# Principal ultrafilters

In "Compositionality in Montague Grammar" (In "The Oxford Handbook of Compositionality"), Markus Kracht writes that "In a standard model (where we allow quantifying over all subsets) there is a biunique correspondence between the individuals of the domain and the set of all subsets of the domain containing that individual (such sets are also called principal ultrafilters)" (p.58).

1. Why is the qualification "where we allow quantifying over all subsets" necessary here? Is this to imply that correspondence he alludes holds only in a full model of second order logic?

2. If the bijection only holds in a full second order model why would that be the case? Why would it fail in first order logic?

3. What is the proof of the biunique correspondence the Kracht mentions (he provides no proof himself)?

Not having the book on hand, I suspect that he means that defining the map "Send $x$ to the set of all sets containing $x$" requires us to quantify over sets of sets. It's not that this bijection would fail to 'work' in too weak a logic, but that it would fail to exist.

Meanwhile, I think "biunique" is probably just a fancy word for "bijection," in which case the proof is immediate.

• Yes, they mean the same. I was wondering whether you could define the map as a schema: "send $x$ to $S$, if $x \in S$". Does that work? What do you mean exactly by saying that it might work but fail to exist? – user65526 Oct 26 '15 at 20:03
• No - you want to send $x$ to the set of all $S$ which contain $x$. – Noah Schweber Oct 26 '15 at 20:05

The map is supposed to send each individual to the set of all subsets of the domain containing that individual.

Consider a Henkin model with individuals $a$ and $b$ and only one set $\{a,b\}$. Then both $a$ and $b$ would map to $\{\{a,b\}\}$, so the map would not give the desired correspondence in this case.

If we add a third individual, $c$, but no new sets, then $c$ would map to $\emptyset$ because no sets in the model contain $c$.

None of this is the desired behavior. In a standard (full) model, all subsets of the individuals are present, and in particular no two distinct individuals can map to the same family of sets, because for each individual $x$ the set $\{x\}$ would be in the model.