Theorem of the Existence of Infinite Non-Funny Mathematical Jokes
There is a “famous math joke” known as The Infinite Mathematical Jokes Theorem, which says the following:
List all math jokes in order of lenght.
Assume there is a largest math joke, L.
Create a new math joke J by appending to L that joke about the pirate who has a wheel on his crotch that is “drivin’ me nuts!”
J is now larger than L, which is a contradiction.
Therefore the set of math jokes is infinite.
Now, assume a good math joke, M.
If M is a good joke, then it is funny.
If a joke is funny then everyone will know it.
If everyone knows a joke, the joke will not be funny.
If a joke is not funny, then it is not a good joke.
Therefor, if M is a good joke, M is not a good joke.
By contradiction, there are no good math jokes.
There are infinitely many math jokes and none of them are good.
In fact, this is a famous math joke, but it’s still not very funny. However, the purpose of this text is not to laugh, but to be reasonably happy as we discuss and correct some errors in this theorem.
First let’s look at the idea that if there is a bigger math joke called L, then the total of jokes has to be finite, that is, N jokes formed with up to L characters. But this is false, as we can insert at the end of a joke with L characters, the joke of “Pirate who has the rudder wheel in the groin”, soon there would be a joke with more than L characters, that is, there is no greater math joke. Thus, there must be infinite mathematical jokes. Although this statement is true, we can demonstrate this in a more “simple” way.
Realize that creating endless math jokes is not as complex as it appears in the famous joke, in fact, it doesn’t need a construction as strange as “the biggest math joke”. To demonstrate, I will create endless math jokes from the structure of this joke:
“What did 2 say to the thousand? You may be great, but it’s not 2!
So, keeping the structure of this joke apart from the word in bold, we can create jokes for any Real number greater than 2, for example:
What did 2 say to 1001? You may be great, but it’s not 2!
What did 2 say to 100? You may be great, but it’s not 2!
What did 2 say to 10? You may be great, but it’s not 2!
What did 2 say to √5? You may be great, but it’s not 2!
What did 2 say to π? You may be great, but it’s not 2!
What did 2 say to 2.1? You may be great, but it’s not 2!
Thus, as the set of Real numbers greater than 2 is infinite, there are infinite mathematical jokes of this type.
This construction can be used to show that there is no longer a joke, as we can always increase the length of a joke by increasing the number of characters that make up the number with whom 2 talks.
Another contradiction in the joke theorem is the definition that if M is a good joke, then M is funny, and if M is funny, everyone knows it. But if everyone knows the joke, then it is not funny, and if it is not funny, then the joke is not good. The problem here is that the set of good jokes will always be empty, since any good joke will be a bad joke, so there could be no good jokes.
The joke theorem was intended to be a joke (maybe even funny) and funny proof that there are infinite mathematical jokes, but that none of them are good, yet it does not form a definition of a good joke. One way to correct this definition and not change the purpose of the theorem is to define a joke as not being funny if everyone already knows it. Thus, a condition for a given M joke is not funny, is that everyone knows M.
Thus we arrive at the desired result. Take the joke:
“What did 2 say to x∈ℝ, x> 2?
You may be big, but you’re not 2! ”
For any x> 2, this will be a different joke, thus, there are infinites not countable (because it has the cardinality of ℝ) mathematical jokes of this type, but that by the known and valid property for any Real number greater than 2, we know the infinites variations that this joke can have. So, there are endless math jokes that are not funny! With this, the Theorem that states that there are infinite mathematical jokes that are not funny, is demonstrated.