Math Forum Discussions

fom

Posts: 1,030
Registered: 12/4/12

when indecomposability is decomposable
Posted: Feb 16, 2013 12:02 AM

Plain Text

I own only a handful of books on constructive
mathematics, and, those are of recent acquisition.
They are from the Russian school -- Markov and Sanin.

In the introduction of Sanin's "Constructive Real
Numbers and Function Spaces" there is a footnote
with the following explanation of constructive
objects:

"By constructive objects are meant objects which
are the results of processes of construction
realizable on the following basis: one assumes that
the objects which figure in the given study as
indecomposable initial objects are clearly
described; one assumes, given a list of rules
of formation of new objects from previously
constructed ones, which in the given study
plays the role of a description of the admissible
steps of constructive processes; one assumes that
the processes of construction are carried out
in discrete steps, where the choice of each s
succeeding step is arbitrary, within the limits
determined by the list of already constructed
objects and the set of those rules of formation
of new objects which can actually be applied to
already constructed objects."

When one invokes the axiom,

Ax(x=x)

by

a=a

there is an ontological interpretation of the
sign of equality corresponding with the sense
of indecomposability. Perhaps one could quibble
over the difference between "ontological invariance"
and "indecomposability." I would probably lose
that argument since I am not particularly adept
in debates.

Now, in the "list of rules of formation of new
objects from prevously constructed ones" there is
the notion of "definite description". In "Word
and Object" Quine goes to great pains just to formulate
an argument that eliminates names because whenever
a name might be needed, a description may be used
to introduce it.

Of course, mathematicians generally do not know
of description theory. But, it is certainly discussed
in metamathematics, and, I believe, it is discussed
in the manner in which I tend to view it. But,
I may be wrong.

Tarski really dodges the issue in "The Concept of
Truth in Formalized Languages." However, in "Some
Methodological Investigations on the Definability of
Concepts" his statements and usage seem to correlate
with my understanding, Section 1 of the paper begins
with:

"The problems to be discussed in this article concern
the specific terms of any deductive theory.

"Let 'a' be some extra-logical constant and B any
set of such constants. Every sentence of the form:

(1) Ax(x=a <-> W(x;b',b'',...))

where 'W(x;b',b'',...)' stands for any sentential
function which contains 'x' as the only real variable,
and in which no extra-logical constants other than
'b',b'',...' of the set B occur, will be called a
*possible definition* or simply a *definition of the
term 'a' by means of the terms of the set B*. We
shall say that the term 'a' *is definable by means
of the terms of the set B on the basis of the set
X of sentences*, if 'a' and all terms of B occur
in the sentences of the set X and if at the same
time at least one possible definition of the term
'a' by means of the terms of B is derivable from
the sentences of x."

The topic of my post is concerned with the syntax
of the sentence

(1) Ax(x=a <-> W(x;b',b'',...))

in which the sign of equality and the sign of
logical equivalence act in coordination with
one another.

One can entangle their respective meanings
even more profoundly with the often-quoted
position expressed by Quine that "identity
is eliminable." That is, when one presupposes
the ontological interpretation that gives
rise to the necessity of

|- (x=y -> Az(zex <-> zey))

and takes as a contextual axiom,

(Az(zex <-> zey) -> x=y) |-

one has properly distinguished contexts.
But, to use those two pieces of syntax
to treat set theory with a signature <e>
rather than <=,e> yields an axiom such
as

AxAy(x=y <-> Az(zex <-> zey)) |-

Observe that in this form, the properties
of an equivalence relation derive from the
fact that logical equivalence is reflexive
and symmetric by truth table semantics, and
transitive relative to the tautologous formula
from propositional logic,

|- ((p<->q) <-> ((p->q) /\ (p<-q))

where, by tautologous I mean its truth table
evaluation. Quine is clear concerning this
meaning:

"The term 'tautology' is taken from
Wittgenstein. The present notion of
tautologous statements, as those true
by virtue solely of truth-functional
composition, seems to agree with his
usage;[...]"

