Infinity in a Nutshell
Infinity has long been an idea surrounded with mystery and confusion. Aristotle ridiculed the idea, Galileo threw aside in disgust, and Newton tried to step-side the issue completely. However, Georg Cantor changed what mathematicians thought about infinity in a series of radical ideas. While you really should read my full report if you want to learn about infinity, this paper is simply gets your toes wet in Cantor’s concepts.
Cantor used very simple proofs to demonstrate ideas such as that there are infinities whose values are greater than other infinities. He also proved there are an infinite number of infinities. While all these ideas take a while to explain, I will go over how Cantor proved that the infinity for real numbers is greater than the infinity for natural numbers. The first important concept to learn, however, is one-to-one correspondence.
Since it is impossible to count all the values in an infinite set, Cantor matched numbers in one set to a value in another set. The one set with values still left over was the greater set. To make this explanation more comprehendible, I will use barrels of apples and oranges as an example. Rather then needing to count, simply take one apple from a barrel and one orange from the other barrel and pair them up. Then, put them aside in a separate pile. Repeat this process until one is unable to pair an apple with an orange since there are no more oranges or vice versa. One could then conclude whether he has more apples or oranges without having to count a thing.
(Izumi, 2)(Yes, it’s a bit egotistical to quote myself…)
Cantor used what is now known as the diagonalization argument. Making use of proof by contradiction, Cantor assumes all real numbers can correspond with natural numbers.
1 ←-----→ .4 5 7 1 9 4 6 3…
2 ←-----→ .7 2 9 3 8 1 8 9…
3 ←-----→ .3 9 1 6 2 9 2 0…
4 ←-----→ .0 0 0 0 0 6 7 0… (Continued on next page)
5 ←-----→ .9 9 9 9 9 9 9 1…
6 ←-----→ .3 9 3 6 4 6 4 6…
… …
Cantor created M, where M is a real number that does not correspond with any natural number. Taking the first digit in the first real number, write down any other number for the tenth’s place of M. Then, take the second digit for the second real number and write down any other number for the hundredth’s place of M.
...nd since from what we know we can imagine things, the fact that we can imagine an infinite, transcendent, omnipresent, omniscient, omnipotent God is proof that He exists, since what can me thought of is real and can be known.” (ch. 2) Saint Thomas Aquinas' rebutting reply would be that it is simply not so, not everything can be known to mortal man and not all that is real is directly evident to us as mankind.
Three of St. Thomas’s arguments - one, two, and five - are established on the observation of the natural world. Arguments three and four are established on rational speculation. All of the arguments, except for the third, theorize that only the existence of God can provide a sufficient explanation for the refutes presented. In argument three, he concludes that God must necessarily exist for his own sake. Thus, arguments one, two, four and five conclude that God exists because the world requires him as an explanation. Meanwhile, argument three concludes that God could not not exist. Yet, still some individuals insist that the proofs are wrong.
Descartes goes on to prove the existence of God in two different ways. His arguments rely on that fact that we have a clear and distinct idea of God. The first way is the cosmological proof where the idea that something cannot come from nothing because something has to exist in order to create something else. As a finite being, it would be impossible for us to come up with an idea for something or someone
How is it that Descartes goes about proving that God exists? His means are limited. He...
St. Anselm begins with a definition of God, argues that an existent God is superior to a non-existent God and concludes that God must exist in reality, for his non-existence would contradict the definition of God itself.
Descartes proves the existence of an all-powerful and perfect being. He reasoned that he is not perfect. If he exists and is not perfect then that which is perfect also exists. He says that this thing, which is perfect, is God. He says God exists because of his thoughts of God as an extension of God's existence. After further philosophical reasoning he proved the existence of God. His proof of God has become the classic ontological proof used ever since.
Firstly, Descartes talks about “proofs” of the existence of God, explained in his third and fifth meditation. Meaning, his proofs are shown by experiment to prove that God exists. He reinterprets Archimedes ' saying, “required only one fixed and immovable point to move the whole earth from its place, I can hope for great things if I can even find one small thing that is certain and unshakeable (Descartes 159).” That he could shift the entire earth
One way that people found to confirm their belief in a God was philosophy. St. Thomas Aquinas used the science of philosophy to prove God's existence. He showed five ways in which the existence of God must be absolutely concluded. His first proof dealt with the mover and...
Descartes proof of the existence of God is derived from his establishment that something cannot come from nothing. Because God is a perfect being, the idea of God can be found from exploring the different notions of ideas. Descartes uses negation to come to the conclusion that ideas do not come from the world or imagination; because the world contains material objects, perfection does not exist.
Thomas Aquinas uses five proofs to argue for God’s existence. A few follow the same basic logic: without a cause, there can be no effect. He calls the cause God and believes the effect is the world’s existence. The last two discuss what necessarily exists in the world, which we do not already know. These things he also calls God.
One way of describing these different levels of infinity is with Cantor's theoretical "Hotel Infinity" which is also an Allegory of his work and struggles in set theory. The story is interesting and also explains the fundamentals of infinity.
The recursive sequence of numbers that bear his name is a discovery for which Fibonacci is popularly known. While it brought him little recognition during the course of his life, is was nearly 100 years later when the majority of the mathematicians recognized and appreciated his invention. This fascinating and unique study of recursive numbers possess all sorts of intriguing properties that can be discovered by applying different mathematical procedures to a set of numbers in a given sequence. The recursive / Fibonacci numbers are present in everyday life and they are manifested in the everyday life in which we live. The formed patterns perplex and astonish the minds in real world perspectives. The recursive sequences are beautiful to study and much of their beauty falls in nature. They highlight the mathematical complexity and the incredible order of the world that we live in and this gives a clear view of the algorithm that God used to create some of these organisms and plants. Such patterns seem not have been evolved by accident but rather, they seem to have evolved by the work of God who created both heaven and
... by assuming that because he experienced the idea of perfection that God must exist. Nonetheless, Descartes was able to provide evidence, if not proof, that God exists and is responsible for the clear and correct aspects of human reasoning.
The history of math has become an important study, from ancient to modern times it has been fundamental to advances in science, engineering, and philosophy. Mathematics started with counting. In Babylonia mathematics developed from 2000B.C. A place value notation system had evolved over a lengthy time with a number base of 60. Number problems were studied from at least 1700B.C. Systems of linear equations were studied in the context of solving number problems.