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Topic: ALL(F):N->R is 2OL! NOT 1OL!!!!!!
Replies: 6   Last Post: Nov 15, 2012 2:39 AM

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Graham Cooper

Posts: 4,495
Registered: 5/20/10
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 <> wrote:
> On Nov 12, 4:13 pm, Graham Cooper <> wrote:

> > -----------------------------------
> > 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
> 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
> 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?


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