Universal Quantifier: Definition & Use

[gotjrgkr]

Homework Statement

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 $\forall$xP(x). This is read "For all x, P(x)". The symbol $\forall$ 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 $\left\{$ x : x$\in$ I for every inductive set I$\right\}$.
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.

Homework Equations

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

Please tell me about the truth if you know .
Thakns..

 

[mighty2000]

Thank you for your post. It is a valid question and something that many people struggle with when first learning about logic and set theory.

First, let's clarify the definition of universal quantifier. As stated in the book "How to Prove It" by Velleman, the universal quantifier is used to express the idea that a statement is true for every possible value of x in the universe of discourse. The universe of discourse, as defined in the book, is the set of all possible values for the variables relating to the statement. So, in order for the statement "for all x, P(x)" to make sense, there must be a set of all possible values for x.

In your example, you mention that there is no set that contains all ordinal numbers. This is true, and therefore the statement "for all ordinal numbers x, P(x)" cannot be expressed using the universal quantifier. However, this does not mean that the universal quantifier cannot be used to express statements about ordinal numbers. It just means that we need to be more specific about the universe of discourse. For example, we could say "for all ordinal numbers x in the set of countable ordinals, P(x)" or "for all ordinal numbers x less than or equal to ω, P(x)".

In regards to your second example about the set of all inductive sets, it is important to note that this set does not actually exist. It is a theoretical concept used in set theory to help define other sets. So, in this case, the statement "for all x in the set of all inductive sets, P(x)" is not meaningful because the set of all inductive sets does not actually exist. However, we could still use the universal quantifier to express statements about inductive sets, such as "for all inductive sets x, P(x)".

In summary, the universal quantifier can only be used to express statements about a universe of discourse, which is a set of all possible values for the variables in the statement. If such a set does not exist, then the universal quantifier cannot be used. However, this does not mean that we cannot use the universal quantifier to express statements about those values, we just need to be more specific about the universe of discourse. I hope this helps clarify your understanding of the universal quantifier. If you have any further questions or concerns, please don't hesitate to ask.

Best