Infinity Essay, Research Paper
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 a greater cardinality, hence its designation as À1.
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