If the universe really is infinite, then...

Great Debates
41

ok, let’s try this. it’s been a while since i’ve done any probability, other than rudimentary notions of probability in game theory and lebesgue integration, so please tell me promptly if i make any mistakes.

claim: for an infinite amount of flips of a fair coin, the probability of getting at least one flip heads is 1.

“proof”:
let it be that the probability of heads on a given turn is 1/2.
then the probability after n turns is 1 - (1 - 1/2)^n or 1 - 1/2^n.
or 1 - 1/(2^n).
the limit of (1 - 1/(2^n) ) as n goes to infinity is equal to 1 - 0.
is 1.

does that work?

42

it seems to me that the reason the probability is 0 for a given outcome is because no outcome is ever reached. you are never done. so you don’t actually have to reach an outcome.

43

Yes, that does prove that the probability is 1. I agree with that.

What you have not shown, however, is that it’s guaranteed to happen. For a finite sample space, the two are equivalent. For an uncountably infinite space, they are not.

44

JstPlainBryan: Try a googlefor the Green Bank Equation. A bit outdated I know, but I like it.

I guess if the OP is true,then The Jet Li movie THE ONE would be realistic Fiction or something now would it?

45

This doesn’t hold water. See my discussion with Spiritus a few posts ago for what we actually mean by “an infinite number of coin flips”. We’re not concerned with the fact that you can’t get done; we only care about what happens if you are.

46

I don’t know how you guys can have this conversation without introducting limits. For example, flipping a coin may have a probability of 0.5 of it coming up heads. Flip it twice, and the chance of it coming up heads at least once is .75. Flip it three times, .875.

More properly stated, the probability of heads coming up at least once approaches 1 as the number of trials approaches infinity. You can never say for CERTAIN that a head will never come up, but the series converges on 1 pretty darn fast.

Back to the universe - I don’t know about an ‘infinite’ universe having to have infinite Earths, because every point of the universe is not necessarily starting from the same state.

In the many-worlds interpretation as I understand it, the universe would branch every time a quantum event occurs, which means that billions of different universes are created each microsecond. So from the beginning of the universe, there is today an extant universe for every single possibility of change. And even though there would be trillions and trillions of universes with man in it in various recognizable forms, and trillions and trillions with a ‘Sam Stone’ in it, they would still be a miniscule fraction of the total.

But what if we discover that there aren’t many worlds, but just one at a time repeated over and over again? What if there is an infinite series of universes, each one marked by a big bang, expansion for however long, a contraction, a big crunch, and then another big bang?

If that big bang starts the universe off in a truly random state, and if the process is infinite, then we’ll all live again, possibly trillions of years in the future (if time can be said to have any meaning when we’re talking about multiple big bangs).

Anyone see a problem with that, other than that today’s data seems to indicate that the universe will not be contracting again?

47

And for a countably infinite space, which is what the coin-toss example would be?

And for the less-mathematically inclined among us, would you care explain what a probability of 1 means in such a case, if not “guaranteed to happen”? I would guess that, in the coin-toss example, the probability of getting either heads or tails at least once would also be 1. It also goes without saying that getting either heads or tails at least once is guaranteed. So how do you distinguish between an event with probability 1 that is guaranteed, and one that isn’t?

Jeff

48

Well, as was pointed out above, you are guaranteed to repeat a given event only if there are a finite number of initial states and initial boundary conditions (or, to appease some of the other posters, “the repetition of an event will have a probability of 1 only if…”).
Jeff

49

Well, since “infinity is not a number” is pretty much what I have been saying all along, it sounds like we are in agreement.

BTW, integers do have reciprocals, it’s just that the reciprocal is not an integer. Transfinite numbers have no reciprocals. Period.

50

Actually, in the coin-toss example there are an uncountably infinite number of possible outcomes. An “outcome” in this case is a particular sequence of flips, and while there are a countably infinite number of flips in each sequence, there are an uncountably infinite number of sequences.

If there are a countably infinite number of possible outcomes, then they can’t all have probility zero (proving that involves some fourth-year measure theory, unfortunately). I honestly don’t know off the top of my head if probability-zero outcomes are guaranteed not to happen, but I suspect the answer is no.

Hmm…yes. Getting heads at least once is not guaranteed in the strictest sense. But getting heads at least once or tails at least once is guaranteed.

I don’t know any hard and fast way to answer that question that will apply to every situation, unfortunately.

In the coin-toss example, for instance, we can say that “probability=1” is the same as “guaranteed” for the example you gave, that is the event “either at least one head or at least one tail”, but only because that event encompasses every possible outcome and hence must be guaranteed. And we can say that “probability=1” is not the same as “guaranteed” for the “at least one head event” because that event doesn’t encompass all possible outcomes. Specifically it fails to encompass the outcome “TTTT…” and if that outcome can’t happen then every outcome can’t happen by symmetry, which is absurd.

Hope that helps a little bit…

51

In fact there’s a number system called the “supernatural numbers” (I kid you not) in which transfinite numbers do have reciprocals. John Conway thought them up a while back; they’re an extension of the ordinals, which are in turn based upon the cardinal numbers.

