Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Topic: An attempt at a finite consistency thesis
Replies: 2   Last Post: Jan 24, 2014 9:14 PM

 Messages: [ Previous | Next ]
 fom Posts: 1,968 Registered: 12/4/12
An attempt at a finite consistency thesis
Posted: Jan 24, 2014 5:51 AM

In December 2012 I began this
series of analyses with the link,

http://mathforum.org/kb/plaintext.jspa?messageID=7935660

which arranges the names for 16
logical connectives as the points
for an affine plane within a
projective plane. The geometric
relations account for the basic
relations of negation, De Morgan
conjugation, and a form of contraposition.
The three operations can be expressed
as a commutative diagram.

A second link established the functional
behavior by combining ideas from Church
and Birkhoff:

http://mathforum.org/kb/plaintext.jspa?messageID=7933608

The semantic structure is provided by
two constructions. First, the incidence
matrix for a trivial affine geometry
corresponds with the incidence matrix
for a tetrhedron -- the simplex for
3-dimensional space. Since affine
geometries are associated with projective
geometries, the intended use of the
affine geometry to establish a semantic
correspondence with the biconditional
connective (LEQ) also permits relation
to a complete connective (NOR).

The orginal first step is in the link:

http://mathforum.org/kb/plaintext.jspa?messageID=7933617

Upon further consideration, I realized
that both of the possible planes in that
construction should be used. The LEQ
selection is bound with the NOR extension.
The second possible plane is semantically
extraneous. It carries, however, the
XOR deselection and the OR extension for
the deselected geometry.

Upon full interpretation, the 8 connectives
of this construction will be configured as
labels for the Vamos matroid,

http://en.wikipedia.org/wiki/Vamos_matroid

The particulars of the matroid specification

{ LEQ, XOR, NOR, OR }

in the sense that this is the only quadruple
containing two pairs of negations admitted
into the matroid basis.

Conceptually, this obfuscates any implicit
fixed truth table representations, making
the system amenable to the notion that
connective meaning is discerned from language
structure through logical analysis.

The second construction uses a De Bruijn
graph to serialize the combinatorial
relations of the 16 points from the
geometry.

The specific graph used can be seen in

http://en.wikipedia.org/wiki/File:De_bruijn_graph-for_binary_sequence_of_order_4.svg

The canonical enumeration for the 16 truth functions
begins with LEQ and ends with NOR. It is given

http://mathforum.org/kb/plaintext.jspa?messageID=7933628

This link contains fixed representations intended
to provide a reader with some familiar sense of
matters. The truth table for NOR is the concatenation
of the last three columns. Irregardless of any
given fixed representation, the first two components
of a truth table are named

FIX LET

to coincide with the interpretation of the canonical
enumeration. Any given system of truth tables can
be put into compliance with this labeling by virtue
of the geometric relations discussed earlier.

To the extent that there is, in fact, a fixed representation,
it corresponds with the sequential interpretation of a
De Bruijn sequence as described in

http://en.wikipedia.org/wiki/De_Bruijn_sequence#Example

Hence, there is no semantic presupposition in the
assignments.

Before moving away from De Bruijn graphs, it would
be prudent to observe that the metamathematical use
of ordinal numbers as algebraic indexes is
coordinated with the combinatorics of power set
operations for De Bruijn graphs on 2 symbols. One
can find an explanation for this in the link,

http://en.wikipedia.org/wiki/De_Bruijn_graph

This correspondence is to be considered in
conjunction with the mathematization of
quantifiers posted recently in the link:

http://mathforum.org/kb/plaintext.jspa?messageID=9370921

That link treats the ordinality of Peano
succession in contrast with the incidence
structure of the natural numbers based
upon aliquot parts and unique prime decompostion.

In addition, there will be a type of incidence
structure for with representatives for every
cardinality 2^n. This incidence structure is
a Steiner Quadruple System. It will play a
significant role.

The point here is that the the combinatorics
associated with power set operations arise
from mathematical structures related to geometry
and topology rather than from naive set theory.
In addition, the role of enumeration corresponds
with traversals of partition lattices. These
lattices are non-Boolean. So, Boolean combinatorics
can be accounted for without Boolean semantics.

