So I've been seeing the concept of infinities cropping up a lot recently in various places and want to see thoughts here at KHV. In particular, how different methods of approaching infinity relate and can one result in a bigger infinity than another. For example take whole numbers, and whole even numbers. On one hand there are less even numbers because every number on the even list is also on the full whole number list plus all the odds. But on the other hand you could consider them the same because if we take the "full" whole numbers list and pass it through the function f(x) = 2x (e.g. 1=2; 2=4; 3=6; 4=8...) we will get a list of all whole even numbers that is the same length as the list of whole numbers. Likewise what if you start looking at fractions, irrational numbers, or complex numbers? Can these also be expressed in lists to be numbered off or do we need consider them somehow bigger?
All that you need to consider is which number grows faster. The concept of different "levels" of infinity isn't new. It's not any different than taking the limit of a ratio as both numerator and denominator approach infinity. To find an answer, you just look at what grows faster. It's intuitive. Of course you could look at some theory to validate your reasoning but the end result is the same. In the end, yes you can consider one infinity to be larger than another.
I know it isn't a new concept. But I question the validity of it in certain contexts. In calculus, sure. If given the limit as x approaches infinity of (x^2)/(e^x) you get infinity over infinity which is indeterminate. But by applying L'Hospital's Rule we get the limit of (2x)/(e^x) and then the limit of 2/(e^x) where we then have 2 over infinity which is 0. I'm okay with this. But this actually doesn't deal with multiple infinities when you get down to it, it actually just describes how rapidly unbounded the functions of the numerator and denominator are as they approach infinity. What concerns me is more of how mathematicians largely claim things like there being more real numbers between 0 and 1 (or really any two arbitrary points) than the number of integers in the whole line. There is one proof dealing with the ability to always produce a new decimal; but my issue is that you can also always produce a new integer, even using the exact same system if you like. So how can we claim more of one?
Between two integers there is an infinite number of (ir)rational numbers. With integers, you only have one infinity. Basically it's like a never-ending ladder. Each rung is an integer. All the space between the rungs are the infinite numbers between them. The more rungs you have, the more infinities you have in the ladder. You always have more space than rungs. At least with this example because I'm sure somebody could physically make a ladder with really closely spaced--That's beside the point. XD If you produce a new integer, you have a new interval which to have more decimal numbers from. It's exactly like the ratio situation I described. When you increase your integer by one, you increase the number of rational and irrational numbers by infinity. (Oh and there's more infinitely more irrational numbers than there are rational numbers...according to a professor of mine who never bothered proving it.)
That' s because we use the infinity symbol as a concept, not as a number. Comparing infinities and wondering which one has more elements than the other is nonsensical : they' re both infinite, period. Do they ? Personally I never heard any of my math teachers say such a thing. I think you might want to read this : http://en.wikipedia.org/wiki/Zeno's_paradoxes
