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In fear of asking an elementary stupid question, I'll say that I am failing to differentiate between the process of Abstraction in set theory and Predicate Logic. I would like to illustrate my confusion with the following example:

I want to outline all the multiples of 2 (universe of discourse: integers): ...,-4,-2,0,2,4,...

In predicate logic, I would characterize it thusly:
$$ (\forall x)[(x \in \Bbb Z ) \wedge (\exists n)[(n \in \Bbb Z) \wedge (x=2n)]] $$

As a set, using Abstraction, I would delineate it so: $$ A= \{ x: (\exists n)[(n \in \Bbb Z) \wedge (x=2n)] \}$$

Can I use them interchangeably? When should I use one and preclude the other? Are they the same thing? Can I use the "all" quantifier ($ \forall $) in Abstraction?

I apologize if my question might not even make full sense. The untidiness of the question exposes an ignorance that I hope you could ameliorate. Thank you for you help.

Sisyphus
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What you wrote in predicate logic was a proposition that says "all objects are even integers" since the domain of discourse in the integers it's redundant to say $x\in \mathbb{Z}$ and even wrong depending on the context. If a formula has no free variables it's a statement. For a characterization you need a free variable or one that is not in the scope of a quantifier. You have that $"x \text{ is even }"\equiv \exists y \,(x=2y)$.

An important distinction is the following. Let $\varphi(x)$ be some formula of predicate logic that has $x$ as a free variable (for example the one above "$\exists y \,(x=2y)$"). The formula itself is not an object in the domain of discourse. In your case the domain of discourse is the integers which do not contain formulas that talk about the integers. Nonetheless the formula $\varphi(x)$ describes a subset (possibly empty) of the integers. This is done by what you call abstraction or in the context of set theory unrestricted comprehension (This can cause some issues if not used carefully). Note that a subset of integers is not an integer so it's not itself an integer. What you can do is broaden the domain of discourse to include subsets of the integers. This means you are going from first order logic to second order logic. The difference here is that $\varphi(x)$ is a formula while $\{x:\varphi(x)\}$ is an element of the second order theory.

I don't know if you encountered any of this yet but I hope this cleared up some confusion.

MIO
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