The review of past work is almost finished.

As noted, the functionality of the names
ideas from Church and Birkhoff. However,
additional work is required in order to represent
the compositional structure associated with
the Fregean development of logic.

The system must be extended by a single symbol.
This is accomplished by restricting attention
to the axioms for which NOR is the principal
connective and introducing a applicative
interpretation in the sense of Shonfinkel and
Curry. There are no combinators here. It
is a simple applicative system base on the
NOR connective.

However, because of the NAND connective, the
applicative system can be extended to a simple
magma. The system is extended by a symbol for
this product, and, in the magma a connective
that forms a product with itself has the value
of its truth-functional negation.

This work is presented in the link:

http://mathforum.org/kb/plaintext.jspa?messageID=7935661

The extended system supports the Fregean notion
of "truth objects" because it admits language
elements to be organized as a minimal Hausdorff
topology. The first attempt at that explanation

http://mathforum.org/kb/plaintext.jspa?messageID=7933648

But, the significance is better expressed in the
analysis of the Tarski-Givant axioms in the link:

http://mathforum.org/kb/plaintext.jspa?messageID=9337125

The two key properties of minimal Hausdorff topologies
are feeble compactness and the embeddability of
semiregular topologies. The topologies used in
Boolean-valued forcing, for example, are semiregular.
And, feeble compactness accounts for a small countable
"surplus". This seems to correspond with the sense
of a set of measure zero that prevents acceptance of
Freiling's axiom of symmetry in set theory,

http://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry

This completes the review of past work needed
for this post. The material moving forward will
attempt to coordinate certain arithmetical
structures with statements made previously in
such a fashion that the 17 symbols discussed above
will be related to the arithmetical forms.

The post mentioned above describing the mathematicization
of quantifiers involves an infinitary assumption of
an infinity of primes. The next step is to offer a
finite arithmetical characterization in terms of a
system of Diophantine equations. There are two things
significant with this system.

First, there are 14 equations. In topology, Kuratowski's
14 set problem is well known. But, few people associate
topology with semantics despite the work of Tarski and
McKinsey. What is also little known is that of the 16
basic Boolean functions that comprise the semantics of
truth tables, 14 are threshold functions. Only LEQ and
XOR fail to correspond with linearly separable systems
of equations in threshold logic.

Moreover, LEQ is significant toward recognizing that
Boolean algebras are not faithful models of the
syntactic structure of the propositional calculus.
This is discussed in the paper by Pavicic and
Megill,

http://arxiv.org/pdf/quant-ph/9906101v3.pdf

To understand the significance of the expression of
14 elements in these systems, one must realize that
the free Boolean lattice on 2 generaors which is
usually associated with truth-functional connectivity
is order-isomorphic with the lattice representaton
for the Cartesian product of the 4-element subdirectly
irreducible De Morgan algebra with itself. When
one applies De Morgan conjugation to all 16 truth
functions, there are 4 which are invariant. These
are the 4 which are different from LEQ and XOR in
the formation of truth table semantics. So, the
"constancy" of logical constants derives from the
De Morgan invariance.

It turns out that a system of 14 symbols forms
4 non-isomorphic Steiner Quadruple Systems. The
thesis here is that it is appropriate to reduce
these systems according to this relation to the
incidence structures.

This is further supported by the fact that the
16 element affine geometry has 20 lines. The
labeling for those lines in the original
post, reproduced here,

http://mathforum.org/kb/plaintext.jspa?messageID=7935660

preserves the geometric relationships in the
dual plane. The only difference is that the
symbol NOT exchanges with the "all false" NTRU
in the point labeling. One may consider the
NOT to correspond with "absurdity" in standard
treatments that use absurdity as an expression
by which to define negation.

