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. [img]/forums/images/icons/grin.gif[/img]

[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.