# Zero-one laws Model Logic, question regarding significance of domain size

Wikipedia informs me that: Essentially (correct me if I'm wrong) the result states that as the domain of objects (domain of discourse) grows (n->inf), a static first order sentence (S) will be either almost certainly validated or almost certainly invalidated for a Model (G_n).

What is it about adding elements to the domain of 'G_n' which forces 'S' to converge to true or false?

• More the merrier! 😄 Ask yerself, like someone seems to have realized, how? Mar 9 at 1:04
• I might be off, but I'm kind of (very hand-wavily, very vaguely) intuiting this as meaning something like if you have enough predicates (an infinite number), you'll be able to deduce S if S can be derived from them or not - but the details of how S relates to G_n would matter. At least S and G_n have to be about the same domain of discourse. So it would mean that if you "cover" the domain of discourse enough, you'll be able to decide if you can deduce S (or not). Do you have a more precise statement for that result? (please no flames for how vague this is) Mar 9 at 1:12
• @Frank, I had a similar intuition. Specifically, I was thinking that: if one is given some fact 'S' about some objects 'D' [where D is a set of objects] then one's ability to prove 'S' given 'D' would depend on whether the specific objects mentioned in 'S' occur in 'D'. While I think this makes sense, I'm not sure if it is relevant in this case. Edit: My reasoning is way off. I don't think what I just wrote applies to this case at all. Mar 9 at 2:55
• There is no easily describable intuition. The 0-1 law fails if you add to FOL with relation symbols a single binary function symbol, which hardly makes a difference "intuitively", and it fails in second order logic. The reason has to do with asymptotic properties of random graphs, but the correspondence between relational FOL and random graphs is far from straightforward or "intuitive", see Abraham's notes for a relatively accessible exposition. Perhaps, you'll get more informative answers on Math SE. Mar 9 at 3:42
• I'm speculating, but one way for this to intuitively make sense is to consider two sentences expressing a property or relation, one existentially quantified and the other universally quantified. The probability that someone is fat increases as the domain of people increases; the probability that everyone is fat decreases. Similarly, the probability that someone loves someone else increases; the probability that everyone loves someone else decreases. In the limit they converge to one or zero, at least under some circumstances. I'm doubtful as to how well this would generalise. Mar 9 at 3:56