wtf

Yes. If you've read my posts you know that. How could you ask? How many times have I agreed? I think I might see what you're saying though. You pick an uncountable set of reals. What is the probability that pi is in the set? Is that a better representation of your question? I believe the answer is that it's not well defined. I'll have to think about that more. To analyze it, first we have to restrict our attention to the unit interval, so that we have a probability space whose total measure is 1. I've been ignoring that point but in this context it matters. So we can say reasonably that, say, if you pick a random real in the unit interval, the probability is 1/2 that it's between 0 and 1/2. The probability is 1/3 that the chosen number is between 1/3 and 2/3, say. So we can assign probabilities to SOME subsets of the reals, but not all. What if you pick an unmeasurable set of reals in the unit interval? Then there's no probability at all. So you'll have to be more specific about this uncountable set of reals. After all: Some uncountable sets of reals include pi; and some don't. Some uncountable sets of reals have a measure; and some don't. So your question is not well-posed. But I do think I understand your question now. https://en.wikipedia.org/wiki/Non-measurable_set

Because there are only countably many such real numbers ending in the digits of pi. You can't get around that. The measure of the set of reals whose decimal ends in pi is zero. > You seem to focus on the probability that a single monkey will make it. No, I have calculated the probability that an uncountable collection of monkeys will make it. The probability is zero. There are uncountably many real numbers, of which only countably many end in the digits of pi. Perhaps you are imagining that countably many strings generated by one monkey times uncountably many monkeys gives you a larger set of possibilities. But the product of a countable set and an uncountable set equinumerous to the reals is has the cardinality of the reals. > Which means, the monkey completes the job successfully if and only if it types π and nothing else, that's it. Well there is only ONE such string, namely pi itself.

> it means smaller than any positive number to which you can assign a value Can you give an example or an explanation of a positive number to which you can't assign a value? I can't imagine what that could possibly mean. Like 14. I can assign the value 14 to it. What does it mean to assign a value to a number? Isn't the value of a number the number itself? I cannot understand this remark at all.

They might get lucky with probability zero. A single monkey might get lucky with probability zero. Remember that probability zero doesn't mean impossible, it just means probability zero. Like the rationals in the reals. Do you agree that there are only countably many reals whose decimal representation ends in pi? Because there can be only a finite sequence of digits preceding the point where pi starts. And there are only countably many finite strings of digits. You agree?

I answered the question correctly, now that I've had a little time to rethink it. How many real numbers end in the digits of pi? Remember, pi must be the ENTIRE TAIL of the number. So only the first FINITELY MANY digits are unconstrained. There are only countably many finite sequences of digits. So the class of all real numbers that end in pi is countably infinite. This seems perfectly correct. In other words, what do some of these reals look like? .123pi, .343424324332pi, .50438590438590430958430pi, etc. Every possible finite sequence, followed by the digits of pi. There are exactly a countably infinite number of these. Out of an uncountable collection of monkeys generating random real numbers, only countably many of those reals will end in pi. Because there aren't any more than that to be found. But there are uncountably many reals. Look at it yet another way. Every such real looks like this: (terminating rational) followed by the digits of pi. There are only countably many terminating rationals.

> I'm sorry there are incorrect statements in your postings in this thread. No, unless minor typos. I stand by everything I've written here. > A Continuum > Def:- A compact connected set. Really? So the real numbers aren't a continuum? Whateva dude. > He pointed out that it doesn't matter that you can't practically go on typing forever. No, it was Russell who invented type theory. [That was a joke]. > The integers are not connected. Um ... what does that have to do with anything? > So consider the monkey typing away forever and consider the set which holds that output. > What do you consider the cardinality of that set to be? We're modeling the output of a monkey as a countably infinite string of symbols, same as the decimal representation of a real number.

Oh, ok. Probability is zero. Same as the idea of a real number "containing all the digits of pi." So now your question is, what if there are uncountably many monkeys. In other words if we have uncountably many real numbers, what's the measure of the set of those numbers ending in pi. Well let's see ... Say a sequence ends in pi. Then its first FINITE number of digits can be arbitrary. There are countably many finite sequences. So there are only countably many possible sequences that end in pi, out of uncountably many possible sequences. Measure zero. Probability zero. What do you think?

No that's not true. The key to the CWS is that the complete works of Shakespeare is a FINITE sequence of symbols. And we know that almost all real numbers are normal, hence disjunctive. So with probability 1, a random countably infinite string not only contains the CWS, but contains the CWS infinitely many times. But now you are asking about an INFINITE string. Why should some random infinite string contain every INFINITE string? And where would such a string appear? Remember the string is contiguous as in the digits of a real number. You can't have the digits of pi appearing as a substring in the middle. It would have to be a tail of the sequence. Your analysis isn't correct here.

