User:Cregox/infinity

First I'll quote the most relevant parts from the infinity article as it is currently presented, then I'll introduce my concept of infinity.

article
Infinity refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. The word infinity comes from the Latin infinitas, "unboundedness".

In theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. In philosophy, infinity can be attributed to space and time, as for instance in Kant's first antinomy. In both theology and philosophy, infinity is explored in articles such as the Ultimate, the Absolute, God, and Zeno's paradoxes.

Three types of infinities
Besides the mathematical infinity and the physical infinity, there could also be a philosophical infinity. There are scientists who hold that all three really exist and there are scientists who hold that none of the three exist. And in between there are the various possibilities. Rudy Rucker, in his book Infinity and the Mind -- the science and philosophy of the mind (1982), has worked out a model list of representatives of each of the eight possible standpoints. The footnote on p.335 of his book suggests the consideration of the following names: Abraham Robinson, Plato, Thomas Aquinas, L.E.J. Brouwer, David Hilbert, Bertrand Russell, Kurt Gödel and Georg Cantor.

Infinity symbol
The precise origins of the infinity symbol ∞ are unclear. One possibility is suggested by the name it is sometimes called &mdash; the lemniscate, from the Latin lemniscus, meaning "ribbon". One can imagine walking forever along a simple loop formed from a ribbon.

A popular explanation is that the infinity symbol is derived from the shape of a Möbius strip. Again, one can imagine walking along its surface forever. However, this explanation is improbable, since the symbol had been in use to represent infinity for over two hundred years before August Ferdinand Möbius and Johann Benedict Listing discovered the Möbius strip in 1858.

John Wallis is usually credited with introducing ∞ as a symbol for infinity in 1655 in his De sectionibus conicus. One conjecture about why he chose this symbol is that he derived it from a Roman numeral for 1000 that was in turn derived from the Etruscan numeral for 1000, which looked somewhat like CI&#390; and was sometimes used to mean "many". Another conjecture is that he derived it from the Greek letter &#969; (omega), the last letter in the Greek alphabet.

The infinity symbol is represented in Unicode by the character ∞ (&amp;#8734;).

Infinities as part of the extended real number line
Infinity is not a real number but may be considered part of the extended real number line, in which arithmetic operations involving infinity may be performed. In this system, infinity has the following arithmetic properties:

Infinity with itself

 * 1) $$\infty + \infty = \infty \cdot \infty = (-\infty) \cdot (-\infty) = \infty$$
 * 2) $$(-\infty) + (-\infty) = \infty \cdot (-\infty) = (-\infty) \cdot \infty = -\infty$$

Operations involving infinity and real numbers

 * 1) $$-\infty < x < \infty$$
 * 2) $$ x + \infty = \infty$$ and $$ x + (-\infty) = -\infty$$
 * 3) $$ x - \infty = -\infty$$
 * 4) $$ x - (-\infty) = \infty$$
 * 5) $${x \over \infty} = 0$$ and $${x \over -\infty} = 0$$
 * 6) If $$0<x<\infty$$ then $$x \cdot \infty = \infty$$ and $$x \cdot (-\infty) = (-\infty)$$.
 * 7) If $$-\infty<x<0$$ then $$x \cdot \infty = -\infty$$ and $$x \cdot (-\infty) = \infty$$.

Undefined operations

 * 1) $$0 \cdot \infty$$ and $$0 \cdot (-\infty)$$
 * 2) $$\infty + (-\infty)$$ and $$(-\infty) + \infty$$
 * 3) $${\pm\infty \over \pm\infty}$$
 * 4) $${\pm\infty}^0$$
 * 5) $$1^{\pm\infty}$$

Notice that $$[{x \over \infty} = 0] \not\equiv [0 \cdot \infty = x]$$. This is because zero times infinity is Indeterminate.

simple
As the word, and the origin of it suggests, the infinity is the outside part of the limit. It's undefined and beyond any limit, in any aspect of it. That's why zero is one kind of infinity, because it's defined exactly by the absence of anything, it's the true opposite of the macro infinity, while being the same thing. Two sides of the same coin.

Being the unbounded part of any boundaries, there are infinite ways to get to infinity. By imagining a 2 dimensional set of natural numbers, there are just 3 ways: $$-\infty, 0, \infty$$. Expanding that to fractions, we have infinite ways inside a 2 dimensional set, since every two natural number has a infinity of numbers between them.

Infinity is a simple concept. Trying to find different ways to get to it, and putting names in those ways, are just attempts to understand each particular way. But that usually lead to misconception of infinity itself.

Due to its own nature, infinity can be seem in infinite ways. Dividing it in three types, or as many types as wanted, is again naming ways to get to it, and it may seem limited because we as humans have limited concepts, names, age, time and each individual life introduces the concept of limit in a very strong way. That might be the main reason why infinity is so hard to understand.

Mathematics must be the simplest way to use an universal language to explain infinity, and mathematical simple operations can also give a better idea of it, that's why I picked mostly that part from the article.

mathematical operations
Defining: $$x = \{ x \in \mathbf{R}|x \not{=} 0\}$$ (x is any real number but zero, keeping in mind infinity isn't a real number)
 * 1) $$\infty + (-\infty) = 0$$ and $$(-\infty) + \infty = 0$$