Different kinds of infinity

By Yuvraj Mamik, Haryana, India.

If you were wondering, Infinity is not a term derived from the new Avengers movie, instead it’s much more than you can ever imagine. Mathematically speaking, infinity is defined as a number greater than any assignable quantity or countable number (symbol ∞).

One standard way of dealing with the different sizes of infinities is to take 

one kind of infinity as a normal infinity or a base infinity, and then compare

its size with other infinities. What we do in practice is take the number of counting

numbers 1,2,3,4,5,… as our normal infinity, and we call it “countably infinite.”   

Then way we compare the sizes of two infinite sets to see whether we can

pair the elements of the two sets up in a one-to-one correspondence.  For

instance, if we match up the even numbers and the odd numbers like this:

2,4,6,8,10,12,14,16,18,…

1,3,5,7, 9,11,13,15,17,…

We can see that there are exactly the same number of even numbers as odd

numbers. What if we try to match up the even numbers and the counting

numbers?

2,4,6,8,10,12,14,16,18,…

1,2,3,4, 5, 6, 7, 8, 9,…

As we can see there are exactly the same number of even numbers

as counting numbers(base set), so we say there are a “countably infinite” number of

even numbers.

Notice that all I’ve done above is write the even numbers in a list,

making sure that I list them all. This is the usual way to show that

there are countably many (a synonym for countably infinite) numbers in

some set: try to find a way to list them all.

What about the rational numbers?  Remember that the rational numbers are

numbers that can be written as the quotient of two integers. I’ll write a

list of all the positive rational numbers; can you find the pattern?  

1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 3/2, 2/3, 1/4, 5/1, 4/2, 3/3, 2/4, 1/5, …

If you’re really ambitious, you could try to find a formula for the nth

number in the sequence!

Here’s a hint about how to look at the numbers in that sequence: look at

the numerators separately, then the denominators.

Do you believe that I’ll get all the positive rational numbers this way?

If so, I’ve just shown that there are countably many positive rational numbers.

How could you use a similar sequence to show that there are countably many irrational

numbers (both positive and negative)?

It is an interesting fact that you cannot write such a listing of all the

real numbers (which includes both the rationals and the irrationals).

If you’re interested, write back and I’ll help you show that.

If that’s true, what does that say about how many real numbers there are?

There’s a lot of them, more than the number of counting numbers. So there are two infinite sets that really do have two different sizes.

Hence infinity having different sizes.

– Here are the Types of infinite numbers and some things they apply to:

Cardinals (set theory, applies to sizes of ordinals, sizes of Hilbert Spaces)

Ordinals (set theory, used to create ordinal spaces, and in ordinal analysis. Noncommutative.)

Beth Numbers (like Cardinals, or not, depending on continuum hypothesis stuff)

Hyperreals (includes infinitesimals, good for analysis, computational geometry)

Superreals (maximal hyperreals, similar to surreals)

Supernaturals (prime factorization matters, used in field theory)

Surreals (Best and most beautiful thing ever, maximal number system, combinatorial game theory)

Surcomplex (surreal version of complex numbers)

Infinity of Calculus (takes things to limits)

Infinity of Projective Geometry (1/0=infinity, positive infinity equals negative infinity)

Infinite Hilbert Space (can be any Cardinal number of dimensions)

Real Line (an infinite line made up of all real numbers)

Long Line (longer than the real line, in topology)

Absolute infinity (self-contradictory, not really a thing)

Here are the types of non-infinite kinds of numbers:

P-adic (alternative to real numbers)

Natural numbers (1, 2, 3…)

Integers (…-3, -2, -1, 0, 1, 2…)

Rationals (1, 1/2, 2/1, 2/3, 3/2, 3/4, 4/3…)

Algebraic (sqrt 2, golden ratio, anything you can get with algebra)

Transcendental (real numbers you can’t get using any finite amount of algebra, like pi and e)

Reals (all possible infinite sequences of digits 0.123456789101112131415…, includes all of the above)

Imaginary (reals times i, where i^2=-1)

Complex (one part real, one part “imaginary,” a consistent, commutative, associative, 2-dimensional number system)

Dual numbers (instead of imagining a number where i^2=-1, make up a number where ε^2=0 and use that)

Quaternions (make up numbers that square to -1, but are different from each other. i^2=j^2=k^2=ijk=-1. 4d, noncommutative.)

Octonions (make up even more numbers, 8d, noncommutative and nonassociative.)

Split-complex (imagine if i^2=+1, but i isn’t 1)

Split-quaternions

Split-octonions

Bicomplex number, or tessarine

Hypercomplex (category that describes/includes all complexy number systems that extend the reals)

References:

J. H. Conway, R. K. Guy, The Book of Numbers, Copernicus, 1996

W. Dunham, The Calculus Gallery, Princeton University Press, 2008, p. 37

K. Kuratowski, A. Mostowski, Set Theory, North-Holland; 2nd revised edition (1976)

E. Maor, To Infinity and Beyond, Princeton University Press, 1991

R. Rucker, Infinity and the Mind, Princeton University Press, 1995

I. Stewart, Concepts of Modern Mathematics, Dover, 1995