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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 4, 2013 9:46 AM
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On Feb 4, 12:25 am, fom <fomJ...@nyms.net> wrote:
> On 2/3/2013 10:19 PM, Charlie-Boo wrote:
> <snip>
>
>
>
> >>>> In PROLOG we use lowercase words for TERMS
> >>>> and uppercase words for VARIABLES
>
> >>>> ATOMIC PREDICATE
>
> >>> ATOMIC PREDICATE meaning relation?
>
> >>> C-B
>
> >> RELATION
> >> p(a, b, e)
>
> > If wffs are built on relations then { x | x ~e x } is not a wff
> > because ~e is not a relation.
>
> Well-formed formulas are built from the alphabet
> of a formal language. If the language contains
> a symbol of negation, then NOT(xex) will be a
> well-formed formula.
You have to define what value a symbol may have - how it is
?interpreted? ? in your definition of a wff. You need to complete B
below to see there is no paradox if you are consistent about what a
wff may contain and what values it may equal after substitution
(interpretation) if it contains variables for functions.
A. Naïve Set Theory
B. Formal definition of a wff including substitution for variables
(aka interpretation.)
C. Statement that x ~e x is not a relation (aka set or predicate.)
[The whole idea of interpretations is also not well designed. It is
an example of generalizing the wrong thing, as Productive Sets
generalize the set of true sentences - a fixed aspect of
incompleteness proofs ? instead of the premises which vary especially
those implicit in the carrying out of the proofs but never stated.
We don?t want to know all functions that satisfy Peano?s Axioms. If
it is done right there is only one set of functions that + * ** can
be. What we have lots of variations of is the properties of N ? that
is what is to be generalized. Saying + is not addition is like
Fortran allowing you to redefine what 1 means. There?s no need for
that either. It only muddies the water.
Godel/Rosser/Smullyan incompleteness theorems include reference to a
wff being true but not provable. But then it must be ?true for which
interpretations??. That is left out and opens up the question, why
the standard interpretation works and which ones work? But we really
don?t care about using other functions for + * ** in the first place!
Certainly not in the middle of an incompleteness theorem.]
> > We don t need ZF - at all. All we need is Na ve Set Theory, a
> > complete formal definition of wff and recognition that x ~e x is not a
> > relation due to diagonalization on sets.
>
> The reason for ZF and other inquiries into the
> foundations of mathematics has to do with a coherent
> explanation for the utility of an otherwise incoherent
> collection of mathematical techniques. If such an
?ZFC is one of several axiomatic systems proposed to formulate a
theory of sets without the paradoxes of naive set theory such as
Russell's paradox.? - Wikipedia
> explanatory role is not forthcoming, such theories
> at least organize mathematical techniques into a
> science (in the sense of Aristotle) connected by
> the argumentation of proofs.
I am not saying to not formalize. (I have personally axiomatized at
least 5 branches of Computer Science/Logic. Every case of
incompleteness is handled by a single axiom to distinguish the sets or
relationships that cannot be characterized in the system. In fact,
that additional axiom is the only difference between the positive and
negative sides of a theory e.g. Universal Turing Machine vs. Halting
Problem in the Theory of Computation.)
I am saying that ZF is a lousy attempt at formalizing and I propose an
alternate formalization ? a simple addition to Frege?s Naïve Set
Theory. Just using predicate calculus instead of a specialized
language to state the axioms makes ZF hard to communicate:
?The precise meanings of the terms associated with the separation
axioms has varied over time. The separation axioms have had a
convoluted history, with many competing meanings for the same term,
and many competing terms for the same concept.? - Wikipedia
> > Logic = Set Theory
>
> If this is true, it is not the logic of which
> you are thinking.
>
> What you are taking for granted is the structure
> of logic without a negation symbol. The negation
> you use in your programming has no reality in the
> underlying computer architectures.
With negation you have all levels of the Kleene Arithmetic Hierarchy,
which means any wff that can be expressed. (Each added ~exists adds a
level.) Without negation you have only Sigma-1 the recursively
enumerable sets, and the negation (complement) of some included sets
of natural numbers are not included.
Set Theory, axiomatic Logic used to express sets with wffs that are
true of its elements, and English all have negation and are
equivalent. Computer programs, proof in axiomatic Logic and the
various bases of computing developed during the 1930s (excluding a
couple of misfires) are Sigma-1, do not allow the complement of every
set allowed and are equivalent.
Is this what you?re referring to?
> More formally, what you are taking for granted
> is that only 14 of the 16 basic Boolean functions
> are linearly separable switching functions. The
> two that are not are logical equivalence (LEQ) and
> exclusive disjunction (XOR).
>
> These particular connectives become problematic
> when considered in the context of classical quantificational
> logic because of the relation of identity, definition,
> and description.
>
> The standard account of identity (for example as
> discussed under "relative identity" at
> plato.stanford.edu) addresses trivial identity,
> that is, x=x, and substitutivity.
>
> What is not addressed is informative identity,
> that is, x=y.