So, while foundational investigations
before Wittgenstein were necessarily
axiomatic, the introduction of truth
tables changed the situation.

As for the sign of equality, Tarski managed
to incorporate the conjunctive propositional
syntax into a quantificational context
with the axiom:

AxAy(x=y <-> Ez(x=z /\ z=y)) |-

Here, the existential operator brings the
investigation back to model theory, names
(or constants (extra-logical constants if
one takes parameters as being the same as
constants)) and definite descriptions.

In this last expression, the sign of
identity is expressing its own transitivity
through the symmetry of the truth functional
semantics of conjunction.

===================

To return to the stated topic of the post,
the problem with all of the above is that
logical equivalence (hereafter, LEQ) is
decomposable where the ontological notion
of identity is not.

Even when authors like Quine and Carnap
may have noticed that truth table representations
were subject to permutable representations,
they ignored it.

There are six column vectors which may, in
various combinations, represent LEQ.

TTTFFF
TFFTTF
FFTFTT
FTFTFT

I am grateful to everyone on sci.logic and
sci.math who may have helped me to sort some
of this out.

===================

To find a ground, both Kant and Frege
looked to "contradiction." In Kant's
case, it took the form of interpreting
the modal notion "impossibility." In
Frege's case, it took the form of individuating
a class having no members. One would think
such a class is indecomposable.

In the theory of orthocomplemented lattices,
there is a notion of orthogonality defined
by:

Two elements a, b of an ortholattice are
said to be *orthogonal* if and only if
a<=b'. For such elements we write a_|_b.

One consequence of this definition is that

Ax(x_|_x -> x=0)

so, the situation above involving six column
vectors may be treated as a single system
satisfying that theorem relative to representation
in terms of the unique self-orthogonal Latin
square on 6 symbols:

134625
625134
463512
512463
246351
351246

To do this, however, one needs a category of
named objects that can be thought of in terms
of that ontological invariance associated with
indecomposability.

That means seeking ways, other than truth tables,
to distinguish the 6 elements.

I will not try to explain my method since it is
a matter of senselessly comparing syntax.

The results, if they may be called that, are
simple.

The "indecomposable ground" is the fact that
any 2-edge coloring of the complete graph on
6 symbols must contain at least one monochromatic
triangle.

That graph has 15 edges. So, by focusing on
constructions that isolated LEQ from the other
15 basic Boolean switching functions, my method
grounded partitions whose cardinal arithmetic
is given by

16=15+1

6=5+1

Another issue -- being that this concerned
itself with logic -- is bivalence and the
law of excluded middle with respect to unary
negation. But, unary negation is eliminable.
What seemed required was to find an assymmetry
by which one complete connective was a syntactic
substitute for unary negation and the other
complete connective was the semantic ground.

This search landed me in the algebra of
miniquaternions, although I could not have
recognized it until I had built the construction
up to where it involved 91-point projective
planes.

One standard set of complete connectives
would be {->,-} which I call IMP and NOT.
If my investigations can be seen to have
established the following relations,

{FIX,LET,LEQ,XOR,DENY,FLIP}{IMP,NAND,NOR}

then those two sets may be given the
algebra of a near-field of order 9.
The assignments

IMP=0
NOR=1
NAND=-1

describe the real elements of the
near field.

If one is willing to entertain the
ability to designate near-field algebras
to this set, then choosing any element
of the non-real elements, say LEQ

{FIX,LET,XOR,DENY,FLIP,{LEQ}}{IMP,NAND,NOR}

permits all nine elements to be representable
in forms such as

a+(b*LEQ)

where a and b are from the set of real
elements.

There is another odd algebra that arises
based on the attempt to isolate XOR from
the 5-set given above.

