Written by haloboy777 on September 7, 2020
How a man destroyed one logicians dream to unify all the branches of mathematics.
So this is a super interesting story. It's about SET theory and basis of all of mathematics. So IDK how you'll respond. But whatever...
So it all started with the two greatest chums who seem to have touched every topic on earth, Plato and Aristotle.
It was 387 BC and Plato was chilling in Akademia[^1]. He was going about his day messing with his students and philosophers after 2000 years from his time, a bloke came to him and asks him a simple question "What are numbers", he got up and then went inside and started having a lukewarm mental breakdown.
After some time he comes out of the Akademia with The Philosophy of Mathematics.
His philosophy basically said. "There are objects and there are forms. And numbers and the relations between them (like addition and stuff) are independent of us and objects of the real world and are in turn objects of world of forms. And we just relate to those objects from world of forms cause they allow us to define our world."
Plato was happy about his explanation, but there was one guy who took that definition not well to his heart, that was his star student Aristotle. Aristotle said there no other world than this, that old man is clearly out of his mind. I can't really tell.
So now he went inside Akademia and had his own version of mental breakdown.
Having mental breakdowns was cool back then - Diogenes.
So he comes out with his own theory about numbers. It goes something like this..
"There are objects in the world and objects have properties and one of the property of the object is a number attached to it. If you see a cube the number of surfaces is the property of the object." - this is what he said in a really broad sense.
And then both of them forgot about this little topic and went on to f@%k with the rest of philosophical society for lulz.
Fast forward to 1870's. Time was wild. First headphone jack was invented, by the way that design is still used to this day. Edison created a light thingy. Bell created phone. Fashion was shit.
In Germany there was a guy named Gottlob Frege[^2]. And he was pissed. He was not happy about how Plato and Aristotle did not do their job properly and never reached a conclusion regarding the deal about numbers.
To be honest he just got stuck with the words "Some" and "All", like a dedicated math freak after getting some good kush. He raised questions about the definition that Aristotle gave out. "Numbers being properties of an object".
He said, "If numbers were the properties of object then A single (1) bouquet and A group 25 of flowers shouldn't be the same thing. And if in-fact they are same, then an object should be having multiple ''number properties' attached to it and that doesn't make any sense."
He then went ahead and proposed his own idea about what numbers are. He said - "Numbers are extensions to a concept. And a concept can be anything you want."
all the cats you know
all the cats you don't know
all the dogs you know
all the dogs you don't know
wheels in your house
rats in your house
all the yellow birds in Amazon forest
So once you define a concept you can just go ahead and assign a number to that concept. And that number can be zero/infinity, whatever you want.
He was damm pleased with his explanations. And was about get them published. But then, an ultimate savage from Britain[^3] wrote Frege a letter. Stating that - "He loves his work and everything he formulated till now is wonderful, but there is one point Frege missed."
Okay this is little tricky, But bear with me on this one..
"Let us call a set 'Normal' if it is not a member of itself, and 'Abnormal' if it is a member of itself."
Simple? Read it again if it's not settled in your brain.
Now consider a Variable R and assume it is a set of all the 'Normal' sets.
Nothing wrong there? now determine whether R is normal or abnormal.
There are two scenarios, R is normal or R is abnormal.
If R is normal then R would contain itself since that's how we defined R - [set of all normal sets] and that would make R abnormal cause that's how we defined 'abnormal'.
And if R is abnormal then R would would be a member of a set whose members are normal cause that's how we defined R.
Thus R is neither normal or abnormal. It's just there to mess with your brain.
This paradox literally caused him to go to hospital. You can search for Barber's paradox for similar kind of reasoning.
Later on we move away from Ferge's theory to Zermelo–Fraenkel set theory. This theory is built in a way, that it just avoids this paradox. And there are contenders to this theory as well[^4]. So let's see what happens in future.
PS: I learned about this paradox from Jades Video. I would highly suggest you to go and watch her. As she explains this in much more detail and much more effortlessly.
In this sort of informal story I took many artistic licences and made up a lot of stuff. Do come and shout on me at twitter if I made any mistake.
[^1]: The academy founded by Plato. Platonic Academy - Wikipedia
[^2]: Our Logician from the title.
[^3]: Bertrand Russell The ultimate savage.