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Infinity and Beyond
Transcript of Infinity and Beyond
*Disclaimer: Infinity is not easy, having a true grasp on the idea can be, well, difficult.
PING PONG BALL CONUNDRUM
George Cantor (1845-1918)
Infinity is not a one size fits all idea.
by: Alissa S.
common knowledge of not countable or innumerable
indefinitely great number or amount (Miriam-Webster Online)
there are also different levels of infinity, some more than others
LET US EXPLAIN ----->
Lets Begin with.....
ALL NATURAL NUMBERS LINE UP IN A ONE-TO-ONE CORRESPONDENCE
Buzz Lightyear Factory ?!?!?!
Otherwise Known AS:
WHICH LEADS TO>>>>
TO THE BOARD!
MIND BLOWN YET?
Certain infinite sets of numbers actually have a greater cardinality than others.
Cantor used one to one correspondence to prove this!
Lets start with Cantor's 2 Lists
Now lets look at his decimals!
Let’s say we have two numbers and both extend to six places past the decimal.
Only the second or hundredth place are given numbers!
The rest of the places we will fill in with question marks ????
Our numbers would look something like this:
We don't know what the numbers are.....
But we know they are different!
Every number behind the various question marks could be the same but the decimal numbers would be different because the hundredth place of each is different.
This is how the decimal system works!
You now have the basis behind Cantors Method!
Cantor created a list of natural and real numbers similar to the list pictured ...
He then used a simple method to produce a number that did not exist on the list.
NUMBER NOT ON LIST
We are creating a decimal number M.
Now, the digits of our number M correspond with those of the real numbers in the list previously shown
The first digit of M past the decimal will correspond with the first digit of the real number across from the natural number 1
The second digit of M will correspond with the second digit behind the decimal of the real number across from natural number 2.
It doesn't stop there.....
We aren’t just plugging any number into M.
Let’s look again at the real numbers on the list.
Every digit of M is different than its corresponding digits on the list. No matter how many real numbers were included in Cantor’s list, we would always be able to create a new one.
In case you forgot!
WE CAN ONLY USE 2 or 4
Giving us better control!
Remembering the digits make up "M"
If the corresponding digit of the real number is a 2, the number we plug into M for that similar digit will be 4. If the corresponding digit is 1,3,4,5,6,7,8,9, or 0,we will plug in a 2.
If we keep creating this number M all the way out to the eighth digit, it would look like: 0.24442424.
M is a real number, but if we are using the numbers already existing on the list to create it, could M exist on the list before we made it?
Real numbers do not have one-to-one correspondence!
This means real numbers have a greater infinity than natural numbers.
In other words, real numbers have a
**THATS STILL REALLY HARD TO GRASP!!!!
ONE MORE DIAGRAM!!!
An easy way to think of it is to imagine a line infinite in both directions.
Pick a point on that line and draw another liNE....
Parallel to the first, which begins at that point and continues infinitely in one direction.
That point is zero and the second line is all the natural numbers
The two lines match up with a one to one correspondence after zero and continue forever in the same direction....
This means that the first line is a larger infinity than the second.
The first line is all of the real numbers.
but the first line has another equally infinite set of numbers heading the other direction that the second line does not.
The idea that infinities can be infinitely more infinite than each other is both confusing and reassuring.
Sure, it may confound the mind momentarily but in a way, the process of proving multiple infinities shows that infinity can be quantified to a certain degree.
It is possible to line up two sets of infinite numbers and determine if one has a greater cardinality
Through the Ping-Pong ball conundrum we see that two sets of infinite items can be equal and this can be proven through one-to-one correspondence.
The fact that infinity can be proven helps make the concept less daunting!!
a certain finite quality!!! :)