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In a book "How to prove it" by velleman, universal quantifier is defined as follows;

To say that P(x) is true for every value of x in the universe of discourse U, we will write [itex]\forall[/itex]xP(x). This is read "For all x, P(x)". The symbol [itex]\forall[/itex] is called the universal quantifier.

And in this book, universe of discourse is defined as a set of all possible values for the variables relating with a statement.

Then, what I want to know is if under a special condition like that the universe of discourse of a statement is not a set, the universal quantifier can't be used to express such the statement.

For example, since there's no set which contains all ordinal numbers, I expect that a statement like "for all ordinal numbers x, P(x)" can't be expressed by using universal quantifier.

In a book " introduction to set theory" by karel hrbacek, the set of all natural numbers is

defined as [itex]\left\{[/itex] x : x[itex]\in[/itex] I for every inductive set I[itex]\right\}[/itex].

In this case, the universe of discourse of the statement is the set of all inductive set.

But , i can't convince of the existence of such a set of all inductive set.

2. Relevant equations

3. The attempt at a solution

In some books described by NBG set theory, universal quantifier is just defined without the use of universe of discourse. I thought that this is because the notion class can be replaced with it. I mean, I thought that the definition of universal quatifier is such that for all x contained in a specified class, P(x).

I'm very confused about this notion and I doubt myself whether asking this question is meaningful or not....

I wanna know what is wrong in my argument and I ask you the exact definition of universal quantifier and also wanna know if the notion universe of discourse is needed in defining universal quantifier.

Please tell me about the truth if you know .

Thakns..

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# Universal quantifier

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