FREE ESSAY ON PHILOSOPHY: MATHEMATICAL NOTION OF INFINITY

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 o. The set of even
natural numbers has the same number of members as the set of natural numbers. The both
have the same cardinality o. 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 o. This correspondence leads to the
conclusion that o+1=o. When we add two infinite sets together, we also
get the sum of infinity; o+o=o. 
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 o, given 1. 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 a greater cardinality, hence
its designation as 1.
Georg Cantor did not come up with the concept of infinity, but he was the first to give
it more than a cursory glance. Many mathematicians viewed infinity as unbounded growth
rather than an attained quantity like Cantor. The traditional view of infinity was
something "increasing above all bounds, but always remaining finite." Galileo (1564-1642)
noticed the peculiarity that any part of a set could contain as many elements as the
whole set. Berhard Bolzano (1781-1848) made great advancements in the theory of sets.
Bolzano expanded on Galileo's findings and provided more examples of this theme. One of
the most respected mathematicians of all time is Karl Friedrich Gauss. Gauss gave this
insight on infinity:
As to your proof, I must protest most vehemently against your use of the infinite as
something consummated, as this is never permitted in mathematics. The infinite is but a
figure of speech; an abridged form for the statement that limits exists which certain
ratios may approach as closely as we desire, while other magnitudes may be permitted to
grow beyond all bounds....No contradictions will arise as long as Finite Man does not
mistake the infinite for something fixed, as long as he is not led by an acquired habit
of mind to regard the infinite as something bounded.(Burton 590) 
Cantor, perhaps the true champion of infinity, built off of his predecessors findings. He
argued that infinity was in fact "fixed mathematically by numbers in the definite form of
a completed whole."(Burton 590) Cantor looked to cardinality, which we looked at earlier,
for his theory on infinity. 
There are an infinite number of ways to think about infinity. This never ending quantity
has many applications to our every day lives. Many of our homes are powered by nuclear
energy. This method of turning the turbines has a potentially lethal side effect, nuclear
waste. Radioactive material that is left behind after the reaction. A radioactive isotope
has a half-life, which is defined as the time it takes for half the original amount of
isotope in a given sample to decay. Uranium 232 has a half-life of 68.9 years. This means
after 68.9 years only half of the isotopes are decayed. This division by two every 68.9
years will never get to zero, it will always be an infinitely smaller number. The nuclear
waste will always be radioactive. This leads us to a problem of disposing of this harmful
substance that will never be benign.
Infinity has many theories and applications. It is a concept that defies regular
arithmetic and has puzzled mathematicians and theologians alike. We are left only to try
to grasp the sheer magnitude of such a concept. 
Burton, David M. The History of Mathematics, An Introduction 2nd ed. W.M.C. Brown
Publisher 1988
Bennet, Briggs, Morrow. Quantitative Reasoning, Mathematics For Citizens in the 21st
Century. Addison-Wesley Publishing Company 1996
Giancoli, Douglas. Physics, Principles With Applications.
Prentice Hall 1995