
Re: mathematical infinite as a matter of method
Posted:
May 4, 2013 2:21 AM


On May 4, 2:17 pm, fom <fomJ...@nyms.net> wrote: > On 5/3/2013 8:20 PM, Graham Cooper wrote: > > > > > You don't just 'choose' a foundational theory, none of ZFC holds up > > to Induction, when it's derived proofs are more closely examined > > there is a lot of convoluted circular reasoning... > > At which point I return to the remark made earlier. The logicians > of the late nineteenth and twentieth century went to great pains > to reject epistemology. They are simply rediscovering it within > their formalisms while continuing along that path of rejection.
Nonsense, Plato wouldn't be so unkind...
the incompleteness theorem is just a note from mum that they couldn't do their homework...
...the homework told them so...
Q1 You can't solve this!
> > There are not many choices... > > http://en.wikipedia.org/wiki/M%C3%BCnchhausen_Trilemma > > At least I know why I make the choices I make. > > AxAy(xcy <> (Az(ycz > xcz) /\ Ez(xcz /\ ycz))) > > AxAy(xey <> (Az(ycz > xez) /\ Ez(xez /\ ycz))) > > Unfortunately, they would do you no good either. >
Well with ordinary subset
xcy <> Aa aex>aey
is sufficient to eliminate quantifiers.
Aa ... and { a  ... }
both mean " ALL A such that ... "
When you see a 'C' in your equations there is a hidden ALL() ranging over all the elements
Quantified Logic and Set Theory are equivalent theories, you don't need the syntax of both.

AxAy(xcy <> (Az(ycz > xcz) /\ Ez(xcz /\ ycz)))
RE: ~ycz
which just means x is PROPER subset of y
I use not(ss(..,..))
in a different way, to remove an ALL() quantifier.
~An p(n) <> En ~p(n)
Proof By CounterExample

e.g p(n) <> x MOD 2 = 0
~An p(n) == not all numbers are even
.........
PROOFBYCOUNTEREXAMPLE.PRO
nat(0). nat(s(X)) : nat(X). even(0). even(s(s(X))) : even(X). odd(s(0)). odd(s(s(X))) : odd(X).
e(A,nats) : nat(A). e(A,evens) : even(A). e(A,odds) : odd(A).
e(A, not(evens)) : e(A, odds). e(A, not(odds)) : e(A, evens).
intersects(S1,S2) : e(A,S1),e(A,S2). not(ss( S1 , S2 )) : intersects( S1, not(S2) ). ********************
This tiny program will work out that
? not(ss(nats,evens)) . >YES
i.e ~nats c evens
by actually testing 0, finding it's both nat & even, testing s(0), and finding not s(0) e evens
Are all natural numbers even > NO!
(this is early work since I didn't have to program ALL() yet!)

This kind of CATEGORICAL NEGATION
if it's ODD its not EVEN if it's BLUE it's not RED
is used in human logic every day!
e(A, not(evens)) : e(A, odds). e(A, not(odds)) : e(A, evens).

PROLOG SET THEORY is the paradigm that early logicians could only glimpse a small part of the program.
Herc  www.BLoCKPROLOG.com

