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Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Posted:
Feb 11, 2013 1:38 PM
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On Feb 10, 6:39 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On Feb 11, 7:24 am, fom <fomJ...@nyms.net> wrote:
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> > On 2/10/2013 2:38 AM, Graham Cooper wrote:
>
> > > On Feb 10, 5:47 pm, fom <fomJ...@nyms.net> wrote:
> > >> On 2/9/2013 6:19 PM, Charlie-Boo wrote:
>
> > >>> On Feb 7, 1:51 am, fom <fomJ...@nyms.net> wrote:
>
> > >>>> How do you see Logic and Set Theory as being the same?
>
> > >>> Both are concerned with mappings to {true,false}. A propositional
> > >>> calculus proposition is 0-place. A set is 1-place. A relation is any
> > >>> number of places. (A relation is a set - of tuples.)
>
> > >>> So you have the same rules of inference: Double Negative, DeMorgan
> > >>> etc. apply to propositions and sets.
>
> > >>> To prove incompleteness, Godel had to generalize wffs as expressing
> > >>> propositions to expressing sets when the wff has a free variable.
>
> > >> Hmm...
>
> > >> This is naive set theory (which you have stated
> > >> as being fine with your views).
>
> > >> I view set theory as being about the existence
> > >> of mathematical objects. Naive set theory failed,
> > >> in part, because of something in Aristotle--do not
> > >> negate "substance". Do not get me wrong. I am
> > >> not planning to run out and buy a number 2 while
> > >> I pick up my next Turing machine....
>
> > >> The problem, however, is that the connection of
> > >> mathematics to any metaphysical truth (if such
> > >> a statement can be sensible) requires that the
> > >> objects represented in physics books (material
> > >> objects) correspond with some sort of mathematical
> > >> notion. So, while mathematics is abstract,
> > >> there must be some sort of interpretation that
> > >> accounts for its apparent ability to model
> > >> real-world situations.
>
> > >> Either physics is a collection of mathematical
> > >> hallucinations or there is a better explanation
> > >> of set theory.
>
> > > Right! the physical world cannot contravene the platonic, so a set of
> > > truths may exist and a set of lies not...
>
> > > ** in Plato land where (angle1+angle2+angle3=pi) **
>
> > > it's the 1 metaphysics principle I subscribe to!
>
> > > I think LOGIC is just applying MODUS PONENS.
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> > > backwards to axioms
>
> > > a1
> > > \
> > > theorem ?
> > > /
> > > a2
>
> > > forwards to contradictions
>
> > > x
> > > /
> > > ~theorem
> > > \
> > > ~x
>
> > > Naive set theory should be able to cope with a SUBSET of WFF that have
> > > been sieved through various checks. if you can formulate what the
> > > real world contradiction is, it can be unstratified.
>
> > Herc,
>
> > What you say here is Kantian.
>
> > Kant called logic the negative criterion
> > of truth (forward to contradiction).
>
> > And he ascribed the discernment of natural
> > laws to presupposition analysis under the
> > presumption of causes (backwards to axioms).
>
> > For what this is worth, your arguments
> > against Cantor's diagonal have been based
> > on transversal designs.
>
> > The march to infinity is most likely taken
> > using finite projective planes described
> > by difference sets.
>
> > If you want to see why, do an internet
> > search on "perfect difference sets" and
> > "neighbor detection"
>
> > Identity requires infinity. Distinguishability
> > in the finitary context of automata is finite.
> > With respect to this, equivalence is defined
> > negatively. Hence, it presupposes infinity.
>
> > Now, identity and diversity are intertwined
> > by negation. Leibniz' principle of identity
> > of indiscernibles relates an object to all
> > of the objects of the system which are not
> > the given object. This is like a geometry
> > where every pair of points define a line.
>
> > Naming is quantization process that requires
> > fewer resources. It is Leibniz' principle
> > of indiscernibles restricted to "landmarks".
>
> > In network analysis they are using perfect
> > difference sets for this purpose.
>
> > Anyway, it may give you a different
> > perspective on some of your thoughts.
>
> > Glad you liked the remark.
>
> I think NAMING-object and EXISTS(object) and SKOLEM FUNCTIONS are all
> part of the same method.
>
> writing a function definition in programming language you just
>
> LET x = 7
>
> all that's required! PUT IT IN WRITING = THEN IT EXISTS(..)
>
> biggerthan(X , Y) <- X = s(Y)
> biggerthan(X , 0 ).
>
> Certain Proofs are viable since an Algorithm Exists to find the
> required solution.
Well, again, there are a few possibilities:
1. Prove (all x)(exists y)P(x,y)
2. Prove (all x) f(x) is defined and P(x,f(x)) for some function f.
3. (2) but also f is recursive.
Axiom of Choice:
x e y => f(y) e y
Is there such an f?
A much better question: Is there an f such that f({x,y}) = x v
f({x,y}) = y for all x,y ? You can't even do that! HA!!
CBL answer: Yes, there is such an f but we can't express it. :(
So all you PHP programmers must know that there is no way to find a
single value in $p[$x] without foreach - but Logic has no foreach. It
deals with non-r.e. sets! (Generalize non-r.e. to any non listable
sets.)
C-B
> Extrapoliate that down to all EXIST(X) commands even for a
> straightforward object X.
>
> A(Y) E(X) X>Y
> /\
> | |
> \/
> A(Y) bigger(Y)>Y
> TRUE
>
> X EXISTS because an ALGORITHM is written.
>
> bigger() is the Skolem Function for EXISTS(X)
>
> you just DEFINE in writing what exists!
>
> Then
>
> ----------------------------------
>
> AXIOM OF SPECIFICATION gives a NAME to the set IFF it can exist.
>
> EXIST(Y) xeY <-> p(x)
>
> X is just a re-usable programming term like i,k,...
> Y is a unique name of the new set, again just by writing it!
>
> ----------
>
> I'm currently pondering using NOT(...) my PROLOG theorem type
>
> not( gt (0 , X ))
>
> as I can write that now in PROLOG
>
> <=> ALL(X) ~0>X
>
> and it gives me an ALL(X) Quantifier without breadth first
> examination of all records although I think it has to examine inside
> the predicate gt(..) coding to return true.
>
> i.e. NOT( NEGATION-AS-FAILURE( gt(0,X) )
>
> would never terminate considering 0, s(0), s(s(0)), ... would all
> have to fail a search branch.
>
> However if it was stated not( gt(0,X) )
> which atleast MEANS ALL(X) not( gt(0,X) )
>
> then the quantifier syntax could be atomic without any quantifer
> predicates added, and humans seem to reason about ALL quantifiers
> without working at the ALL RECORDS level... perhaps most PROLOG
> recursive programs, usually 2 lines can have a proof by induction
> method applied automatically.
>
> Herc
> --www.BLoCKPROLOG.com
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