00:02
So what is the quantified statement for every, there are infinite many prime numbers? this means that for every n, which is a prime number, there exists some m, which is also a prime number, and m is strictly bigger than n.
00:31
So this is what it means.
00:32
This is a quantifier statement for a quantifier statement for saying that there are infinitely many primes.
00:46
Many primes quantified is going to be this.
00:51
Okay, so let's do a proof by contradiction.
00:56
So suppose not.
00:59
This means that, well, there exists some big m, which is prime, such that for every m prime, we have m is less than equal to n.
01:27
So what does this? tell you this will show you that there just is a large number such that every prime number is always below it.
01:37
Now we'll show that this cannot happen.
01:39
That's a proof by contribution.
01:41
Why? well, so let's list out all the primes below n.
01:48
So this will be m1, m2, all the way to at most n.
01:56
So n is the largest prime and we have, you know, all these primes are in order.
02:04
The first thing to note is that the number of primes is finite.
02:14
So the number of elements in the set, let's call the set p.
02:22
So the cardinality of the set or the number of elements in the set is less than plus infinity.
02:31
So it's a finite set...