>
> In classical model theory, however one has
> determined an object in a model and a name
> for that object has consequences for the
> satisfaction map. That is how the classical
> model theory interprets x=y.
>
> In 1971 Tarski directed his attention to the
> representation of first-order logic in the context
> of algebraic logic. In those deliberations, he
> introduced the axiom
>
> AxAy(x=y <-> Ez(x=z /\ z=y))
>
> In the formulation of these "cylindrical algebras"
> the formula
>
> x=y
>
> corresponds geometrically to a 2-dimensional subspace
> of the domain. In a 3-dimensional domain, this is
> a hyperspace separating the domain into two regions.
>
> This suggests that there is a fundamental geometric
> reason for LEQ and XOR to not be represented in the
> underlying propositional logic by linearly separable
> switching functions.
>
>
>
> > Logic = NOT AND OR EXISTS simple, easy
>
> What happens if I take NOT away?
Interesting question. Assuming you can express without quantifiers
all recursive sets, since all wffs can be put into prenex normal form,
you can express the same sets. Neither the quantifiers nor the
relations need the negation symbol.
(I did go through a period of writing software in search of minimal
bases (subsets of the 16 binary Boolean functions) for propositional
calculus.)
> There is no real way to post this picture to a newsgroup.
> It is the ortholattice which is an atomic amalgam of a
> Boolean lattice with 4 atoms (the usual 16 element lattice
> associated with basic Boolean functionality) and a
> Boolean lattice with 3 atoms.
>
> ....................................TRU....................................
> ............................./.../..//\...\.................................
> ......................../..../.../../....\...\.............................
> .................../...../..../..../.........\.....\.......................
> ............../....../...../....../...............\......\.................
> ........./......./....../......../.....................\.......\...........
> ....../......./......./........./...........................\........\.....
> .....IF......NAND.......IMP.....OR.........................ALL........NO...
> ..../.\.\..../.\.\..../..|.\..././\.\\..................../...\.....././...
> .../...\./\......\./\....|./..\./..\...\...\...................../.........
> ../../..\...\.../...\./.\|...../.\..\....\............/....../...\../......
> .//......\.../\.../....\.|.\../....\.\......\...\......../.................
> LET.......XOR..FLIP....FIX..LEQ.....DENY........./.../............/\.......
> .\\....../...\/...\..../.|./..\...././......./...\.......\.................
> ..\..\../.../...\.../.\./|.....\./../...../...../.............../....\.....
> ...\.../.\/....../.\/....|.\../.\../.../.../...........\.........\.........
> ....\././....\././....\..|./...\.\/.//.....................\./.......\.\...
> .....NIF......AND......NIMP.....NOR........................OTHER......SOME.
> ......\.......\.......\.........\.........................../......../.....
> .........\.......\......\........\...................../......./...........
> ..............\......\.....\......\.............../....../.................
> ...................\.....\....\....\........./...../.......................
> ........................\....\...\..\..../.../.............................
> .............................\...\..\\/../.................................
> ...................................NTRU....................................
>
> The two lattices share TRU, NTRU, OR, and NOR. The structure of
> the unfamiliar lattice has
>
> SOME=EXISTSWITH=Ex
> OTHER=EXISTSWITHOUT=Ex-
> ALL=Ax
> NO=Ax-
>
> with
>
> ALL=NOT(OTHER)
> NO=NOT(SOME)
>
> on the basis of the order relation alone.
>
> This construction, while described specifically
> for quantificational logic here, actually characterizes
> the geometric (in the sense of an atomic lattice) structure
> of any extension to propositional logic with negation.
> Modal extensions are four-fold interrelating POSSIBLE and
> NECESSARY. Deontic extensions are four-fold interrelating
> OBLIGATORY and PERMITTED.
>
> For quantificational logic, each variable has this
> four-fold structure. This corresponds with the indexing
> of quantifiers found in the cylindrical algebras of
> Tarski's later work.
>
> The negation symbol masks this four-fold structure in
> the formation rules for formulas.
>
> Yes. Logic in the absence of NOT--just like in the
> hardware of your computer systems--is not easy.
>
> > ZF Set Theory = a dozen messy axioms for which people can t even agree
> > on the specifics ??
>
> There are actually an infinity of axioms (damn those
> first-order logicists) because well-formed formulas are
> separately generated and present in the axioms of separation.
Yes, but this schema ranges over wffs (rather than sets) so the set
defined is aleph-0 and in fact recursively enumerable. Better to
range over sets and pull in that needed aleph-1.
> > There are a dozen set theories and a dozen interpretations of the most
> > popular set theory, and 2 or 3 versions of it (with or without Choice,
> > etc.) none of which decide any of the important questions of set
> > theory due to exhaustive work (a waste!) by Godel and Cohen.
>
> That is an odd thing to say. While I find forcing to be nonsense
> in set theory (but I reject the axiom of extension as foundational)
> it is extremely important to recursion theory, is it not? And that
> has consequences for the practical application in computational
> contexts, does it not?
You are referring to the advancements in prosthetics made during war.
C-B
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