On Nov 13, 11:29 am, George Greene <gree...@email.unc.edu> wrote: > On Nov 12, 4:13 pm, Graham Cooper <grahamcoop...@gmail.com> wrote: > > > -----------------------------------http://en.wikipedia.org/wiki/Second-order_logic > > > First-order logic uses only variables that range over individuals > > (elements of the domain of discourse); second-order logic has these > > variables as well as additional variables that range over sets of > > individuals. For example, the second-order sentence > > > A(P) A(x) ( xeP v ~xeP ) > > > says that for every set P of individuals and every individual x, > > either x is in P or it is not (this is the principle of bivalence). > > Second-order logic also includes variables quantifying over functions > > --------------------------------- > > Set theory can EMULATE all the higher orders AT FIRST order because > THE INDIVIDUALS (elements of the domain of discourse) INCLUDE *ALL* > sets, > and functions can be represented AS SETS (of ordered pairs). > > This fact has called some of the best philosophers of math (famously > W.V.O. Quine) > to say that "set theory is the wolf in sheep's clothing", i.e. is > second-order logic > being "clothed" as (phrased in the language of) first-order logic. > But THIS DOES NOT CHANGE the fact that the proof of Cantor's theorem > in ZFC > is a proof IN FIRST order logic. It is a proof ABOUT the KIND of sets > that CAN be talked > about in ZFC. There are certain "large" collections ("proper > classes") that ZFC canNOT talk about, > at least not as elements of its first-order domain of discourse. But > N and R are NOT two of those > "large" collections. THOSE AND ALL RECURSIVE FUNCTIONS OVER THEM > really are small enough > that first-order ZFC really can prove things about them. > > More to the point, the fact that Af is 2nd-order is actually OUR > point. The fact that it "is" > (underlyingly) 2nd-order IS PRECISELY *WHY* IT *MUST* be BIGGER THAN > the first order N !! > So you started out by trying to say that our proof was flawed because > it is a first-order > proof, when the class of all these proposed/possible bijections is > actually *2ND*-order. > So you think THAT means that WE have the wrong "kind" of proof. BUT > THAT'S NOT WHAT THAT MEANS. > What THAT means is that OF COURSE WE ARE RIGHT. > OF COURSE the collection of functions (as well as the powerset) IS > MUCH bigger. >
The logic string ALL(f):X->Y
is about all FORMULA f ... whether you interpret it about sets of ordered pairs or not.
If you want to formulate a conjecture about naturals and reals regarding setwise properties then state that specifically and use those methods exclusively.
Why are you citing ZFC to prove Cantor's proof 1 day and FOL the next?