Philosophy: Mathematical Notion Of Infinity

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Philosophy: Mathematical Notion Of Infinity

The mathematical notion of infinity can be conceptualized in many different ways. First, as counting by hundreds for the rest of our lives, an endless quantity. It can also be thought of as digging a whole in hell for eternity, negative infinity. The concept I will explore, however, is infinitely smaller quantities, through radioactive decay
Infinity is by definition an indefinitely large quantity. It is hard to grasp the magnitude of such an idea. When we examine infinity further by setting up one-to-one correspondence’s between sets we see a few peculiarities. There are as many natural numbers as even numbers. We also see there are as many natural numbers as multiples of two. This poses the problem of designating the cardinality of the natural numbers. The standard symbol for the cardinality of the natural numbers is Ào. The set of even natural numbers has the same number of members as the set of natural numbers. The both have the same cardinality Ào. By transfinite arithmetic we can see this exemplified.
1 2 3 4 5 6 7 8 …
0 2 4 6 8 10 12 14 16 …
When we add one number to the set of evens, in this case 0 it appears that the bottom set is larger, but when we shift the bottom set over our initial statement is true again.
1 2 3 4 5 6 7 8 9 …
0 2 4 6 8 10 12 14 16 …
We again have achieved a one-to-one correspondence with the top row, this proves that the cardinality of both is the same being Ào. This correspondence leads to the conclusion that Ào+1=Ào. When we add two infinite sets together, we also get the sum of infinity; Ào+Ào=Ào.
This being said we can try to find larger sets of infinity. Cantor was able to show that some infinite sets do have cardinality greater than Ào, given À1. We must compare the irrational numbers to the real numbers to achieve this result.
1® 0.142678435
2® 0.293758778
3® 0.383902892
4® 0.563856365
:® :
No mater which matching system we devise we will always be able to come up with another irrational number that has not been listed. We need only to choose a digit different than the first digit of our first number. Our second digit needs only to be different than the second digit of the second number, this can continue infinitely. Our new number will always differ than one already on the list by one digit. This being true we cannot put the natural and irrational numbers in a one-to-one correspondence like we could with the naturals and evens. We now have a set, the irrationals, with ...

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