Definition and Properties
The simplest Henkin quantifier is
- .
It (in fact every formula with a Henkin prefix, not just the simplest one) is equivalent to its second-order Skolemization, i.e.
- .
It is also powerful enough to define the quantifier (i.e. "there are infinitely many") defined as
- .
Several things follow from this, including the nonaxiomatizability of first-order logic with (first observed by Ehrenfeucht), and its equivalence to the -fragment of second-order logic (existential second-order logic)—the latter result published independently in 1970 by Herbert Enderton and W. Walkoe.
The following quantifiers are also definable by .
- Rescher: "The number of φs is less than or equal to the number of ψs"
- Härtig: "The φs are equinumerous with the ψs"
- Chang: "The number of φs is equinumerous with the domain of the model"
The Henkin quantifier can itself be expressed as a type (4) Lindström quantifier.
Read more about this topic: Branching Quantifier
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