Consider the proposition "There are infinitely many prime numbers". Express the proposition as a

1. We want to express that there are infinitely many prime numbers.

Consider the proposition "There are infinitely many prime...

Consider the proposition "There are infinitely many prime numbers". Express the proposition as a predicate formula using quantifiers, logical connectives and the predicate P(x). Assume the domain to be natural numbers. Note that you don't need to use the answer from the previous part in this problem; you can write your answer in terms of P(x). Hint: For inspiration, consider the following game: you pick any natural number x (as large as you want) and I have to find a number y such that y is larger than x and is also a prime number (I win if I can find such a y, otherwise you win!). Now notice that the proposition is in some sense equivalent to saying that I can always win the game!

Transcript

00:01 Alright, so we have a proposition.

00:03 There are infinitely many prime numbers.

00:06 We want to express this proposition as a predicate formula using quantifiers, connectives, and the predicate p of x, where p of x can be interpreted as x is prime.

00:22 Now, how can we do this? well, we're going to assume that our domain is all natural numbers, so when we say for all x, that means that for all natural numbers x, not for all possible objects x.

00:43 And then, let's see what we can do here.

00:44 There are infinitely many prime numbers.

00:46 We can't just say, like, there exists infinitely many x such that p of x is true.

00:54 Because this, like, what does this even mean? there exists infinity x? no, that doesn't count.

01:00 We need to be a little bit more subtle here.

01:02 In particular, what does it mean if there are infinitely many prime numbers? well, i'm going to claim what that means is that if we have a set of prime numbers, x1, x2, dot dot dot, xn, such that we have p of xi for all i, that is, all of these are prime, then there also exists some x prime, some additional x here, such that that is prime, and it is not in the set.

01:39 That is to say, any finite set of prime numbers does not contain all the prime numbers.

01:45 There is another one.

01:48 So let's see here.

01:49 An important part of the natural numbers as well is that they are well -ordered, which means that any set has a lowest bound.

02:00 And in particular, if we say that the set of prime numbers, so all those x such that x is prime, has no upper bound, the set is not bounded.

02:19 We know it's not bounded above.

02:21 We know it's bounded below by 1 or 0, depending on how you take your natural numbers.

02:26 But it's not bounded above, because otherwise there would be a largest prime number, and that's not acceptable.

02:34 We can't have a largest prime number.

02:38 So what we can do here is then for all x, if we have that x is prime, we can conclude that there exists a new prime number, a new y, such that y is greater than x, and y is prime.

03:02 That is, there exists a y such that y is greater than x, and y is prime.

03:08 So let's go through this a little bit...

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