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Re: ALL(F):N->R is 2OL! NOT 1OL!!!!!!
Posted:
Nov 14, 2012 3:28 AM
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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?
Herc
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