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[rlbell]

Originally Posted by DoomYoshi

I accidentaly figured out two different sized infinities. There are an infinite number of prime numbers. There are also an infinite amount of numbers which are not prime. However, at any given point after 10, the number of prime numbers will be smaller than the number of not prime numbers. Hence, a smaller infinity. Trippy, huh?

Actually, there are as many integers as there are primes. They are both countable sets. Between any two integers, or any two primes, there are only a finite number of elements. A bigger infinity is the set of real numbers. Not only is there an infinite number of them, but there is an infinite nmber of distinct real numbers between any two elements.

The bit with Achilles and the tortoise is that while there are an infinite number of slices of time with Achilles not cathing up to the tortoise, the sum of the durations of that infinite number of instances is a finite number. If the tortoise has a 100 meter head start and scampers at the un-tortoiselike speed of one meter per second to Achilles leisurely (for Achilles) ten meters per second, the total duration of all those instants of time one hundred nineths of a second, for a total distance of one nineth of a kilometer

[SlowCoder]

Originally Posted by rlbell

Actually, there are as many integers as there are primes. They are both countable sets. Between any two integers, or any two primes, there are only a finite number of elements. A bigger infinity is the set of real numbers. Not only is there an infinite number of them, but there is an infinite nmber of distinct real numbers between any two elements.

The bit with Achilles and the tortoise is that while there are an infinite number of slices of time with Achilles not cathing up to the tortoise, the sum of the durations of that infinite number of instances is a finite number. If the tortoise has a 100 meter head start and scampers at the un-tortoiselike speed of one meter per second to Achilles leisurely (for Achilles) ten meters per second, the total duration of all those instants of time one hundred nineths of a second, for a total distance of one nineth of a kilometer

Is there a smiley dictatiing an exploding head? Oh, well, let's have a go ... :boom:!

[arete]

Oh dear *goes crosseyed* ... I think I've just gone crosseyed...

[Kugerfang]

Infinity is not a number, it's a concept.

[_r.u.s.s.]

kugerfang how about you read the topic before posting and making yourself look funny

[rlbell]

Originally Posted by Kugerfang

Infinity is not a number, it's a concept.

It really is a number, it is just a really big number. One of the early problems in quantum mechanics is that infinity kept showing up in the calculations, normally of the form infinity = infinity. As this did not allow for any meaningful analysis, they 'renormalised' the equations by dividing both sides by infinity. Since the infinity on each side of the equation was of the same order, this allowed for meaningful calculations with finite results-- that matched experimental observations. Quantum gravity had the problem (I took a course in elementary particle physics eighteen years ago [and discovered that the particles were elementary, not the physics], and they may have solved it by now) that dividing both sides by infinity still had infinities on both sides.

It should be mentioned that dividing by infinity was not the straightforward task like dividing a number by two, but first coming up with a creative method of composing an expression that equals infinity and dividing by that (similar to constructing innovative ways of multiplying by one, or adding zero, in Algebra)

Never let anyone tell you that a degree in mathematics is entirely useless!

[SlowCoder]

Originally Posted by rlbell

It really is a number, it is just a really big number. One of the early problems in quantum mechanics is that infinity kept showing up in the calculations, normally of the form infinity = infinity. As this did not allow for any meaningful analysis, they 'renormalised' the equations by dividing both sides by infinity. Since the infinity on each side of the equation was of the same order, this allowed for meaningful calculations with finite results-- that matched experimental observations. Quantum gravity had the problem (I took a course in elementary particle physics eighteen years ago [and discovered that the particles were elementary, not the physics], and they may have solved it by now) that dividing both sides by infinity still had infinities on both sides.

It should be mentioned that dividing by infinity was not the straightforward task like dividing a number by two, but first coming up with a creative method of composing an expression that equals infinity and dividing by that (similar to constructing innovative ways of multiplying by one, or adding zero, in Algebra)

Never let anyone tell you that a degree in mathematics is entirely useless!

tssssss ...
:BOOM:!

[DoomYoshi]

Here is a neat site on the set of prime numbers.
infinity
I found it by exploring this incredible site