My previous post left one important issue unresolved. Second order logic needed to make use of set theory in order to work its magic, pin down a single copy of the reals and natural numbers, and so on. But set theory is a first order theory, with all the problems that this brings—multiple models, of uncontrollable sizes. How can these two facts be reconciled?
Quite simply, it turns out: for any given model of set theory, the uniqueness proofs still work. Hence the proper statement is:
For any model of set theory, there is a unique model of reals and natural numbers obeying the second order axioms.
Often, different models of set theory will have the same model of the reals inside them; but not always. Countable models of set theory, for instance, will have a countable model of the reals. So models of the reals can be divided into three categories:
The standard model of the reals, the unique field that obeys the second order axioms inside standard models of set theory (and some non-standard models of set theory as well).
Non-standard models of the reals, that obey the second order axioms inside non-standard models of set theory.
Non-standard models of the reals that obey the first order axioms, but do not obey the second order axioms in any model of set theory.
Second order logic, in first order set-theory: what gives?
With thanks to Paul Christiano
My previous post left one important issue unresolved. Second order logic needed to make use of set theory in order to work its magic, pin down a single copy of the reals and natural numbers, and so on. But set theory is a first order theory, with all the problems that this brings—multiple models, of uncontrollable sizes. How can these two facts be reconciled?
Quite simply, it turns out: for any given model of set theory, the uniqueness proofs still work. Hence the proper statement is:
For any model of set theory, there is a unique model of reals and natural numbers obeying the second order axioms.
Often, different models of set theory will have the same model of the reals inside them; but not always. Countable models of set theory, for instance, will have a countable model of the reals. So models of the reals can be divided into three categories:
The standard model of the reals, the unique field that obeys the second order axioms inside standard models of set theory (and some non-standard models of set theory as well).
Non-standard models of the reals, that obey the second order axioms inside non-standard models of set theory.
Non-standard models of the reals that obey the first order axioms, but do not obey the second order axioms in any model of set theory.
And similarly for the natural numbers.