ultrafilter: I can’t quite remember the terminology, but wouldn’t it make sense to talk about an event as {x subset S | x gives the desired outcome}? Then the event ‘2222’ corresponds to the empty set.
Thanks Ultrafilter and Jabba for your last posts.
can I just check …
for infinitesimal x nx<1 for all positive integers n
If this is so doesn’t it follow
x<1/n for all positive integers n
but aren’t 0, and 1/n, rational numbers?
I seem to remember that there are an uncountable infinite number of reals between any two rational numbers. If my memory was right then doesn’t it follow that.
There must be an infinite number of reals between the 1/n and 0 for all positive integers n. So where do we get that x is positive and less than any positive real number, not just less than any positive rational number?
Bippy: Because any positive real number r has some 1/n less than it (for instance, pick any n greater than 1/r), so x<1/n<r.
But that contradicts the ‘there are an uncountable infinite number of reals between any two rational numbers’ 0 and 1/n being rationals. Though I will agree if that remembered fact is just plane wrong. (and if it is wrong, or wrongly used in this case, please tell me so I can learn).
For counter-claim
for every 1/n where 1/n > 0 there is a real r such that 0<r<1/n by generating r from n by changing any non zero digit in the expression of 1/n to the digit 0.
How does it contradict it?
Maybe it will help to think of it this way, Bippy.
Forget for the moment all of the construction of the hyperreals and that sort of thing. To see that it’s possible to have a positive infinitesimal in the reals (and by that I simply mean a “number” smaller than any positive real, yet greater than zero), just consider the reals as a linearly ordered set (to simplify things, just ignore any algebraic structure, since it seems to be the just ordering structure that you’re unsure about). Now introduce an element, call it e. We can simply declare two things:
- e > 0
- if x is a positive real number, then e < x.
(then extend the order transitively).
Do you see that we can do this? I can define the order structure any way I want (as long as it’s consistent).
We still have an uncountable number of reals between any two reals. There are no reals between e and 0, but that’s not a problem, since e wasn’t a real in the first place.
Also, if I start at zero, and jump up to any positive real, no matter how small, I’ve skipped over uncountably many reals (along with skipping over e, of course).
Does that make it any clearer?
There certainly are. I may be wrong here but could it be what you haven’t realised is that there are different infinties of real numbers - that is, that given any positive real number, there’s a rational less than it, but, given any rational, there’s some real number which it’s greater than, but specifically which real number depends on the rational…?
Sorry I thought that the e>0 and e<1/x for all integers x was a definition for the infinitesimal e. Which lead me to believe e was bound by rationals not reals.
If e>o and e<x for all positive reals x, then it makes sence to me that you say e is postitve and smaller than any positive real.
Again thanks for helping me catch up on some of the interesting Math I missed by doing a joint math/physics honours followed by physics post grad.
I’ve never seen it done that way, but I guess it makes sense…
Really, it depends on what you pick as the universe of events. If the universe is the set of infinite length binary strings, then it’s not sensible to talk about the probability of getting 2… But if the universe is the set of infinite length ternary strings, then it has probability 0.
AHA! I just remembered what I was thinking of: random variables. That is, the probability space is often something fixed, but what one thinks of as the actual outcomes are in the codomain of the function.
So, in a case like this the r.v. is essentially the constant function, but you can naturally extend the codomain to anything without changing the function, so X[sup]-1/sup has an obvious meaning.
Does that make any more sense?
I don’t see anything obviously wrong with it. I’ve never studied random variables at that level, so I’ll bow to your expertise on the matter.
Please don’t; I have technically studied measure theory, but have forgotten a lot, and this is based on hazy recollections of probability in first year of degree and at A-level. I was kind of hoping to defer to somebody else on the matter, but no-one seems forthcoming 
As you said, 000… has a positive probability for any finite sequence and thus can happen. An infinite sequence has zero probability and cannot happen, because an infinite sequence is not reachable. When we say that the limit of the sequence is zero, what we are really saying is that, given any value e > 0, if we take enough terms in the sequence we will find a value x with the property e > x > 0.
