# Three kinds of infinities [duplicate]

In the beginning, I thought of infinity as something inaccessible. Which can't be reached. And when something can't be reached ( with 'reached' i mean can't be known) its properties were automatically considered as weird.

But in the book " one two three...... Infinity" by George gamow it is written that there are three kinds of infinities :

1. Infinity no. 1- corresponding to number system ( one dimension)
2. infinity no. 2- corresponding to set of points in plane ( two dimension)
3. Infinity no 3- corresponding to three dimensional space. When three infinities are there, then the so called inacessibility of infinity is somewhat hurt.

Second guess is that it is a large number, then why are it's properties different?

• The three infinities you list does not correspond to the three infinities mentioned in the book (and would actually all fit into the second category). – Arthur Oct 18 '17 at 9:04
• This book is entirely nonsense. Yes, there are many different kinds of infinities. No, the descriptions given by the book are awful, misleading, and outright false. – Asaf Karagila Oct 18 '17 at 9:08
• See these infinitely many questions that were asked before you (the following is a partial list): math.stackexchange.com/q/1/622 (literally the first question on the site); math.stackexchange.com/q/182171/622; math.stackexchange.com/q/2377865/622; math.stackexchange.com/q/5378/622; math.stackexchange.com/q/614561/622 – Asaf Karagila Oct 18 '17 at 9:12
• @AsafKaragila That's a bit harsh if you ask me. From the part of the book that OP is posted, I see nothing that I would call false. $\aleph_0$ is, just like the book says, the number of fractions. $\aleph_1$ is, just like the book says, the number of points in a cube (or on a line). – 5xum Oct 18 '17 at 14:59
• @5xum: No, that's entirely not true. $\aleph_1$ is not the cardinality of the reals, not unless you assume CH. And $\aleph_2$ is not the cardinality of "geometric lines on the plane", and even if the author meant all subsets of the plane, it is still not $\aleph_2$ if you don't assume something extra like GCH (or explicitly state that $2^{\aleph_1}=\aleph_2$). So yeah, it's riddled with misleading and terrible suggestions. It's not unfair. – Asaf Karagila Oct 18 '17 at 15:28

1. Infinity no. 1- corresponding to number system ( one dimension)
2. infinity no. 2- corresponding to set of points in plane ( two dimension)
3. Infinity no 3- corresponding to three dimensional space.

You are misunderstaning this.

All three infinities you describe fit "infinity number 2" in your case. As the book you cite clearly state, infinity number 2 is the number of points on a plane, or on a line, or in a cube.

So, to go over it once again:

Infinity number one does not correspond to a number system or one dimension. The infinity corresponds to all integers, or equivalently, to all rational numbers. This is called "countable" infinity, and it is the "smallest" infinity possible. This infinity is the "size" of the set $$\{1,2,3,4,\dots\}$$

Infinity number two corresponds to the number of numbers on a line, in other words, it is the size of the set $\mathbb R$. It is possible (not too difficult) to show that this is also the size of the set $\mathbb R^2,\mathbb R^3$ and so on. Yes, infinities are weird. The set $\mathbb R$ (a line) has exactly as many points as the set $\mathbb R^2$ (a plane).

It is also possible to prove that infinity number two is bigger than infinity number one. There are more real numbers than there are rational numbers.

Infinity number three corresponds to all possible (curvy, not just straight) lines on a plane. Again, this infinity can be shown to be bigger than infinity number two.

• Is it a number, because only numbers can be compared? – Pranjal Rana Oct 18 '17 at 14:54
• @PranjalRana It is not a number, but it can still be compared. Specifically, two sets (even if they are infinite) are defined to be equally sized if and only if there exists a bijection between them. If there exists an injection $f:A\to B$, but no bijection from $A$ to $B$, then $A$ is smaller than $B$ (and so, the size of $A$ is smaller than the size of $B$). – 5xum Oct 18 '17 at 14:56

This behaviour is a fruit of Cantor's theory (nowadays generally accepted as the basis of mathematics). Actually, there are very many more different infinities (see any book on set theory)..

For Greeks, the infinity was "potential", i.e. just the possibility of going always further, but without possiblity of seeing the whole infinite thing (e.g. the set of natural numbers) at one moment. While the clasiical axiomatic construction of mathematics today permits you looking at such an infinite thing - you can "think the infinity" as completed, hence the name "actual" infinity.

The whole question is in fact very deep and is all about philosophy (in the best sense) and very often goes closely to almost theology..

(There are also approaches to mathematics that forbid infinity - google for finitistic mathematic.)