Is the infinite content in different ways?

34 Views Asked by At

If between $ 1 $ and $ 2 $ there are infinite numbers, then the infinity is contained between $ 1 $ and $ 2 $?

And also, the infinite of the reals, is greater than, that of the natural ones? Since the real contains the natural, integers, etc. Therefore, the real ones contain the infinite of the natural, integers, etc.

It is not a specific term, but to understand it further, instead a simple symbol.

1

There are 1 best solutions below

0
On

Both these observations are poignant and are a good start to thinking about how to properly capture the concept of infinity (something that, historically, took mathematics and mathematicians quite a while to grasp at all rigorously).

Your first observation points out that there are different ways to measure the interval $[1,2]$ (i.e., the set of all real numbers lying between $1$ and $2$ on the number line). On the one hand, that interval has a length of $1$. On the other hand, if we consider how many individual numbers are in the set, we see there are infinite such numbers. The lesson here is that counting the individual members of a set isn't the only notion of size; something can have size $1$ in some sense (in this sense, length) even if it has infinite members as a set.

As to your second observation, if you are considering the size of the set as just counting how many members are inside, then mathematics has concluded that a good way to compare whether two sets have the same size is to see if we can put the two sets in one-to-one correspondence. This is intuitive when considering finite sets -- if I grab a pile of forks and knives, I have the same number of forks as knives if I can match each fork to one knife with no knives left over. But this has some counterintuitive implications for infinite sets; e.g., famously, we can put the set of positive integers $\{1,2,3,4,\dots\}$ into one-to-one correspondence with the set of even positive integers $\{2,4,6,8,\dots\}$ via the rule of doubling, even though the latter set is contained within the former set! So, in some sense, these two sets have the same size, even though one is contained within the other. Turns out that, by this notion, the sets of rational numbers, integers, and natural numbers all have the same size, but the set of real numbers is bigger -- for more information on that last point, you can look up Cantor's diagonal argument.