Quote:
Originally posted by Poloboy
No dude, infinity doesn't have sizes. I know what you mean, I understand the different number sets, but the problem is that infinity is a concept, not a number. You can't think of infinity as a limitless set of values, such as natural numbers, certain fractional intervals, or even the set of real numbers. It's not a limitless set of values, but simply is the concept of a continuing function. There are no increments associated with it. Using infinity in algebra and calculus is ok, but you have to remember that when you write it, what you're writing is just a symbol representing the concept on the graph you're interpreting.
Sorry about the hijack.
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Actually, I'm not sure that you do understand what the original poster was talking about. But, first thing's first...
Using infinity in algebra and calculus is
not okay. There are some number systems that incorporate an infinity as an element but they are not typically used by scientists, engineers, or joe schmoe. They are certainly not groups, rings, or fields.
Calculus simply doesn't use inifity at all - at least, not as a number. The word is used colloquially, or as shorthand, but it is certainly not a number. I think this is what you were thinking of when you replied. Calculus is defined using limits. Values that are arbitrary or unbounded, but not infinite!
Now, back to the main point. There
is a definition of the "size" of a set, even if it is infinite. Although we can't define the
cardinality of an infinite set in terms of non-negative integers, we can define their relative sizes.
Set A is defined to have a strictly greater cardinality (or simply put, is bigger) than set B if there exists a bijection from B to a strict subset of A while no bijection between A and B exists.
For example, the power set of A, denoted P(A), is defined as the set of all subsets of A. Now, there is a famous (and very clever) proof that the set P(A) is strictly bigger than A.