# What is the theorem that shows that second-order logic is the ceiling of model characterization?

I was reading this blog posting and the following claim was made:

...[T]here's tricks for making second-order logic encode any proposition in third-order logic and so on. 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.

I think I've heard similar claims before -- that second-order logic provides us all the resources we could have for characterizing models.

Is this claim correct? What is the theorem associated with this claim?

• I have been taught that second order logic is nothing more hat first-order (multi-sorted) logic. Apr 29, 2015 at 6:09
• @PrimoPetri That is true for second-order logic restricted to Henkin models (as opposed to full models). With Henkin semantics, SOL is no more expressive than FOL. Once you move to full models (which differ from the Henkin models in allowing the second-order quantifers to range over the full powerset of the domain) you get a logic which is more expressive. It's also incomplete. In the SEP article linked to in Jonny's answer, this is discussed in sections 2 and 3. Your claim is true of what they call the "General Semantics" (section 3). Apr 29, 2015 at 12:24