

@@adb012 Could you elaborate on what you mean when you say "probability that a random program halts"

Opps, yes you are correct! Maybe I need to start a campaign for omicron = pi/8

No, no, no. There are people that DISCOVERED those things. You only have to UNDERSTAND. "I never understood" is not even acceptable.

Yes, these would all be computable. One way to think about this is to considering how you would compute them using the infinite series that converges to pi. If you can manipulate that series using rules of algebra to produce a series that converges to pi^pi that would be a way (not efficient, but that doesn't matter) to compute pi^pi with arbitrary precision.

Well of course it would @@DavidEvans so my question essentially was can you actually manipulate that series using rules of algebra to produce a series that converges to pi^pi? I would have no clue as how to do that.

​@@jimsteinmanfan80 you don't need to find a power series specifically for pi^pi, even though you could. Just observe there's a power series for pi. I can find pi to whatever accuracy as I want. Whatever I find is strictly a non repeating(in base 10) rational number. I can raise that to the power of itself, no problem. That gives me pi^pi to some measure of precision. Whatever measure of precision I want, there exists another measure of precision I need to calculate pi to in order to make sure pi^pi gets within the precision I want. Therefore it just is computable.

​@@DavidEvans This does raise an interesting question though. Pi is defined as the ratio between the circumference and the diameter of a circle. That's a proper and unambigious definition that we know can only describe one number. Now I can make multiple infinite series that converge to pi. But none of those are immediately trivial when looking only at the definition of pi. Couldn't it be that some other number has a similar definition, but that there exists no finite algorithm that allows us to compute it up to any desired degree of accuracy? Pi 'could have been' that number so to speak. That we could only get more precision on pi by physically making a large enough circle and using state of the art equipment to measure the ratio between the diameter and the circumference. We are 'lucky' pi isn't like that, but couldn't there be any numbers that are?

That's about the depth I would have written to if I had cared to @@dekippiesip but I thought completely meaningless since I have no idea without thinking a whole bunch what that measure of precision is and then it is not an algorithm, just a hunch that there is probably some. Of course it gets much harder if you take pi to the pi to the pi to the pi to the pi even though I suspect that is true also. The computation need to get just to one decimal would be absolutely enormous.

I think you are misunderstanding what it means for a number to be computable. Numbers can exist and be precisely described (in a non-constructive way) without being computable - but if the description is constructive (that is, it says how to mechanically construct the number), then it corresponds to an algorithm that can be described by a TM so much be computable. Chaitin's constant is not computable - the way you and others describe it does not give a method of actually constructing it, even though it precisely describes the number. The number you've written out is computable - assuming that the "..." means repeat the pattern forever, we can easily construct a machine that does that (for any number that can be written this ways), so the number must be computable. In this case, only a few states are needed - you just need to use part of the tape as a counter, and write out the number of ones, followed by a 0, and increase the counter.

Well I disagreed with the statement "if it can be described, that makes it computable" and judging by your answer you seem to agree. Of course, if it can't be described algorithmically it's uncomputable, but that's by definition. But it is a common misconception that TMs are the limit of what is describable. TMs have limits that we understand quiet well. E.g. it's easy to describe constructions with infinite states and TMs can't have that. Regarding the number I posted: Your intuition was mine as well but when I tried to write it out I realized that TMs can not simply read counters if they grow infinitely in size. Again, can't remember a proof of uncomputability, so very happy to stand corrected.

@@iamtheguitar I don't think I actually say that - at least looking at the part of the transcript that is closest to this (starting at 7:52: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Cuf4zkYwGlQ.html), I say that if we can describe a number in a "straightforward way" then we know how to compute it. I agree this is a bit ambiguous, and maybe there are ways to describe Chaitin's constant that some would think are straightforward, and important to point out that what it means to "describe a number" is not well defined. Regarding your claimed uncomputable number, there is no limit to how high a TM can count - they have an infinite tape. The definition of computable is precise, and any number that you can write one with a repeating pattern is most definitely computable - it can be output to arbitrary precision, by generating any number of digits of the number.

Your number is irrational and transcendental and computable like pi and e, but unlike sqrt(2) and the goldenrod ratio which are irrational but algebraic. There is no polinomial with rational coefficients that will have your number (or pi, or e) as a root. But is perfectly computable because there is an algorithm that can approximate it in finite steps with any arbitrary precision you like. IfvI told you write the first n digits, you know how to do it doesn’t matter if n is 10, 1000 or a goggleplex. It may take millions of years to complete the task but the number of steps is finite.

@@adb012 Are you talking about Chaitin's constant? It's proven to be uncomputable. It would solve the halting problem