basic question

[hutz]

My friend and I disagree about something and hopefully some of you experts will be able to help resolve this. Is 1 infinitely more than 0? For example, if a food product contains one grain of salt and another has none, does the first one have "infinitely more" salt in it? Thanks in advance for tolerating this no-doubt basic question. /forums/images/icons/grin.gif

[marbles]

From the definition of infinity, per an online dictionary (I deleted the other definitions, as they weren't relevent to the question)
Infinity:
2. Unbounded space, time, or quantity.

Since the space between 0 and 1 is bound (by one unit), the space between the two is clearly finite.

On the other hand, there are an infinite number of numbers between zero and one, provided you don't limit the number of digits after the decimal point. This concept of "infinite smallness" is referenced in a conversation between Wormser and Poindexter in Revenge of the Nerds II: Nerds in Paradise. Funny movie.

[BruceZ]

It doesn't have infinitely more, but it has infinitely times more.

marbles "what if cat spelled dog" ha ha ha go tri lams

[marbles]

That's heavy, Ogre... Dog.

Was hoping someone would get the RON2 reference! lol.

[RocketManJames]

I agree with BruceZ. 1 is infinite times more than 0. Now, here's something interesting about infinity...

True statement: There are infinitely many more irrational numbers between 0 and 1 than there are rational numbers between 0 and 1.

Food for thought,

RMJ

[Mason Malmuth]

How about the measure of irrational numbers between zero and one is 1, while the measure of rational numbers between zero and one is 0. Also, the rational numbers are countable and the irrational numbers are not.

MM

[Mano]

I think a simpler way to state it would be that the raitio of salt in the one product to salt in the other product is infinite (1/0).

[RocketManJames]

Yes... exactly. Countable vs. uncountable infinities. Fascinating notion, really. I'm no expert in the subject, unfortunately. The first time I learned of such a thing, I was truly amazed.

RMJ

[BruceZ]

It's quite simple really. When two sets cannot be put into one-to-one correspondence with each other, they are said to have different cardinalities or "sizes". This extends the notion of the size of a set to sets with an infinite number of elements. If a set can be put into one-to-one correspondence with the integers, it is said to have a "countable infinity" of elements. The set of integers has a countable infinity of elements. So do just the odds, so do just the evens, so do just the primes, so do the rationals. It may seem odd that a subset of a set can have the same size as the set, but that's what happens when you deal with transfinite sets. Infinity + 1 is still infinity, and it is the same size infinity. However, it is easy to show that the set of irrationals cannot be put into one-to-one correpondence with the set of integers, hence it has a higher infinity of elements, an "uncountable infinity". So does the set of all real numbers. In fact, there are an infinite number of different infinities. The set of all subsets of a set (called the "power set" of a set) always has a greater size than the set itself.

Yet even though the irrationals cannot be put in one-to-one correpondence with the rationals, there is still a rational as close as you please to any irrational, and there is an irrational as close as you please to any rational. In fact, there are an infinte number of each as close as you please to each other.

If you pick a truely random number between 0 and 1, the proability is 0 that it will be rational and 1 that it will be irrational. That doesn't mean that it must be irrational and can't be rational. Probability of 1 and dead certainty are two different things.

[BonJoviJones]

"Probability of 1 and dead certainty are two different things."

Really? What's the definition of "Probability of 1" then?

(I swear, I learn as much on the Probability forum as I have in some of my math courses)

[RocketManJames]

Yes, it's not difficult to prove that there isn't a one-to-one correspondence via Cantor's diagnolization trick. But, I know one thing for sure... there's no way I'd figure it out on my own unless I read about his diagonalization and what it meant about cardinalities of infinite sets.

In any case, the reason I brought the whole thing up was to perhaps stir up interest in those who had never heard of the idea of varying degrees of infinity.

RMJ

[RocketManJames]

BruceZ may have a much better answer to this, but this is how I understand it...

In terms of finite sets, then yes, probability of 1 implies 100% certainty, and 100% certainty implies probability 1.

However, if you're not talking about finite sets, then the implications hold only in one direction. That is, 100% certainty implies that the probability is 1. But, probability of 1 does not imply 100% certainty.

Similarly, you can see how probability 0 is not the same as impossibility.

RMJ

i hate infinity
if you believe the universe is infinite in all directions, then you can say we are at the center of the universe
also, the percentage of all rational integers that contain the number 3 is 100% (because of that damned infinity again)

Well, in my vocabulary, probability 1 and 100% certainty amounts to the same thing. As pointed out above, the problem arises when considering infinite spaces where probability zero of an event, strangely does not correspond to the event being impossible in a theoretical sense.

Anyway, the subject is mostly an interesting mind job, since picking a truly random element from an infinite set cannot be done in the real world. We can only make (almost) random selections of sets of elements from a finite set.

[Candyman]

I posted this /forums/images/icons/smile.gif

[Mason Malmuth]

This guy knows his stuff.

MM