Question

Wikipedia informs me that the probability that a given structure (G-subN) with Domain {1,...,n} models S where 'S' is a first order sentence converges to either 0 or 1 as n->inf. I have two questions regarding this result. First, am I to understand the 'Domain' in this context to be the Domain of Discourse which is stipulated by G-subN, and is given by the non-empty set {1,...,n}? Second, what does it mean for n->inf in this context? In the context of the probability that a given structure (G-subN) with Domain {1,...,n} models S, is the 'Domain' referring to:

A) The Domain of Discourse stipulated by G-subN.
B) The entire set of natural numbers.
C) The set of prime numbers.
D) The set of real numbers.

Answer 1

The 'Domain' in the context of the probability that a given structure (G-subN) with Domain {1,...,n} models S refers to the A) Domain of Discourse stipulated by G-subN and is given by the non-empty set {1,...,n}. 'n->inf' means n tends to infinity, and it signifies the behavior of the probability as n approaches infinity. In this context, the 'Domain' refers to A) The Domain of Discourse stipulated by G-subN.

Step-by-step explanation:

The 'Domain' in the context of the probability that a given structure (Gn) with Domain {1,...,n} models S refers to the Domain of Discourse stipulated by Gn.

It is given by the non-empty set {1,...,n}. As 'n->inf' means n tends to infinity, it means that we are considering the behavior of the probability as n approaches infinity.

In this context, the 'Domain' refers to A) The Domain of Discourse stipulated by Gn.