Is my expression for this quantifier correct?

Question

Quantifier: “for exactly $n$ distinct $x$ $P(x)$ holds.” (Where $n \in \Bbb{N}$.)

My expression: $$ \exists x_1, \ldots ,x_n ((P(x) \rightarrow (\exists i \in \{1,\ldots ,n\} )(x=x_i)) \wedge (\forall i,j \in \{ 1, \ldots ,n\} )(i \neq j \rightarrow x_i \neq x_j))$$

Is it correct? Is it okay to use quantifiers for indexing as I’ve used above?

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Is it now correct? $$ \exists x_1, \ldots ,x_n (P(x_1)\wedge \cdots\wedge P(x_n)\wedge (P(x) \rightarrow \bigvee_{i =1}^n x=x_i) \wedge \bigwedge_{i,j=1}^n (i\neq j \rightarrow x_i\neq x_j))$$

Answer 1

If you're trying to work within a formal system of predicate logic (I presume you are, since otherwise you might as well just use the sentence in words which gets the idea across just as precisely and much more readably) then you can't do this since the natural numbers $\mathbb{N}$ might not be things that exist in the structures you're working with. The natural numbers are things that when we meta-reason about the logic, not elements in the structure that $\forall$ can draw from.

In some scenarios, it might even be dangerous to mix this up: if you're working on, say, first-order set theory, this looks like the perfect recipe to start conflating the natural numbers that exist in a particular model of set theory and the natural numbers that exist in the real world which we use when reasoning about the model: this is sure to end up with you missing some important distinctions between them and ending up very confused.

If we're treating $n$ as a fixed known number, we can work this with a chain of disjunctions: something like $\forall x: P(x) \rightarrow \bigvee_{i=1}^n x=x_i$ where that big V is just shorthand for the chain of disjunctions $x=x_1 \vee x=x_2 \vee ...$

P.s: Since you're trying to translate 'Exactly $n$ distinct $x$,' you'll also need to include a clause saying that each $x_i$ has property $P$.