predicate logic truth value

Question

I have a structure of predicate logic given by taking the set of natural numbers, 0,1,2... as the domain of discourse

∃X · ∀Y · X < Y

Do you read this as there exists an X such that for all Y X is less than Y?

Is the domain of discourse the possible values for X, and those values are the set of natural numbers?

is this false (I think it is but cant quite get my head around it) If someone could explain it would be very much appreciated. thank you.

Answer 1

Yes, you are right in both questions: it reads as

There is an $X$ such that for all $Y$, $\ X$ is (strictly) less than $Y$.

And, this is false: for any (natural number) $X$ we can find an $Y$ such that $X$. Alternatively, in this context, $Y:=0$ would also be fine for all $X$ to show that the statement is false.

Answer 2

Yes, the domain of discourse is the set of natural numbers. And yes, the sentence $\exists X\forall Y(X\lt Y)$ is false in the natural numbers. Informally, it says that there is a natural number $X$ which is *less than every natural number.

Let $Y=0$. It is not true that there exists a natural number $X$ such that $X\lt 0$.