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"Infinity" Defined

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Posted by Aurino C Souza on August 7, 2002 13:34:28 UTC

" How do you define "infinity"? "


It's been a while since I left school, but I still remember the definition of infinity. It basically arises from limits. Consider, for instance, every student's favourite misunderstanding of math: 1/0 = infinite. Is that right or wrong? Some people think it's right, some people think it's wrong, but few people realize the real problem, hence all the confusion and claims pro and against the usage of infinity in math.

Contrary to what some people say (including some authorities on the subject) there's nothing wrong or illogical about the concept of infinity if you define and use it the right way, "right" in math meaning "formal". A formal definition of infinity can be stated thus:

infinity = lim(1/x) when x -> 0

As far as I can tell, very few people understand what that means, yet the meaning is quite straightforward to me: infinity is a statement that some functions don't have limits! The closer x gets to zero, the greater 1/x gets. But x can never be zero, therefore 1/x can never have a known value. The source of confusion here seems to be closely related to people's difficulty in understanding what randomness is. In both cases what is being said is that what we are trying to say cannot be said. It may appear a bit confusing but it's really a very simple concept. Most people's minds seem to warp under that kind of semantic strain though.

This goes back to a question I posed a while ago, which no one, not even the math gurus here, answered to my satisfaction. You can't find infinities in physics for the same reason you can't find anything physical which has no dimension. Entities whose measurements are zero don't exist, by definition! Only abstract entities, such as "point" and "universe", can have no size or infinite size, and there's absolutely nothing wrong or illogical about it.

It was a great vacation! I took some time to read Einstein and was quite surprised by it, as it made me agree with Bruce on his opinion of Dick's work.



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