What is infinity mathematically?

[kurt.physics]

What is infinity mathematically? What type of number is infinity, i.e., which number system does it belong to? Is there any good books/text books on infinity and its weirdness? I am hoping for a book that has no philosophy, mathematics of infinity


Thanks,

 

[HallsofIvy]

There is no one definition of "infinity". When we speak, in analysis, of "x going to infinity", that is just a "shorthand" for "x increases without bound" and doesn't qualify as a "real" infinity. The "real" infinities do not exist in the normal rational, real, or complex number systems but different kinds of "infinity" are defined in terms of cardinality or in the "ultra" numbers, "hyper" numbers, etc.

 

[Hurkyl]

What is infinity mathematically? What type of number is infinity, i.e., which number system does it belong to? Is there any good books/text books on infinity and its weirdness? I am hoping for a book that has no philosophy, mathematics of infinity

The first thing to do is to forget the English word 'infinity'; it's incredibly misleading because:
(1) It's a noun -- the adjective 'infinite' is often what's really meant.
(2) It's a proper noun -- there is not a single, specific object called 'infinity'.


Two common examples are:
(1) The extended real numbers $\pm \infty$
The extend real numbers are formed by adjoining the two objects $\pm infinity$ to the set of real numbers. Geometrically, they are simply added as endpoints to the real line, completely analogously to how 0 and 1 are the endpoints of the open interval (0, 1).

Other things can be extended similarly; e.g. you can add many points "at infinity" to turn the ordinary Euclidean plane into the Euclidean projective plane.


(2) Infinite cardinality
Cardinality is a generalization of the 'size' of a set; two sets are the same size if there is a way to pair together the elements of your two sets.

Each natural number is a cardinal number, but there are (mind-bogglingly many) cardinal numbers that are larger than any natural number. Those are called 'infinite cardinals'.

 

[ice109]

they are the top and bottom of the extended reals thereby making them, the extended reals, a lattice. if you look up top, bottom, and lattice you will understand that they are simply convenient.

 

[sadhu]

infinity is just a hypothetical number whose properties are determined by the context it is used for like infinity + infinity can be infinity or 2*infinity depending where are you using it.

 

[andytoh]

Each natural number is a cardinal number, but there are (mind-bogglingly many) cardinal numbers that are larger than any natural number. Those are called 'infinite cardinals'.

Here's an example. Suppose A has an infinite number of elements. How many subsets does A have? An infinite number greater than the infinite number of elements of A. How many subsets does P(A) have? An even greater infinity! And what about P(P(A))? What about P(P(P(A)? What about P(P(P(P(P(... (forever) P(P(A))))))...))) ? And how about we apply this chain the same amount of times as all the infinities just described! Infinity is beyond comprehension.

 

[Gib Z]

Why is infinity beyond comprehension? Your entire posts dilemma is solved if you reread what Hurkyl said about there not only being one infinity etc.

 

[hedayat]

infinity is related to frame of refrence

 

[Hurkyl]

infinity is related to frame of refrence

Which infinity, what do you mean by 'frame of reference', and why the heck would it be related?

 

[HallsofIvy]

infinity is related to frame of refrence

Which infinity, what do you mean by 'frame of reference', and why the heck would it be related?

Notice that the statement has no capital letters and no punctuation. Clearly it was not intended to mean anything!

 

[Gib Z]

Hmm well a a general relativist, or a physicist in general, would argue hedayat is correct in the right context, but they use the term "infinity" in a different manner to us mathematicians. That statement alone, however, is incorrect in any context.

 

[hedayat]

Friends ,when we work in electro physics (calculation of potential difference),even few centimeters have to be considered as infinity.It is no wrong to say that,in physics or better say in practical sense infinity is very closely related to frame of reference.And sorry for punctuation errors.

 

[ice109]

they are the top and bottom of the extended reals thereby making them, the extended reals, a complete lattice. if you look up top, bottom, and lattice you will understand that they are simply convenient.

correction

complete lattice

 

[Dragonfall]

The most intuitive extension of natural numbers to infinite numbers is to think of the natural numbers as describing not size, but order. 1 comes before 2, etc. Then it makes sense to add a smallest infinity $$\omega$$, which comes after every natural number, and then say $$\omega +1$$ to be the first element which comes after $$\omega$$, etc. Viz ordinals: http://en.wikipedia.org/wiki/Ordinal_number

 

[CRGreathouse]

Gosh, the ordinals were always confusing to me. The cardinals made more sense. $\aleph_0+\aleph_0=\aleph_0$ and such, where you get to keep nice things like commutativity that you lose in the ordinal framework.

 

[Hurkyl]

Friends ,when we work in electro physics (calculation of potential difference),even few centimeters have to be considered as infinity.

It would be more accurate to say that such distances are approximately infinite.

 

[Gib Z]

It would be more accurate to say that such distances are approximately infinite.

It doesn't really make a difference, physicists don't usually care for mathematical technicalities anyway.

 

[Hurkyl]

It doesn't really make a difference, physicists don't usually care for mathematical technicalities anyway.

There's a big difference between ignoring the difference for practical reasons, and thinking that there isn't a difference. For the expert, it's fine to use shorthand to expedite your work. For the student, it's important that you know what you're doing if you want to learn it correctly.

In my estimation, thinking that there isn't a difference is one of the more common sources of misunderstanding mathematics -- e.g. many of the "0.999... is not 1" arguments are based on treating infinity (more specifically, the ordinal number $\omega$) as a large, finite number.

 

[Gib Z]

I wasn't saying the physicists are correct for doing it! =[ Just saying that they do.

 

[Dragonfall]

Gosh, the ordinals were always confusing to me. The cardinals made more sense. $\aleph_0+\aleph_0=\aleph_0$ and such, where you get to keep nice things like commutativity that you lose in the ordinal framework.

In return you get the impossibility of defining the successor cardinal in terms of cardinal arithmetic. Not so with ordinals!

 

[CRGreathouse]

In return you get the impossibility of defining the successor cardinal in terms of cardinal arithmetic. Not so with ordinals!

True -- but then I always saw the successor cardinal function as a 'complicated' function, and I'm satisfied with a 'complicated' (ordinal) solution for it.