The same is true for the zero probability of having a particular outcome in a continuous pdf.
The probability of having a value in the interval (a;b) is the integral from a to b of x.p(x).dx, where p(x) is the pdf of the random variable x.
So, the probability of having a value in the interval (a-e;a+e) is the integral between those limits. The limit of the integral when e tends to zero is zero.
We have to make a difference between being zero and having limit zero. For instance, everybody knows that the limit when x tends to zero of sin x/ x is 1. But the limit of 0/x is zero and the limit of sin x/0 does not exist, since division by zero is not allowed in the field of the reals.
Please, let’s stay in the reals! As any textbook defines:
*Probability is a * real number in the interval [0;1].
If someone wants to go in the hyperreals he should create a totally new theory.
Interesting thread. I think some of you misunderstood my previous point. I’m well aware that the probability of a continuous random variable measured on a single point is zero yet that outcome might be possible. But that completely destroys the interpretation of probability as a frequencist measure. Either it is a bad question to begin with (like “what is the color of hunger” or something like that) or the pdf cannot answer that problem, it can only give an approximation on a given interval centered on that point. Because the only definition that could apply to an answer of probability zero for a possible event is meaningless. No? What is the interpretation or the usefulness of that result? If the answer is none, and it destroys the “proper” interpretation of probability, do away with it says I!
ultrafilter, if you know of such a text that explains why my previous argument about flipping coins is false do try me. I’m not convinced that a “straightforward” thing like taking a well defined limit should break an otherwise consistent theory.
The fact that possible events have probability 0 is perfectly consistent with the frequentist interpretation: over any finite amount of time, we expect that event not to happen.
You’ll have to remind me which of your previous arguments you mean.
This is the one you mean, right?
Your error is still in the assumption that probability 0 implies impossibility.
I don’t know of any text offhand that addresses this, but as you may have noticed, all of the mathematicians here have accepted this point.
The proper domain for dealing with infinite sample spaces is measure theory. A measure is a function from the powerset of the sample space* to the extended positive real numbers [0, infinity) which is countably additive: the measure of a countable number of pairwise disjoint sets is the sum of their measures.
For a uniform probability distribution, the probability of an event E is [symbol]m/symbol/[symbol]m/symbol. A particular outcome is the set consisting of a single point, and the measure of that is always 0. So the probability of any particular point is 0. But if you pick a point, you’ll get something.
There are any number of textbooks that will teach you measure theory, but unless your math background is pretty good–like, really good–you’re going to struggle with it. At my school, a lot of the professors didn’t want to teach measure theory because they felt it was too complicated.
*Yeah, I know. [symbol]s[/symbol]-fields and all that. Too complicated for the point I want to make.
Forgot to add–S is the sample space, and [symbol]m[/symbol] is the measure function.
I think we lost ourselves discussing high level math and forgot to answer the OP.
Let’s come back to the plain! First of all we must define who is Meta-Gumble in order to see if there is another one in an infinite Universe, with an infinite number of human beings.
IMO, you are the product of your genes and your experiences.
Since the gene pool is large, but finite, there is a finite probability that with an infinite number of human beings two of them could share the same genes (it is true for identical twins).
The number of experiences is infinite. Since identical twins have different experiences they are different persons.
The probability that there exists someone with the same genes and the same experiences of Meta-Gumble is zero, even in an infinite Universe.
Hey wow its nice when the boards actually load
Sergio - you’re attacking the problem from a philosophical frame of mind and I see where you’re coming from. I think thats a completely valid objection to the “theory”.
But theres an even more fundamental objection that I was trying to make - that you just can’t predict a thing in an infinite universe because as theres an infinite number of outcomes theres always something more probable that can happen. Of course this is making the assumption that there is an infinite number of mutually exclusive results for every event.
Its easy just to say: “Oh whatever I can think of will be the case in an infinite Universe” but I really don’t think this is the case - it seems to me a fallacy to say whatever is possible will happen in an infinite Universe. We know that many possible things do not actually happen. Its as simple as that.