Infinity: Clarifying required

[Moses]

Yep, so infinity is a number or not? Does is exist in reality or not? If it exist can we reach it or not? I have someopinions about this issue, which i cannot promotew it to be "unarguable facts". I want to hear others views on this issue...

Cheerz,

 

[Hurkyl]

Yep, so infinity is a number or not?

The question is vague, but generally the term "infinity" is not used by mathematicians to denote any sort of number. Numbers may be described as being finite or infinite, but there won't be a number called "infinity".

None of the "standard" examples of number systems (such as integers, real numbers, complex numbers) contain any infinite numbers.

Does is exist in reality or not?

This isn't a math question; you'd have to turn to philosophy or science for an answer.

 

[Integral]

Mathematically speaking infinity is not a number. BUT! The Real number system can be extended to include the concept of infinity. The resulting set is referred to as the extended Real numbers. Generally the definition goes something like this:
Start with the basic definition:
$$\infty > x \forall x \in \mathbb R$$

In english this says, infinity is greater then x for all x in the Real numbers.

note that this is positive infinity, you would need to make a similar definition for negative infinity.

Once you have the definition you must define how it behaves under the basic arithmetic operations:
For addition:
$$\infty + x = \infty \forall x \in \mathbb R$$

For Multiplication:
$$\infty * x = \infty \forall x \in \mathbb R$$

$$\frac 1 \infty = 0$$


etc.

A significant result of these definitions is that the Extended Real Numbers are not a Field.

You may want to search one of Scientific reference ( Wikipedia) for what it means to be a field.

 

[Moses]

Well, thanks for help.
Still, there is some points needed to be clarified for me..
so "Infinity"+1="infinity"
But we know that "X number"+1 > "X number"
Thus, could we claim that: For ANY limited quintity "This what numbers and their wieghts represents, if i am not wrong.." if we added 1 it will greater that that quintity, since "infinity" concept does NOT fit in this definition, thus "infinity" is just imginary concept, i.e cannot exist...

This question i saw it in the math bottom, which might intersect a iwth philosophy, but i do not I think philosphy guys will "kick" the topic out as they did with other one similar in general idea...I do not want my thread to me homeless

 

[Sirus]

The reason infinity+1=infinity is because, as discussed above, infinity is not a number. Rather, infinity is often thought of as a direction. The idea of infinity is somewhat philosophical in nature and not always easy to grasp.

 

[Moses]

Well, if it is "some what"philosophical...and thus "could be wrong" to thikn that it is exist by the proof above "In philosophy... if A is different than B, so A should at leat differ in one attribute from B" since infnity+1=infinity thus infinity could not exist, since two "infinites" are equal even if one is different in attribute than the other one... so can we say [For sure: for limited existing thingys..infinity does not exist]

We are walking on the line between math and philosopy at the moment, but this is still a math thread

 

[Chronos]

I think it would be accurate to characterize infinity as those cases in which a mathematical model is unable to yield a quantifiable result. Renormalization is a powerful tool that is often used to cancel out the infinities in sets of mathematical models. It has proven to be a very successful approach to such problems.

 

[kreil]

The word infinity refers to the fact that if I have an infinite set, no matter how big of a number you point to, I can point to a larger one.

George Cantor did some extraordinary work on infinite sets. I was lucky enough to find a website with an excellent summary of his work & conclusions:

http://www.cis.nctu.edu.tw/~wuuyang/Lecture.DiscMath/Cantor001.htm

Summary for those that don't want to read the above:

1. Two sets will have the same cardinality if and only if they can be placed in a one-to-one correspondence to each other.

2. There are only two different types of sets:
a) An Infinite Set is a set that can be placed in a one-to-one correspondence to a proper subset of itself.

The Basic Infinite Set is the set of positive whole numbers {1, 2, 3, 4, 5,……}.

b) A Finite Set is a set that cannot be placed in a one-to-one correspondence to a proper subset of itself