Wouldn't it make more sense to say that some infinite sets are more dense than others, rather than bigger?

Whether or not the numbers of an infinite set could theoretically be listed both sets are still infinite in size.

If, for example, we imagine two infinitely sized parallel universes, one with 1 star per billion cubic light years and the other with 10 stars per billion cubic light years. We wouldn't say that one of the universes was 10 times bigger than the other - they are both infinite.

Georg Cantor is one of my favorite people in the world, I voted before I watched! Time to watch!

>> ^heathen:

Wouldn't it make more sense to say that some infinite sets are more dense than others, rather than bigger?
Whether or not the numbers of an infinite set could theoretically be listed both sets are still infinite in size.

If you take the set of natural numbers (1, 2, 3, ...) and the set of integers (.. -2, -1, 0, 1, 2, ...), you will see that the infinite set of natural numbers is encompassed within the infinite set of integers. Thus the infinite set of integers is "bigger" by any measure.

If you prefer to look at the set of integers within the set of real numbers, you could reasonably use the term "denser" since there is an infinite and uncountable set between any given pair of integers, but bigger is still a reasonable term.

>> ^maestro156:

>> ^heathen:
Wouldn't it make more sense to say that some infinite sets are more dense than others, rather than bigger?
Whether or not the numbers of an infinite set could theoretically be listed both sets are still infinite in size.
If you take the set of natural numbers (1, 2, 3, ...) and the set of integers (.. -2, -1, 0, 1, 2, ...), you will see that the infinite set of natural numbers is encompassed within the infinite set of integers. Thus the infinite set of integers is "bigger" by any measure.
If you prefer to look at the set of integers within the set of real numbers, you could reasonably use the term "denser" since there is an infinite and uncountable set between any given pair of integers, but bigger is still a reasonable term.


What if we were to talk about the numbers between 0 and 10, instead of 0 and infinity?

Some sets would be listable, and some not. In one number line there would be 10 numbers, and in others an infinite number of numbers between 0 and 10.

However we wouldn't say that some tens are bigger than other tens, as 10 always equals 10.

Even though it would take longer to list all the three decimal place numbers between 0 and 10 than it would to list the integers between 0 and 10 it doesn't make the first 10 larger than the second.

>> ^maestro156:

>> ^heathen:
Wouldn't it make more sense to say that some infinite sets are more dense than others, rather than bigger?
Whether or not the numbers of an infinite set could theoretically be listed both sets are still infinite in size.
If you take the set of natural numbers (1, 2, 3, ...) and the set of integers (.. -2, -1, 0, 1, 2, ...), you will see that the infinite set of natural numbers is encompassed within the infinite set of integers. Thus the infinite set of integers is "bigger" by any measure.
If you prefer to look at the set of integers within the set of real numbers, you could reasonably use the term "denser" since there is an infinite and uncountable set between any given pair of integers, but bigger is still a reasonable term.


Not exactly, as you can pair up every integer with any natural number in a one to one relationship. Bigger usually always means "more of". There is no meaningful way to say that any set of numbers that can have a 1 to 1 relationship with any other set of numbers is larger. The fact that one set can eat up another set and still be of the same "magnitude" infinity is pretty amazing, but not really indicative of "size" in the typical sense. For instance, take the set of numbers infinite half's from 0-2, and the set of numbers from 0-1 inclusive of ends. So, you have

(0:2)0...1/4...1/2...3/4...1,...5/4...3/2...7/4...2

(0:1)0...1/8...1/4...3/8...1/2...5/8...3/4...7/8...1



Notice 2 things; one, every time you go and make a new set of halfs for the 0-2 set, it eats up all the similar numbers in the 0-1 set, however, set 2 will also create a new set that doesn't match up...ad infinitum. We didn't have to start at these particular halfs, this is just the list order I chose, there are infinite other sets with the same numbers but different orders. Also note, that the every number from the top set is a 1/2 multiple of the set on the bottom. Every number from set 1 can get a number in set 2 by way of multiplication by .5, the revere is true of the other set. In fact, if you don't include the end point for the set of numbers from 0-2, then 0-1 has more numbers in it than 0-2 because a one to one relationship can't be established for 1 (because in set 2, if you multiply 1 by 2 to get a number by our multiplication factor you get 2...which is not included!)...so the set of numbers from [0-1] is greater than the set of numbers from [0-2).

Infinity will drive you nuts! I have heard that the human minds has evolved to think to the number 7 without abstraction. Beyond 7, you venture into a realm for which your mind has no naturally evolved tools.

This post has been removed from the Education channel by channel owner Sagemind. Please review the FAQ to learn about appropriate channel assignments.