Date: Feb 16, 2013 12:02 AM Author: fom Subject: when indecomposability is decomposable

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))

)

)

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