Consider the first order predicate formula 𝜑: ∀𝑥 [(∀𝑧 𝑧|𝑥 ⇒ ((𝑧 = 𝑥) ∨…

Interpretation of the formula φ: For every element x in the domain, if every divisor z of x is either z = 1 or z = x, then there must exist some w greater than x such that every divisor of w is either 1 or w. In other words, whenever x has only trivial divisors (like a prime), there must be a larger element with only trivial divisors.

• S1 = {1,2,3,…,100}: Fails. Take the largest prime in the set (for example 97). That x has only trivial positive divisors, so the antecedent holds, but there is no larger element in S1 with the same property, so the required w does not exist.

• S2 = set of all positive integers: Satisfies φ. An x that meets the antecedent must be 1 or a prime (within positive integers). By Euclid's theorem there are infinitely many primes, so for any prime x there is a larger prime w > x whose only positive divisors are 1 and w; thus the existential part is satisfied. For x where the antecedent is false (composite x), the implication holds vacuously.

• S3 = set of all integers: Fails under the standard integer divisibility relation (which includes negative divisors). Example: take x = -1. The divisors of -1 are {1, -1}, so for every z dividing x we have z = x or z = 1, so the antecedent holds. But any integer w > -1 is 0 or positive, and such a w has −1 as a divisor (or many divisors) which is neither 1 nor equal to w, so no w > -1 has the property that all divisors are only 1 or w. Hence the existential part fails for x = -1.

Conclusion: Only the set of all positive integers satisfies φ under the usual interpretation of divisibility over integers; the finite set {1,…,100} and the set of all integers do not satisfy φ.

Remark: If divisibility were intended to consider only positive divisors even when the domain is all integers, then the behavior of S3 would mirror S2 and S3 would satisfy φ. Because the problem statement explicitly says the variables range over integers and divisibility is the standard integer relation, the correct mathematical reading treats negative divisors as valid, so S3 fails.