Universal and Existential Quantifiers - Variables as Natural Numbers

Question

It is said that an proposition with a universal quantifier represents a possibly infinite conjunction, and a proposition with an existential quantifier a possibly infinite disjunction. How can one illustrate this approach, where the quantified variables can take as values the natural numbers?

Answer 1

If the language has the numerals, i.e. the terms corresponding to the natural numbers, we can write e.g. $\forall x (x \ge 0)$ as :

$(0 \ge 0) \land (1 \ge 0) \land (2 \ge 0) \land \ldots$

The same for e.g. $\exists x (x=0)$ :

$(0=0) \lor (1=0) \lor (2=0) \lor \ldots$