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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?

  • Can you think of a predicate on the natural numbers that is satisfied by infinitely many of them? Can you think of a predicate on the natural numbers that is satisfied by at least one of them? Having concrete example predicates in mind may help you... – Eric Towers Jun 17 '19 at 09:42

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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. $\exists x (x=0)$ :

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

Mauro ALLEGRANZA
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