I handed in my thesis a few days ago and I've always wanted to make a longer post about what it is about. But also I know that most people on here that I interact with are not mathematicians and those who are are probably not set theorists. So if I just told you "It's about how you can prove the relative consistency with ZFC of the splitting number being strictly less than the bounding number using a finite support iteration of Laver forcing with the Frechet filter of length ω2" then that wouldn't mean anything to anyone reading this. It certainly wouldn't have meant anything to me a year ago.
So how about I leave out the details and talk big picture. I hope this will be accessible to anyone with high school-level math knowledge and curiosity to learn more about it. I'll tag this and all subsequent posts in this series with # grey's thesis explained
Sets and where to find them
Real quick: 'sets' is just what mathematicians call their collections of things (yes, it's more complicated but we'll get to that). Think the set of real numbers which is just the collection of all real numbers (or decimal numbers for the programmers out there). But you can just as well have the set of all points in 3d space. Or maybe the set of all colors, or the set of all kinds of meows a cat could possibly make (yes, yes. terms and conditions apply).
Set theory is, de facto, the foundation of modern mathematics (category theorists dni). While I doubt it's really taught anywhere in high school when you get into university this is basically the first thing you'll learn (the basics) because everything else is built upon it.
But set theory is a fairly recent invention. While geometry and number theory has been studied in some form or another by ancient Greek mathematicians like Euclid and Pythagoras (whether or not he actually was a person), set theory was essentially invented by Georg Cantor in the 1870s.
Collections of things are, of course, not a new concept. You could have a set of 13 apples or a set of ∞ bananas for example. But Cantor made a discovery about those sets of infinite bananas: Some infinite sets are bigger than others o_o
You can't count the real numbers
Cantor published a different proof for this first, but here's a later proof by him that is much more famous now. We want to show that there are more reals numbers than there are natural numbers (so 0, 1, 2, 3, 4, ...). Well, that's obvious right. Every natural number is a real number but numbers like 0.5 or π are real numbers but not natural numbers. QED
Okay, not so fast. That's not what Cantor meant by "there are more real numbers than natural numbers". Natural numbers are the counting numbers (because that's how you count (imagine you started counting at 0)). What Cantor says is there's no way for you to name every real number.
Let's say it like this: Imagine I give you an infinitely strip of paper and infinitely much time to write down, one-by-one, every real number. Then, no matter what you do, there will always be a real number that you missed. No matter what you do. (In fact there will be infinitely many numbers that you missed. Actually, there will be infinitely many times more numbers that you missed than there are numbers that you wrote down. But let's start at just one number that you missed.)
Don't believe me? Here, I'll show you. First, we make the problem easier for you. Instead of writing down every real number, I'd be okay with you only writing down every real number between 0 and 1. And let's also say you're writing them down as a decimal expansion (you could do binary or base 12 or whatever and the proof would still work). Now let's say your list looks something like this (I'll count from 1 to make it easier)
- 0.5852195621244589...
- 0.903661666216237...
- 0.3577217461557666...
- 0.552045814913943...
- 0.9501837853404622...
- 0.3839465375043226...
- 0.48136503796489816...
- 0.402722451940127...
- 0.13022626956925853...
- ...
Your list will go on forever. It's infinite. Okay, now let's construct a new number that appears nowhere in this list.
We'll construct it step by step. First, look at the first decimal place of the first number. If it's a 0 then our new number will have a 1 in the first place and if it's not a 0 then our new number will get a 0. So in this case 0.5852195621244589... has a 5 in the first decimal place so our new number starts with a 0:
0.0
Next, for the second number we'll look at the second decimal place. Here it's 0.903661666216237... which has a 0 in the second decimal place, so our new number will have a 1 in the second decimal place.
0.01
We continue this and get in this case 0.010100100... (I think). Your results may vary. However, this new number won't appear anywhere on your infinitely long list! Here's why: It can't be the first entry of the list because our new number has a different first digit than the first number. It can't be the second number because it has a different second digit. And it can't be the third either, nor the fourth, nor the fifth, nor the 27387891498437th. It can't be any of those. You missed this one :(
"Oh hon hon", I hear you say, (you have a French accent in my imagination) "zhat would have been zhe zerozh number of my list." (also you're a yinglet). Well played! But now replace all the 1s in the new number by 2s and try to find that number somewhere in the list.
Okay, so you can't count up all the real numbers. The set of real numbers is actually bigger than the set of natural numbers. (Fun fact, the set of rational numbers (that is fractions) isn't bigger than the set of natural numbers. They're the same size! Somehow the only simple sketch for this I could quickly find was on a homeschooling website o_0)
Now this was BIG at the time. Infinities that are bigger than other infinities?? Gee whiz! Cantor needed some way to work with all this and to compare the sizes of various collections and to this end he invented set theory.
Last little tangent here: You can generalize an argument like this for general sets and a result of this is that for any set there exists a set which is bigger. But then that means there are infinitely many infinite sets which are all bigger than the last :D
So long and thanks for all the fish
All right, that's it for the first part. I'll write more and actually get to what my thesis was about in more detail. Next time, we'll forget everything I just said and count the real numbers anyway :D
I hope this made sense. If you don't understand something, it's not your fault, I probably explained it poorly. Please comment on the chost to ask me about it :)
