# Expressing the existence of irrational numbers

## Homework Statement

Express the following using existential and universal quantifiers restricted to the sets of Real numbers and natural numbers

## The Attempt at a Solution

I believe the existance of rational numbers can be stated as:

##(\forall n \in \Re)(\exists p,q \in \mathbb{N}) \ni [(p \mid q)=x]##

So to say that there are irrational numbers is the negation of this being:

##\neg (\forall x \in \Re)(\exists p,q \in \mathbb{N}) \ni [(p \mid q)=x]##

Which becomes

##(\exists x\in \Re)\ni(\forall p, q \in \mathbb{N})[(p\nmid q )\ne x]##

Is my rationality correct

Last edited:

WWGD
Gold Member
If I understood correctly, the right-most expression should be ##p|q \neq x ##. Just slide the negation parenthesis by parenthesis according to usual negation rules.

If I understood correctly, the right-most expression should be ##p|q \neq x ##. Just slide the negation parenthesis by parenthesis according to usual negation rules.

WWGD
Gold Member
No problem, sorry for the necessary nitpick.

No problem, sorry for the necessary nitpick.
But one question:

when the negation moves inside the expression it becomes:
##[(p\nmid q )\ne x]##
Correct?

WWGD
Gold Member
But one question:

when the negation moves inside the expression it becomes:
##[(p\nmid q )\ne x]##
Correct?

Well, no, because the predicate , or the content of the original statement is that the expression p/q =x . The statement is not about whether p divides q.
The predicate of the original is that there is an equality between the object p|q and the object x. Here p|q is sort of the subject of the sentence, not part of the content of what is being asserted; the content is that the expression p|q equals x. Maybe you can make this more rigorous by using a predicate of something like ##R_{pqx} ##. More formally, ##p|q## is a member of your universe of discourse, i.e., it is an element of
the collection of objects about which you are stating something. p|q is 'the subject' and = is the predicate, i.e., what is being stated about p|q. You negate the predicate/relation , not the subject.

Well, no, because the predicate , or the content of the original statement is that the expression p/q =x . The statement is not about whether p divides q.
The predicate of the original is that there is an equality between the object p|q and the object x. Here p|q is sort of the subject of the sentence, not part of the content of what is being asserted; the content is that the expression p|q equals x. Maybe you can make this more rigorous by using a predicate of something like ##R_{pqx} ##. More formally, ##p|q## is a member of your universe of discourse, i.e., it is an element of
the collection of objects about which you are stating something. p|q is 'the subject' and = is the predicate, i.e., what is being stated about p|q. You negate the predicate/relation , not the subject.
Ah I understand, this makes more sense to me now. Thank you.

WWGD