Infinities (1)
February 22nd, 2008 | 
Author: 
Suppose there is a people somewhere that has not yet been corrupted by what we like to call civilization.
Their mathematical skills allow them to count up to five, but not beyond. Anything over 5 is considered ‘many’.
Does that mean they consider all collections of over 5 items as quantitively equal?
No, of course not. They fully realize that there are different kinds of ‘many’: a handful of peanuts is not quite as ‘many’ as a bucket full of peanuts!
Somewhat similarly, ‘civilized’ mathematicians have come to realize that there are different kinds of infinities. Even though we can’t count them, we know that there are infinite sets that differ in ‘size’.
But things get a bit weird when it comes to infinities.
Take for instance the infinite set of all integers.
It’s obvious, and one can also show mathematically, that the set of all EVEN integers is as ‘big’ (or strong) as the set of all ODD integers.
To get you to think a bit about infinities, I’ll end this short introductory post to a (finite!) set of posts on infinities here with a small ‘test’.
I intend to explain later where I’m going with all this. For now, I’d like to keep it simple.
Question:
Let ‘I’ be the set of all integers, ‘E’ the set of all EVEN integers and ‘S(s)’ the size of set ‘s’.
What can be said about the size of I?
Answers 1)
a. S(I) < S(E)
b. S(I) = S(E)
c. S(I) > S(E)
d. Huh?
Bonus question:
What is the most dangerous remark in the above post, especially with regards to the correct answer?
| 1) | For the equation challenged: Is the set of all integers a) smaller, b) equal or c)Â larger than the set of all even intergers, or d) no clue? |
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Posted in Math
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“It’s obvious” sounds like the most dangerous part.
I won’t embarrass myself by trying to answer the math question 🙂
I agree that “it’s obvious” sounds dangerous, but then… it really is obvious that sets S(E) and S(O) are “equally big” (for every even integer you can imagine pairing an odd integer). Where it gets less obvious, is for humans to imagine what “big” then means, in terms of “infinitely big”. Different rules apply there; result being that S(I) being equal to S(O) is the really not-so-obvious, but correct, answer.
All this just to find out where this story is leading 🙂
… and you’re going to have us wait until, what, infinity, for the follow up article? 🙂