Infinite Possibilities: Exploring the Different Types of Infinity in Mathematics

[Maatomaat]

How many kinds of infinity does math have?
in my point of view "THERE ARE INFINITY KINDS OF INFINITY"
What's your point?

 

[Simon Bridge]

Welcome to PF.
Define "infinity" and explain, in terms of this definition, how there is more than one "kind", with an example. Then, perhaps, there can be meaningful POVs.

Maybe you are thinking of Cantor sets and the ideas around them:
http://www.scientificamerican.com/article.cfm?id=strange-but-true-infinity-comes-in-different-sizes

 

[Maatomaat]

Welcome to PF.
Define "infinity" and explain, in terms of this definition, how there is more than one "kind", with an example. Then, perhaps, there can be meaningful POVs.

Maybe you are thinking of Cantor sets and the ideas around them:
http://www.scientificamerican.com/article.cfm?id=strange-but-true-infinity-comes-in-different-sizes

Are you sure ? only "one" infinity?
We know "ℝ" is infinity.also "(1,0)" is infinity.
We can also say that we are able to place (1,0) on ℝ.so ℝ is more powerful than (1,0) so they are different.
I mean ℝ-(1,0) is also infinity.
Please think about what I said. Are'nt you agree?

 

[espen180]

It may look that way to you, but R and (0,1) are actually the same size!

Define the map $(0,1)\rightarrow \mathbb{R}$ such that $a\in(0,1)$ is sent to $\frac{1}{1-a}-\frac{1}{a}$.

This map gives a bijection between the two sets. Each and any point in (0,1) corresponds to one unique point in R. We can also reverse the map such that each and any point in R corresponds to one unique point in (0,1). Therefore the two sets have equal size.

 

[Simon Bridge]

"We know "ℝ" is infinity.also "(1,0)" is infinity."

Technically these statements are incorrect according to the usual understandings of what "infinity" means. This is why it is important to define your terms in philosophical discussions. What you have shown is the set of real numbers and an interval on a numberline. Neither of these is infinity.

You have responded to my request to provide examples, but you have failed to define your terms or explain how these two examples illustrate the idea that there are different kinds of infinity.

espen180 has shown you by using a 1-1 mapping that these two sets are the same size. In this sense, they both have the same infinite number of members. So if you intended to distinguish different "types" of infinity by different sizes of infinity, that won't stand up.

Have another go.

 

[Maatomaat]

How about "N"?
N is not equivalent to R.
So they are different.

 

[gb7nash]

So they are different.

The cardinality (size) of N and R are different, mainly due to the fact that N is countable and R is uncountable. Another reason is that there is also no 1-1 correspondence between N and R

 

[Simon Bridge]

You still have to define your terms.

If you mean to demonstrate that there are different sizes of infinity, then please say so.

You seem to be having trouble communicating the ideas you want to discuss.
Have a look at how Cantor handled the same problem.

 

[Maatomaat]

The cardinality (size) of N and R are different, mainly due to the fact that N is countable and R is uncountable. Another reason is that there is also no 1-1 correspondence between N and R

As a result we can say infinitys are different.

 

[Maatomaat]

You still have to define your terms.

If you mean to demonstrate that there are different sizes of infinity, then please say so.

You seem to be having trouble communicating the ideas you want to discuss.
Have a look at how Cantor handled the same problem.

Are you able to prove that there is only one infinity or can you reject the flowing statement:
"N , R are infinity but R is not equivalent to N"

 

[Deveno]

it is a well-known theorem, first demonstrated by cantor, that:

if A is a set, then |A| < |P(A)|, whre P(A) is the power set of A, or set of all subsets of A.

if |A| is finite, this is obvious, and not very interesting.

if |A| = |N|, where N is the natural numbers, this yields the surprising result, that there exist uncountable sets. so that's 2 kinds of infinite: countable, and uncountable.

but if one continues with P(A), where A is uncountable, then one gets a set is that is "even more uncountable". it is possible to carry this reasoning on indefinitely, each time winding up with a "bigger infinity than before".

but there are other ways of thinking of "infinity" besides just "something not finite in size". for example, one can imagine that the distant horizon, is all one single point: the "point at infinity". this one single point (which is strange because we can approach it from any direction) acts like a boundary for what seems an endless plane, and it makes a flat euclidean plane act like a sphere (or the real number line act like a circle).

in this strange geometry, the hyperbola xy = 1 suddenly becomes a closed curve: the points at infinity connect. and this is yet another kind of infinity, which has nothing to do with number, per se, but has to do with space.

and here, again, we aren't limited to a single choice: instead of wrapping the plane into a sphere, we might want to distinguish between ±∞, which gives us a different kind of geometry.

and it gets weirder, still: we might declare some large number R to be the "largest finite number" (computers actually do something akin to this, called "overflow error handling"). now infinity is looking a lot smaller, and has "absorbing properties" similar to 0 (and, unfortunately, plays havoc with our usual algebraic rules).

these aren't the only possibilities. in some sense, infinity represents a choice: we might mean different things by it, and each meaning we assign to ∞, has different consequences. some of these definitions "break" the structure we add ∞ to: if we call ∞ a real number, we lose some of the field structure (algebra) of R. sometimes adding ∞ to our structure enhances it: if we add infinite numbers in a certain way, we can adapt induction to cover finite AND infinite cases.

i hope this gives you a small idea of how infinity comes in different flavors, it's not enough just to say: "∞".

 

[Mark44]

Are you sure ? only "one" infinity?
We know "ℝ" is infinity.also "(1,0)" is infinity.

Do you mean the interval (0, 1)? If that's what you mean, the interval (0, 1) is NOT infinity, but the cardinality of this interval is infinity.

We can also say that we are able to place (1,0) on ℝ.so ℝ is more powerful than (1,0) so they are different.

I can't see some of the symbols you wrote - they show up as squares in my browser. If you are saying that the interval (0, 1) can be placed on the real line, then what you are saying is incorrect. The cardinality of the interval (0, 1) is the same as the cardinality of the entire real number line.

I mean ℝ-(1,0) is also infinity.
Please think about what I said. Are'nt you agree?

No, I aren't.

 

[Simon Bridge]

Are you able to prove that there is only one infinity or can you reject the flowing statement:
"N , R are infinity but R is not equivalent to N"

I cannot do any of these things because you have yet to define your terms.

For instance, neither sets are infinity - though the cardinality of both is infinite.
Why are these different "infinities"?

I take it you've read a pop-math book that covered the concepts recently?