> Yes, a single monkey will type the works of Shakespeare on repeat with probability zero. Because the single monkey will be restricted to type a single sequence of letters, whereas there is continuum many possible sequences to choose from. No, the probability is 1. I already explained that.. Remember the key fact is that the CWS is a finite string of symbols. It's exactly the same question as asking whether a random real number is disjunctive. It is, with probability 1. As well as normal with probability 1. So a random countably infinite strings of characters from the English alphabet contains the CWS infinitely many times with probability 1. > But that was not the question. The question was what happens if you engage not a single monkey, but continuum infinitely many. The number of monkeys will be the same as the number of possible sequences. You expect that the expected number of monkeys to achieve the correct sequence will be equal to 1, no? I'd expect the number of monkeys to type the CWS to be an uncountable infinity. That's because the chance of each one doing so is 1, as we've already seen. And now there are uncountably many monkeys. Almost all of them will type the CWS. If I'm misunderstanding your question please let me know.

It's a popularized fable to give people a feel for the mathematical infinite. Sort of like Hilbert's hotel, which is an infinite hotel that can nevertheless accommodate infinitely many new guests without having to displace anyone. Sort of like a rubber sheet with gridlines drawn on it, and a bowling ball in the center pulling the sheet down, showing how mass distorts space to create gravity. Of course the question is ... what pulls the bowling ball down? So these stories should not be taken for science; but rather, as science-y fables.

Great article, thanks. Yes that's a really interesting point. We're so much more familiar with the everyday rationals, even though they are the great outliers in the reals. That's another interesting thing about the normal numbers. The normal numbers are real numbers whose decimal representation has every finite string a statistically equal number of times. It's been proven that almost all (all but a set of measure zero) real numbers are normal. However, very few real numbers have been proven normal. When people speculate that "Pi contains the names of all the people you will ever know," which is a meme that was floating around a while back, they're assuming that pi is a normal number. But in fact that's an open problem. Nobody has any idea if pi is normal. [Edit - as with Shakespeare and the monkeys, that requires pi to be a disjunctive number, a point many online are confused about. I didn't want to add to that confusion. If all we need is for every finite sequence to appear at least once, that's disjunctive].

Hope that was helpful. I searched for a simple online explanation but found nothing. The Wiki article on probability theory doesn't say anything about this topic, and searches on "infinitary probability theory" brought up advanced academic articles. https://en.wikipedia.org/wiki/Probability_theory But the Wiki article did mention the connection to abstract integration theory. So if you took calculus, then you know that even for the Riemann integral, changing the value of one point doesn't matter. If I have some function defined on the unit interval whose integral is 47, say; then another function that differs from the first at only one point has the same integral. It's also not hard to prove that you can change the value at any countable set of points without changing the integral; and in fact you can change the value on any set of measure zero without changing the value of the integral. [I don't recall and didn't feel like looking up whether that last statement applies to the Riemann integral or only to the more advanced Lebesgue integral]. Modern probability theory is based on modern abstract integration theory. Everything you can say in probability theory is only true "up to a set of measure zero." This situation comes up so much that we have the terminology "almost all," or "almost surely," attached to a lot of theorems. As a familiar example, it's mathematically correct to say that almost all real numbers are irrational. "Almost all" is a technical term meaning, "all but a set of measure zero." We can view this as a probability. If we "randomly pick a real number," it will almost surely be irrational. Even though the rationals are sitting there saying, "Pick me, pick me!" There just aren't enough of them to make any difference in the scheme of things. [This is true even though the rationals are dense in the reals; that is, there's a rational between any two reals. This is an intuitive paradox but not actually a mathematical paradox. It's just one of those strange things you get used to]. So all of this is just a natural extension of the fact from freshman calculus that integrals are about behavior in the limit; and they allow for many individual points of exception to the statistical certainties; which should always be understood to be almost certainties. Here's a math.stackexchange thread on the subject. https://math.stackexchange.com/questions/41107/zero-probability-and-impossibility It's worth mentioning that these ideas bear on the speculative physics of "many worlds" and related ideas. People say that if there are infinitely many universes then "everything must happen." This is false for the same reason. You can flip infinitely many coins and have them be all heads. There's no reason why not. Likewise you can have many exceptions to the rule of "everything" happening in a multiverse; likewise measure zero events can happen. Like life on earth, maybe. It's important to remember that all such infinitary arguments are essentially statistical arguments and do not prohibit individual exceptions.

