Set Size Comparison, Continuum, 3 Essence of Set Theory

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Powerset, general Cantor's diagonal argument. Proof that the powerset of the set of all the natural numbers is equipotent to the set of real numbers using Cantor-Bernstein Theorem.

[toasteduranium]

Never thought I’d see a John Green quote in a video somewhere else. Cool!

[luizfelipegarcia]

Very nice video! Good work with the animation!

[erikstephens]

Here's an existential crisis inducing thought: The number of characters we use in the union of all human languages is finite (English, Chinese, all natural languages, and all symbols used to make well-formed formulas in formal maths). If we take every possible string of such characters and put them into a set, it would only be countably large (and would include all computable numbers and probably lots of uncomputables like the Chaitin constants). With the reals being uncountable, there must exist an uncountable about of numbers that individually cannot be defined with well-formed formulas. I like to call these numbers "ineffable numbers." The fact that this argument works for any uncountable set also tells us that ineffable numbers are dense within the reals, stretching across the infinite like an eldritch horror, yet seeping their Lovecraftian tentacles into every crevice of the known universe. And somehow, despite all of this, these numbers are practically invisible to the human mind.

For anyone wondering if this could be formalized: all you need to do is use a finite set A to represent an alphabet, and, for each natural n, create a map from {1, ..., n} to A. Such maps are strings. Take the set of all strings of size n, and pick out the gibberish strings (there are grammars that can be used to keep the sensible strings. These are essentially operations that let one take small wff's and inductively combine them to make each of the bigger ones.), then take each sensible wff with one free variable: P(x), and keep the ones that which are satisfied by exactly one real number. These can be thought of as "definitions" of said reals. Since the set of "definitions" of size ≤ n must be finite, the set of all definitions is countable.

Also, If you are thinking you can get around this by formally defining the set of definable real numbers D, and creating its complement R\D, and saying let x be a member of R\D (which you can totally do), note that you haven't really defined an undefinable number. Yes, you can follow each of these steps, but the number you make isn't a specific number, it's a generic number, meaning it isn't truly rooted to an actual value. It's a stand-in, much like letting p be prime creates a stand-in for a prime and doesn't define a particular number like 2. I guess, in a way, these ineffable numbers can be talked about, but they cannot be constructed.

An interesting consequence of this fact is that the existence of a number implying its constructability is in direct conflict with the existence of uncountable sets. This would also put such exotic beings in conflict with Cantor's theorem and with the ZFC axioms that allow uncountable sets to exist.

I also have to wonder how many of math's "choice functions" (which are used for bizarre things like the well-ordering theorem, the Banach-Tarski Paradox, the construction of sets with undefined Lebesgue Measures, and every other reality-breaking thing mathematicians have built) would no longer work without these ineffable entities?

Feeling small yet? Well, with the subsets of the real line, continuous nowhere-differentiable functions (Check out the Baire-Category Theorem for more info on this b.s.), higher cardinal numbers in general, Conway's surreal numbers (They make the proper class of ordinals look like a joke), and inaccessible cardinals (outside the reach of ZFC), we've only just begun.

Great video btw.

[emilioyared]

There is something I can't wrap my head around, we can map every real number to a subset of P(N), but why does the subset of P(N) map another number and form this zigzag pattern?

[polfosol]

I really enjoyed watching these lectures. Thanks for making them and keep up the good work.
By the way, some parts could have been a little more expanded, as they weren't clear at first sight. For example, Cantor's diagonal argument at the end of the 2nd video was a bit vague.

[abc]

I'm not sure I follow the proof at 5:00. How do we know the subset M is not contained later on in the matching? What about a matching makes it so that it cannot contain this M?

[maurocruz]

I was completely lost in 7:00.
What is the relationship between (10, 1, 11, 5) has to be ordered and the fact you begin to add "hundreds" in the codification?

[wiggles]

In the beginning, you talk about the bijection between the set of all subsets of ω, and the set of all binary sequences. However, when you map each binary sequence to a real number by putting a "0." in front of it, you do get all the real numbers y such that 0<=y<=1, but this mapping is not injective. It's not injective because for example = 0.1. The sequence 01111... doesn't equal the sequence, or in other words, x1 didn't equal x2, yet f(x1) = f(x2), which is the counter example to injectivity. If a mapping is not injective, it's not bijective, but we need a bijective map to conclude two sets are the same size. I think there's a way (but I don't know the details) to modify the mapping from binary sequences to real numbers so that it's injective, and then since we already have that the mapping is surjective it gives a bijection to {y in R : 0<=y<=1}. I don't know that way though.

[DrizzleWoolf]

To show an isomorphism between the Reals and the binary sequences, isn't it just as easy to say that the infinite binary sequence represents a float with infinite precision ?

[zuko]

Hey really nice videos but i cant wrap my head around to why 42 cant be expressed using the system 7:50

[tim-cca]

Any set has cardinality greater than continuum?

[Lauschangreifer]

Could you provide the sources of your videos?