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Topic: mathematical infinite as a matter of method
Replies: 25   Last Post: May 4, 2013 11:24 PM

 Messages: [ Previous | Next ]
 Graham Cooper Posts: 4,495 Registered: 5/20/10
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...

>
> 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 Counter-Example

----------------

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

Date Subject Author
4/21/13 fom
4/21/13 Virgil
5/2/13 Hercules ofZeus
5/2/13 fom
5/2/13 Virgil
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 Brian Q. Hutchings
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 fom
5/4/13 Graham Cooper
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/3/13 Graham Cooper