The objects FIX,LET,DENY,FLIP are distinguished
in that they are invariant with respect to
DeMorgan conjugation. It should be obvious
how they will be assigned in the ortholattice L12
near expression (29) in the link,

http://www.clas.ufl.edu/users/jzeman/quantumlogic/generalized_normal_logic.htm

All that is important is that

c=XOR

Although one might consider interpreting the bottom of
the lattice with a 6th symbol, its position in the
larger framework means that it should be left null.
Consider this an application of Leibniz' law.

But, the algebraic system that goes with
these five objects is obtained using the
alternating group on 5 symbols. Once again,
the assignments are made to convey a sense
of relation through purely syntactic
construction -- therefore, senselessly.

G=LET
H=XOR
I=FLIP
J=FIX
K=DENY

The algebra is the icosian group. Provided this
link works, you should be able to see the relationship
from the assignments above:

http://books.google.com/books?id=upYwZ6cQumoC&pg=PA207&lpg=PA207&dq=icosians+leech++lattice&source=bl&ots=_L1P4YjbCT&sig=xMmrxwlFUT09KI266toGazHZROs&hl=en&sa=X&ei=kwkfUdq9C-SQ2AXTlYGIDg&ved=0CEUQ6AEwAg#v=onepage&q=icosians%20leech%20%20lattice&f=false

When looking at the icosians, observe that they
are defined with respect to what would be a single
field extension to the rationals if the coefficients
were the rational numbers.

I point this out because statements such as

"infinity is a necessary assumption
for identity"

reflect the fact that some representation of a
process of quantization is needed to

======================================

I made a claim suggesting that I was a set theorist
(although, I neglected to say non-professional) who
thought about his work in terms of quantum mechanics.
Here is a paper whose introduction mentions these geometries
in terms of entangled qubits

http://arxiv.org/abs/1002.4287

As for the impression that I could not possibly "know"
all of this math. I cannot and never will know it the
way you know your subjects. I am simply a guy who was
interested in a hard problem in set theory and followed
his nose.

My mathematics, being impredicative is not a "set theory".
Impredicativity is fractal.

My mathematics, admitting a null class is not a "mereology".
Leibniz law needs a halt.

If anyone is interested, here are the sentences again...

...and, thanks.

===========================================

We take the consequences of the
following as the basic theory.

It's signature is given by

<<M, |M|>, <c, 2>, <e, 2>>

with models interpreted coherently
according to

M=V()

in the extended signature

<<M, |M|>, <c, 2>, <e, 2>, <EQ, 2>, <=, 2>, <V, 0>, <null, 0>, <set, 1>,
<S, 1>, <P, 1>>

Definition of proper part:
AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))

Lesniewski's First Axiom is provable:
AxAy(xcy -> -ycx)

Lesniewski's Second Axiom is provable:
AxAyAz((xcy /\ ycz) -> xcz)

Lesniewski's First Theorem is provable:
Ax(-xcx)

Definition of membership:
AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))

Definition of grammatical equivalence:
AxAy(xEQy <-> (Az(xcz <-> ycz) /\ Az(zcx <-> zcy) /\ Az(xez <-> yez) /\
Az(zex <-> zey)))

Definition of object identity:
AxAy(x=y <-> Az(xez <-> yez))

Assumption that equivalence of mereological filters imply equivalence of
neighborhood filters:
AxAy(Az(xcz <-> ycz) -> Az(xez <-> yez))

Assumption that objectual inclusion implies mereological covering:
AxAy(Az(zex -> zey) -> Az(ycz -> xcz))

Assumption that proper parts are collectible:
AxEyAz(zey <-> zcx)

That proper parts imply objectual inclusion is provable:
AxAy(xcy -> Az(zex -> zey))

That mereological covering implies objectual inclusion is provable:
AxAy(Az(ycz -> xcz) -> Az(zex -> zey))

That proper parts are expressible in terms of object extension is provable:
AxAy(xcy <-> (Az(zex -> zey) /\ Ez(zey /\ -zex)))

That object inclusion implies mereological inclusion is provable:
AxAy(Az(zex -> zey) -> Az(zcx -> zcy))