The smallest infinite ordinal number is generally denoted w (omega), and corresponds to Aleph0, and in the supernatural numbers 1/w=epsilon, which is greater than zero but smaller than every positive real number. (I’ve seen iota used instead of epsilon occasionally.)

So transfinite numbers have inverses too (in the right number system), it’s just that their inverses aren’t transfinite numbers. Same as with the integers.

52

Math Geek: Are you sure that it’s the transfinites that have reciprocals in that system? I thought that there was a separate class of numbers that were larger than all natural numbers, but not transfinite. Or maybe I’m thinking of the non-standard reals. This isn’t really my area of expertise.

53

This is a terribly minor quibble, but we’re already dealing with some heady math here, so why not? Integers don’t have reciprocals. Rational numbers with a denominator of 1 do. Those two sets are completely disjoint. But since members of the second set behave exactly like those of the first, we treat them as if they were the same thing.

54

forgive me for my slowness here…this just doesn’t make much sense to me.

specifically, i’m having a hard time getting just why p(at least one head) = 1 doesn’t mean that it’s a guaranteed event.

my trouble probably stems from my apparent misunderstanding of infinity. suppose, for example, i decide to flip a coin until i get a heads. it is possible that i never achieve a heads for any finite number of flips. but i always sort of envisioned limits as the embodiment of infinity. like, what would happen “at infinity”. obviously the rigorous way of dealing with it is with limits, but that was my intuition of what a limit was. where was i wrong?

and by the way, perhaps this is not the place, but i’m not scared of 4th year measure theory. it’s been a few years, but i’d like to be reminded of those things, and i wouldn’t mind figuring it out again.

anyway, to conclude, it seems that the distinction between “any finite number of” and “infinitely many” are hanging me up. i don’t think you can claim that an infinite sequence is an outcome. i think that you can only speak of in “the set of all possible outcomes” the set of all outcomes after any finite number of trials. it seemed to me that when we considered instead infinity, the limit applied. why doesn’t it?

55

Ramanujan, you might want to go back and read this thread.

56

math geek
I confess to confusion about how a tranfinite number could have a reciprocal when the set contains no unity element? Has Mr. Conway changed the definition of reciprocal?

Ultrafilter
I’m not certain of the distinction, here. By what quibble does one assert that the integer 2 multiplied by the rational [sup]1[/sup]/[sub]2[/sub] does not yield the unity element for integers? Is the quibble based upon the idea that multiplication is well-defined only within a given set? If so, what would prevent us from defining an operation “*” which operates upon one rational and one real?

57

Yes, multiplication is currently defined only on like objects. Nothing prevents us from defining such an operation–it’d be quite simple–but we don’t need it. Like I said, that was an extremely minor quibble.

58

Hey, no worries, I like minor math quibbles. I just wanted to make certain that I knew where this one lay.

Math Geek
Does measure theory deal with ideas like, “The real numbers between 0 and 1 have exactly as many members as the real numbers between 1 and 100, and the points are no more or less densley distributed, but the second set is “larger” than the first?”

It has been a long time since my undergrad days (when most of my mathematical explorations took place), but that idea always struck me as beautifully counterintuitive.

59

added to the list of beautifully counterintuitive things for me is that p(1) doesn’t mean “guaranteed to happen”. and p(0) doesn’t mean guaranteed not to happen. i guess since the last math course i took was game theory, these items just didn’t fit in with my way of thinking. it makes sense, though, that i was extending my intuition of something somewhere it didn’t belong. on a side note, what if we required of all probability functions that p(0) meant “guaranteed not to happen” and/or p(1) meant “guaranteed to happen”? would that essentially destroy all of our mathematics of probability?

thanks for the link, ultrafilter. i think i sort of get it now, though it is quite counterintuitive. also, it seems rather arbitrary, but what do i expect delving deep into mathematics?

so, the set of all outcomes in the coin tossing experiment is equivalent to the set of all strings of 0s and 1s. which i recall is uncountably infinite. but i can’t remember why. doesn’t any given string of 0s and 1s represent a natural number? isn’t that the required mapping?

anyway, in my class that dealt with measure theory, we were more concerned with integration and never really got into probability. so thanks for the info. i think it’s about time i break out rudin again.

60

It’s not my area of expertise either…for example, I’m not certain what the non-standard reals are…but I do know that every ordinal number is also a supernatural number and has a reciprocal in that system.

The transfinite numbers have no unity element, but the supernatural numbers do: the supernatural numbers contain all of the real numbers (including 1) as well as the ordinal numbers.

Somewhat. A measure on a set is a particular way of defining the “size” of subsets of that set. The probabilities of the outcomes of an experiment, for example, can be thought of as a measure on the set of outcomes.

An infinite string of 0s and 1s represents a real number between 0 and 1, actually. The same “Cantor diagonalization” proof that shows the real numbers between 0 and 1 are uncountable works for infinite binary strings as well.