Thread by @toorandom

"Exploring different infinities" (since @mauropm & @gnuowned) told me they did no understand when they invited me to their podcast @openenchilada).

The idea of this thread is to show you easily that there are different infinities [...] #mathematics #mathschat #math #maths

Lets start with the set of natural numbers IN={1,2,3,...} the size of this set is ℵ0 (aleph-0) which is the first infinity. We want to compare this infinite set with others. But first, how do you know when a set A has the same size as a set B without counting? [...]

Imagine a dance party, in one side you have the set M of men & in the other side the set W of women. You would like to see dancing pairs of people of different sex. If you generate the pairs and EVERYBODY is dancing then you can say that |M|=|W| (both have the same size) [...]

Consider the set E={2,4,6,.2n,....} of even numbers, that set has "half of the elements" of IN. Lets try to make E and IN dance. Consider the pairs (2,1), (4,2), ... (2n,n),.... Every element of E has a unique partner in IN and viceversa (bijection). Therefore, |E|=|IN|. [...]

This means there are as many as even natural numbers as natural numbers! (weird huh?). This is kind of weird since E is strictly contained in IN. You make a perfect match for dancing even for multiples of 1000 with IN or multiples of every k>0 no matter how large is k. [...]

With a little more effort you can also prove that the set of rational numbers, which are the fractions a/b with a,b whole integers can dance perfectly with IN. You just arrange the fractions as a zigzag & enumerate them to get the bijection with IN (red) as in the picture
[..]

So the idea is to arrange your infinite set in such a way that you can enumerate their elements. You'll be automatically generating the "dancing" pairs. The proof that the rationals and IN have the same size had a lot of controversy in the XIX century thanks to Georg Cantor [...]

What about the real numbers IR which include numbers like √2 or π?. These numbers have infinitely non-periodical digits, so IR includes IN, together with all fractions & all the numbers you can build with any infinite sequence of decimals and its permutations! (loots more) [...]

Lets prove that IN & IR cannot have the same size (ℵ0).
Assume the opposite to get a contradiction, so IR has the same size as IN. Therefore each element R_i of IR can be enumerated in a list L by assumption. This list grows vertically & horizontally.

Consider the diagonal digits [a_ii] of all the real numbers R_i as in the list L containing the whole IR. Consider a function F(m)=1 if m is even and F(m)=2 if m is odd. Construct the following number by concatenating all the F(a_ii):
X=0.F(a_11)F(a_22)...F(a_nn)... [...]

Observe closely this number X which is clearly an element of IR. We used all the real numbers (listed in L) to construct it. This special constructed X cannot be in L since at least the digit i-th digit will be wrong (we changed the parity). Therefore L cannot have all IR [...]

This is a contradiction to the fact that IR can be enumerated meaning that contradicts the fact that IR and IN have the same size.

Since IR contains strictly IN, o IR\IN is non-empty, & we proved that they don't have the same size, then we say that IR has size |IR|=c>ℵ0 [...]

There are bigger infinities, for instance, the set of all subsets of IR or the set of all functions f:IR->IR is bigger than |IR|. This topic is really delicate in terms of axiomatic mathematics and touches really delicate fibers of it.

Ask more interesting (& philosophical) questions, like "Is the set of all theorems in mathematics in bijection with IN or IR ?" If a theorem is by definition a finite set of symbols in a set S (human+math) that is derived by some proof-system using some logic inferences [...]

Since you can enumerate all the strings that are possible in the language, the collection of theorems is indeed a set & can be put in bijection with IN. This tell us that our language is kind of a "limitation". Our logic & mind may be no enough to cover all math (mindblow). [...]

Nasty stuff is that the proof to the statement "There is an infinity that is strictly lower than |IR| and strictly greater |IN|" is proved to be unprovable (yes, proved that you cannot prove it!!). In fact is a consequence & application of Gödel's incompleteness theorems. [...]

However, Georg Cantor was too advanced for his era, he committed suicide. A lot of his problems were not solved at his time & was rejected by the math community. However David Hilbert (my great-great-great-great Ph.d father according to my math genealogy haha) defended his legacy

I hope you enjoyed! maybe @mathematicsprof , @logicians or @johncarlosbaez can have better historical opinions on this thread! or maybe can complement it.
The images here were taken from:
quora.com/What-is-a-bije…
kazuhikokotani.wordpress.com/2011/12/05/an-…

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