That equivalent neighborhood filters imply equivalent mereological
filters is provable:
AxAy(Az(xez -> yez) -> Az(xcz -> ycz))

That grammatical equivalence is expressible in terms of mereological
filters is provable:
AxAy(xEQy <-> Az(xcz <-> ycz)

That grammatical equivalence is expressible in terms of object extension
is provable:
AxAy(xEQy <-> Az(zex <-> zey))

That grammatical equivalence is expressible in terms of neighborhood
filters is provable:
AxAy(xEQy <-> Az(xez <-> yez))

That grammatical equivalence is equivalent to object identity is provable:
AxAy(xEQy <-> x=y)

Assumption of Aquinian individuation:
AxAy((Az(ycz -> xez) /\ Ez(xez /\ -ycz)) -> Az((xez /\ -ycz) -> (Ew(xew
/\ wcy) \/ Aw(zcw -> ycw))))

Assumption of singletons via pairing:
AxAy((Ez(xcz) /\ Ez(ycz)) -> EwAz(zew -> (z=x \/ z=y)))

Definition of top:
Ax(x=V() <-> Ay(-(ycx <-> y=x)))

Assumption of top:
ExAy(-(ycx <-> y=x))

Assumption of almost universality:
Ax(Ey(xcy) -> Ey(xey))

Definition of set:
Ax(set(x) <-> Ey(xcy))

Definition of bottom:
Ax(x=null() <-> Ay(-(xcy <-> x=y)))

Assumption of bottom:
ExAy(-(xcy <-> x=y))

Assumption of context separation (regularity/foundation):
Ax(Ey(ycx) -> Ey(yex /\ -Ez(zex /\ zey)))

Assumption of arbitrary unions:
AxEy(Az(zey <-> Ew(wex /\ zew)) /\ (Ez(xcz) -> Ez(ycz)))

Assumption of arbitrary intersection:
AxEy(Az(zey <-> Aw(wex -> zew)) /\ (Ez(zcx) -> Ez(ycz)))

Definition of power function:
AxAy(x=P(y) <-> (Ez(ycz) /\ Az(zex <-> (zcy \/ z=y))))

Assumption of power set:
Ax(Ey(xcy) -> Ey(Az(zey <-> (zcx \/ z=x)) /\ Ez(ycz)))

Definition of successor function:
AxAy(x=S(y) <-> (Ez(ycz) /\ Az(zex <-> (zey \/ z=y))))

Assumption of successor set:
Ax(Ey(xcy) -> Ey(Az(zey <-> (zex \/ z=x)) /\ Ez(ycz)))

Assumption of transfinite cardinalities:
Ex(Ey(xcy) /\ null()cx /\ Ay(ycx -> Ez(zcx /\ ycz)))

Let the restricted quantifier

Ap[pEQp]

be interpreted as

Ap[pEQp](phi(p)) <-> Ap(pEQp /\ phi(p))

Then for each n and each well-formed formula phi(y, p_0, ..., p_n),

assume

=================
Ap_n[p_nEQp_n]...Ap_0[p_0EQp_0]
AxAy(
Ew(ycw) ->
(Ez((Ew(zcw) /\ (yez <-> (yex /\ phi(y, p_0, ..., p_n))))) <-> Ew(xcw))
)
=================

and assume

=================
Ap_n[p_nEQp_n]...Ap_0[p_0EQp_0]
(
AxAyAz(
(
((Ew(xcw) /\ Ew(ycw)) /\ (phi(x,y, p_0, ..., p_n)) /\
((Ew(xcw) /\ Ew(zcw)) /\ (phi(x,z, p_0, ..., p_n))
) -> (y=z)
)
->
AxAy(
Ew(ycw) ->
(Ez((Ew(zcw) /\ (yez <-> Ew(wex /\ phi(z,w, p_0, ..., p_n))))) <-> Ew(xcw))
)
)
=================