Infinity

[herme3]

What exactly is infinity? Isn't it just a term to describe a number that is beyond our calculation ability? Since the numbers go on forever, wouldn't it be impossible to have a value that is too big to be a number?

[Kyrisch]

Infinity is a term coined not for one exact number, but for the concept that the numbers go on forever. It's a hard concept to grasp.

[Asimov Pupil]

yes but please confirm this with me. You cannot use infinity in a mathematical system unless it is derivatives in calculus?

[CPL.Luke]

no you can use infinity anyware

 

(infinity)(1)=infinity

[ydoaPs]

no you can use infinity anyware

 

(infinity)(1)=infinity

iirc' date=' operations on [imath']\infty[/imath] are undefined. you can, though, have infinity in limits [math]\lim_{x{\to}\infty}f(x)[/math], integrals [math]\int^{-\infty}_{\infty}f(x)[/math], and sums [math]\sum^{\infty}_{j=1}x_j[/math]. you can also have it in intervals [math](-{\infty},{\infty})[/math].

[Psion]

what i dont like about infinity is that infinity plus one is still infinity. so how can the new infinity be equal to the old infinity if one has been added?

(sorry if these seems pathetic and childish to some other more experience mathematian, im 17, and my mind is just curious)

 

moreover, i do think that this thread deals with physics too because i thought about it in a physics state of mind also. is the universe infinate? from basic GCSe knowledge (I nolonger study at A Level Physics) i rememebr that the universe is infinate and is always expanding. i therefore ask that how can something which have no end increase in size. for it is the largest possible value. please excuse me if all this nonsense is due to lack of knowledge.

[ydoaPs]

as i stated before, i am pretty sure that operations on infinity are undefined, but i will ask my math teacher tommarrow to get a definite answer.

[DQW]

as i stated before, i am pretty sure that operations on infinity are undefined

...they are only undefined in the field of real numbers because infinite numbers are not members of the set of reals.

[matt grime]

And here we go again. Firstly let me make clear that I am about to explain the way infinity appears in mathematical literature. If you want a layman's interpretation you are in the wrong place.

 

There is no one such thing as a simple unqualified infinity in mathematics. It is better instead to think of things as being finite or infinite where infinite means not finite.

 

In general we use finite as an adjective to mean that some measurement of some object is an element of the real numbers.

 

In terms of sets, where we have discrete elements then a set is finite if it contains a finite number of elements in the usual sense (ie its cardinality is an integer). If a set is not finite then it is infinite. This is the first time we meet the use of the word infinity when people say such things as there is an infinity of integers. Now, in maths Cantor came up with the infinite cardinals, these are attempts to compare sets with an infinite number of elements. For instance we know that a set with 4 elements has fewer elements than a set with 5 elements. But how do we make this rigorous with infinite sets? He chose to use the distinction that two sets *wil be declared* to have the same cardinality if there is a bijection between the two. More easily understood is perhaps that S and T have the same cardinality if there is an invertible map/function from S to T. In this sense we have aleph-0 as the first infinite cardinal (it is used to describe sets with the same cardinality as the natural numbers). Sets that have this cardinality are the integers (ie the natural numbers and their negatives) and the rational numbers. In this sense we can define an addition on cardinals, and in this sense 'infinity plus one equals infinity' or if X is any infinite cardinal then X=X+1. For example the natural numbers and the natural numbers with zero have the same cardinality yet one has 'one more element than the other'. The bijection is given by

 

[math]f:\mathbb{N}to \mathbb{N}\cup\{0\}[/math]

 

where f(n)=n-1, its inverse is g(m)=m+1.

 

 

It should be noted that there are infinte ordinals too wherein adding 1 to an infinite ordinal produces a different ordinal.

 

Another place in infinity is the more usual one of 'something larger than any real number' so it is simply a symbol that we can add to the real numbers and we can think of it as being a useful symbol for doing analysis: something tends to infinity if it grows without bound. We may also add in negative infinity as smaller than all real numbers. we can now extend certain algebraic operations to infinity (but not all) from the real numbers. and again in this case infinity plus one is infinity, since if we are to have infinity plus one a sensible number in this extended system it must be larger than all real numbers, and the unique element larger than all real numbers is infinity.

 

note again that there are extensions of the reals (called hyperreals or surreals depending on whether you prefer robinson's or conway's flavour) where there are 'infinite' numbers and adding one to an infinite number produces a disticnt infinite number.