I am interested in the following paper of M. Magidor:
"On the role of supercompact and extendible cardinals in logic", Israel Journal of Mathematics, 05/1971; 10(2): 147-157.
The abstract (which I got from www.researchgate.net) reads as follows:
"It is proved that the existence of [a] supercompact cardinal is equivalent to a certain Skolem-Lowenheim Theorem for second order logic, whereas the existence of [an] extendible cardinal is equivalent to a certain compactness theorem for that logic. It is also proved that a certain axiom scheme related to model theory implies the existence of many extendible cardinals."
I have several questions related to this paper:
i) What is the axiomatization of second-order logic he uses in this paper?
ii) Do the "Skolem-Lowenheim Theorem for second-order logic" and the "compactness theorem for that logic" hold for the 'full semantics' for second-order logic or just the Henkin models?
iii) As it has been claimed by some (Quine, for instance) that second-order logic is 'set theory in sheep's clothing', is the "axiom schema related to model theory" he refers to in that paper derivable as a theorem in the second-order logic he refers to in the paper?
Any help you could give me regarding answering these questions will be greatly appreciated. Thanks in advance.