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kocykl comments on Second-Order Logic: The Controversy - Less Wrong
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"If there's a collection of third-order axioms that characterizes a model, there's a collection of second-order axioms that characterizes the same model. Once you make the jump to second-order logic, you're done - so far as anyone knows (so far as I know) there's nothing more powerful than second-order logic in terms of which models it can characterize."
You clearly can state Continuum Hypothesis in the higher order logic, while a 2nd order formulation seems elusive. Are you sure about it?
Eliezer is correct. See SEP on HOL.
Link is good, but I guess direct explanation of this simple thing could be useful.
It is not hard to build explicit map between R and R² (more or less interleaving the binary notations for numbers).
So the claim of Continuum Hypothesis is:
For every property of real numbers P there exists such a property of pairs of real numbers, Q such that:
1) ∀x (P(x) -> ∃! y Q(x,y))
2) ∀x (¬P(x) -> ∀y¬Q(x,y))
(i.e. Q describes mapping from support of P to R)
3) ∀x1,x2,y: ((Q(x1,y)^Q(x2,y)) -> x1=x2)
(i.e. the map is an injection)
4) (∀y ∃x Q(x,y)) ∨ (∀x∀y (Q(x,y)-> y∈N))
(i.e. map is either surjection to R or injection to a subset of N)
These conditions say that every subset of R is either the size of R or no bigger than N.