IA Project for Math Class

1. Is it really possible for there there be different sizes of
infinity?
Ricky Chubbs

2. Introduction
I decided to take advantage of the sample topics
and I found one that really caught my eye. Sizes of
infinity? How can you compare two things that are
infinite? How would you measure infinity? Is it
really possible to have different sizes of infinity?
∞ - ∞ = 2? :S

3. What is infinite and finite?
Infinity essentially means limitless, as oppose to it’s
opposite anything finite, infinity is unbounded.
Infinite can represent just about anything, but just
to make things a little simpler we’ll just be treating
infinity as a set of numbers.
Infinity and well.....Not infinity :P

4. Georg Cantor
Georg Cantor is one of the main contributors to the theory of having sizes
of infinity. Instead of accepting the traditional theories concerning
infinity, in the 19th century he and some other mathematicians formulated
the set theory. The set theory suggests that the cardinality of different sets
of infinity may differ. George Cantor invented a collection of transfinite
ordinal and cardinal numbers that would describe the order type and a
number set that doesn’t cover an absolute infinity.

5. Cardinality
Cardinality represent the number of elements in a
set. For example:
We have the set [1, 2, 3,]. What is the cardinality?
The Cardinality Is 3! There are three numbers in the
set, therefore, the cardinality is 3.
But now how will this apply to infinite sets.
More examples:
If we have [2, 4, 6, 8,] Cardinality is equal to 4
If we have [3, 4, 5,] Cardinality is equal to 3

6. Cardinal and Ordinal numbers
I mentioned Georg Cantors Cardinal and Ordinal
numbers earlier but I thought I improve of that just
to help understanding.
An ordinal number would be the order or way of
counting the number within a set.
A cardinal number is the number of elements
within the set.
Example using the positive set of integers (natural numbers)
Ordinal number under the traditional order is: ω
Cardinal number is: Aleph Null (unfortunately I couldn’t find the
appropriate symbols)

7. Dedekind’s Approach
Another very accomplished mathematician concerned
with the set theory is Richard Dedekind. Dedekind’s
approach takes place mainly by using one to one
correspondence in order to formulate a size using the
comparison of the 2 sets. The way I like to think about
it is to think that you’re partnering each number up
with itself then eliminating them. Which ever set has
the most remaining cardinal numbers was the bigger
size of infinity.
Set 1: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
Set 2: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23...
Bigger

8. Still a little weird?
Imagine having two enormous containers one
holding apples and the other holding oranges. If you
remove one apple and one orange at a time, at the
end, we will either be left with apples, oranges, or
nothing. Which ever crate contained more will have
leftovers and it’s this set that will be greater in size.
I know it’s a poor drawing :P
Notice the same amount of apples and oranges are removed yet some oranges remain in the
crate. This obviously shows that there were more oranges than apples.

9. Countably infinite?
The are many different sets of infinity, the smallest
one being the infinite set of natural numbers;
(1, 2, 3, 4). The infinite set of natural numbers are a
set of ordinal numbers that can easily be
counted, although cardinality may not be a specific
number this set could be counted one after another
without a problem. One the other hand we also
have sets of infinity that are too extensive to count.
Few examples:
Infinite set of natural numbers is countable
Infinite set of both positive and negative integers is countable
Infinite set of all the real numbers is uncountable.

10. Reals vs. Naturals
As you may know both real numbers and natural numbers
are sets of infinity when unbounded. Cantor used the
argument that there is a higher cardinality for an infinite set
of all real numbers than there are is for an infinite set of all
natural numbers. Since the natural numbers miss both the
rational numbers and the negative integers, the infinite set of
real numbers is obviously greater in size the natural numbers.
Just for the sake of saying it, there is an infinite amount of
irrational numbers just between the numbers 1 and 0.
And I think we just answered our question. So far the theory looks good, it
makes sense and I personally agree, it could be possible to have different sizes
of infinity.

11. Conclusion
In conclusion we now have a basic understanding of
how there can possibly be more than 1 size of
infinity. All this theory points out that there are
different ways to interpret infinity that will have
different values. I understand think I’ve covered the
basic concepts of the set theory of infinity, and I’ve
discovered yes it is possible that there is more than
one size of infinity.


12. Main Sources
Browser: Internet Explorer
Search Engine: Google
Wikipedia: Infinity, finite, cardinality.
http://www.math.grinnell.edu/~miletijo/museu
m/infinite.html
http://www.scientificamerican.com/article.cfm?
id=strange-but-true-infinity-comes-in-
different-sizes