The bolded part? The measure of the rationals in the reals is zero. If you "throw a dart at the real line," or randomly pick a real number out of a hat, the probability that the number you pick is rational is zero. Yet you might get lucky and pick a rational. In infinite probability theory, probability zero doesn't mean an event is impossible; and probability 1 doesn't mean that it's certain. As a (perhaps) intuitively visualizable example, suppose I flip a fair coin infinitely many times. The probability that they're all heads is zero. Yet there's no reason why they couldn't all be heads. It's just really really unlikely. As unlikely as the sequence of coin flips representing a rational number in binary [with a binary point in front of the sequence so that it represents some real number in the unit interval].

The probability of choosing one particular real number at random from the set of real numbers in the unit interval (so that the total measure of the probability space is 1) is zero. As always that doesn't mean it can't happen ... just that the probability is zero.

Not at all. As I pointed out, one monkey generating a countably infinite sequence of letters is exactly equivalent to infinitely many monkeys doing the same. A countable union of countable sets is countable.

You're getting up and going to work tomorrow, aren't you?

I have a proof that any finite string will almost surely be generated by infinitely many monkeys typing forever. First, we have to make precise the idea of infinitely many monkeys typing forever. If we regard striking a key (or randomly choosing a character) as a discrete event, it makes sense that a single monkey, typing forever, will generate a countably infinite string. Now if we have countably infinite monkeys, they will generate countably many such countably infinite strings. Since a countable union of countable sets is countable, we gain no more strings than if there is just one monkey generating a countably infinite string. Now the complete works of Shakespeare are certainly a finite string of symbols. So the question comes down to this: What is the probability that a given countably infinite string contains every possible finite string as a substring somewhere in it? Note that as long as our alphabet is finite, it makes no difference whether these are strings of alphabetic symbols, or just the symbols 0 and 1. A real number with the property that its decimal (or binary, same difference) expression contains every finite sequence at least once, is called disjunctive. A sequence is called normal if it contains each finite sequence an asymptotically equal number of times. In other words as we go to infinity, each finite string occurs infinitely often. It's clear that being normal is stronger than being disjunctive. And it's known that the measure of the normal numbers in the reals is 1. This does not mean that ALL reals are normal; only that if you pick a random real, it will be normal with probability 1. There could be exceptions, in the same sense that the rationals have measure zero in the reals. A randomly chosen real is irrational with probability 1, yet the rationals still exist and you might get lucky and pick one. In the same way, it's possible that a randomly chosen real may not be normal, but the probability is zero. Remember that in infinitary probability, 0 doesn't mean impossible and 1 doesn't mean certain. We use the phrase "almost surely" to mean the probability is 1. Since being normal implies being disjunctive, the probability of picking a random real that's disjunctive is also 1. However it's conceivable that we may be really unlucky and pick a measure zero event. So with probability 1, a countably infinite string of letters of the English alphabet will contain every possible finite string; including the finite string that matches the complete works of Shakespeare. There are sequences that might not contain the works of Shakespeare, but they're as rare as the rationals in the reals. Note that it makes no difference whether you generate that infinite sequence with one monkey typing forever or countably many monkeys typing forever. It's the exact same thing. https://en.wikipedia.org/wiki/Disjunctive_sequence https://en.wikipedia.org/wiki/Normal_sequence

You can track an IP back to an ISP. But at that point you need a court order to get the ISP to reveal who is the account behind that IP.

No, that is exactly wrong. A limit is what you get when the increment is ARBITRARILY small. It's always strictly positive but gets as close as you like to zero. I now see clearly the source of your confusion. You don't know what a limit is. You have a freshman calculus understanding at best. If you would take the trouble to learn the actual definition of a limit, you would see that no infinitesimals are involved.

Can you say more about second order logic in this context? My understanding is that nonstandard analysis is an alternative model of the FIRST order theory of the real numbers. If you go to second order logic you can express the completeness theorem (every nonempty subset of reals bounded above has a least upper bound). And any system containing infinitesimals is necessarily INCOMPLETE. So second order logic would seem to preclude infinitesimals entirely. Not an expert but would appreciate context.

There's a pdf of Bell online.

Hardly bears on the history of the limit concept and whether smooth infinitesimal analysis was prefigured in the 17th century.

And then I need a button that locates my keys.They can invent the Internet but I still can't find my keys.

You know what they need to invent? A button on the tv that makes the remote beep so you can find it.

So how would you extract 12 and 13 from (12,13)? You can work this out by experimenting with the syntax.