[FOM] What is second order ZFC?

Harvey Friedman hmflogic at gmail.com
Sat Aug 31 18:27:28 EDT 2013

I have always found that a good way to deal with issues of first order
versus second order theories is to step back and ask

          what is second order ZFC?

We all agree on what ordinary ZFC is. We can call this first order ZFC. We
know its syntax, what it means to be an axiom of ZFC, what it means to be a
proof in ZFC, what it means to be a theorem of ZFC.

I invite those who do not readily agree with what I have said about second
order theories, to answer this question, and compare it with ordinary ZFC.

To make this worthwhile, do not use, without complete explanation, phrases
like "second order logic".

I will jump start the ensuing discussion by asking these questions:

1. Is CH an axiom of ZFC?
2. Is CH an "axiom" of second order ZFC"?
3. Is CH a theorem of ZFC?
4. Is CH an "axiom of second order ZFC"?

Harvey Friedman
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