Now, the free Boolean lattice on 2 generators
is also order-isomophic to a Boolean subblock
in the lattice N_1 x 2 on page 433 (pdf pg 7)
in the paper,

http://www2.latech.edu/~greechie/2008%20Coverings%20of%20%5BMOn%5D%20and%20minimal%20OMLs.pdf

If you examine N_1, you can see the relation
of 6 interconnected elments organized into
3 orthocomplemented pairs and 2 additional
orthocomplemented elments. This should be
compared with the semantic relations discussed
earlier. When this lattice is put into a
Cartesian product with the 2-element De Morgan
algebra, it forms a 20-element ortholattice
having a Boolean subblock with 4 atoms and
16 elements. It is this subblock that is
order-isomorphic with the free Boolean algebra
on 2 generators.

All of the 14 element systems are taken to
relate to these structures by way of the
4 non-isomorphic SQS(14)'s in relation to
the 4 invariant truth functions that ground
the semantic interpretations.

The arithemtical system upon which the finitary
description of prime numbers rests is to be
found in the Diophantine equations discovered
by Jones,

http://en.wikipedia.org/wiki/Formula_for_primes#Formula_based_on_a_system_of_Diophantine_equations

This system is to be associated with the
affine geometry on 16 points by virtue of
its system of 26 variables.

Without concern for a specific choice, one
variable is taken as a distinguished element.
Thus, the ground for this construction is
a pointed set. It is significant that the
distributivity laws do not hold for the
category of pointed sets. In the paper
discussed above in which Boolean algebras
are shown to not be faithful to the syntactic
structure of the logic, it had been upon
the failure of the distributivity laws
that that conclusion had been reached.

In addition, there is the relationship of
a pointed directed complete partial order
to the structure of arithmetic in terms
of a base element and a fixed point under
order-preserving maps into itself,

http://en.wikipedia.org/wiki/Directed_complete_partial_order#Properties

This is reminiscent of the Dedekind formulation
of arithmetic.

To obtain 16 objects that may be organized
into the elements of a finite affine geometry
as in the first post of December 2012, the
use of order 5 pandiagonal squares is employed.

There are exactly 16 such squares that are
self similar. Mitsumi Suzuki has compiled
those squares here:

http://mathforum.org/te/exchange/hosted/suzuki/MagicSquare.5x5.selfsim.html

Our objective, however, is to obtain a system
with 17 squares.

Now, with respect to a De Morgan transformation,
the 5 elements on the line at infinity are
not fixed. Only one point on the line at
infinity is fixed because it is the
center of the collineation. In addition, a
De Morgan transformation exchanges TRU and NTRU.

This duality is captured in the base 5 representation
of the Graeco-Latin pandiagonal magic square

http://www.grogono.com/magic/5x5.php

With respect to the involution described above, let
two copies of the center of the collineation be the
center of the pandiagonal square labeled '33'.
Observe that in the secondary diagonal, digits are
exchanged when reflected across the main diagonal.

The scheme used in labeling the points at infinity
had been the term logic quantifiers. These are
mere schemes reflecting any extension of the
logic by operators. Any extension, in the context
of exclusion negation, has De Morgan conjugations.

In the case of mathematical logic, let ALL correspond
to Ax and let SOME correspond to Ex. Then they form
a De Morgan conjugate pair. Similarly, let NO correspond
to Ax~ and let SOME correspond to ~Ex. This forms
another pair.

Situating these pairs on the secondary diagonal to
reflect their exchange in a De Morgan transformation
will populate this pandiagonal square.

Now, the 17 desired objects have been obtained.

All of these constructions have been directed at
formulating the truth table for the complete
connective NOR.

Associated with the construction of a truth table
is the combinatorial complexity of its formation.

There are two symbols from which one must be
the preferred symbol to represent truth with
respect to transformations in a deductive
calculus.

There are two symbols which will correspond
to propositional variables. They can have
any order.

And, there are four rows which may also
be in any order.

The possibilities correspond to a 96 element
group,

Z/2 x S_2 x S_4

Based on a group representation capable of
representing both a 3 symbol group and a
6 symbol group, a 96 element block design
had been formulated. By virtue of the
canonical enumeration, the block design
has a lexicographical ordering. By construction,
it is a (96,20,4,4) strongly regular graph.

But, the lexicographic ordering permits
it to form a labeling for the 96 element
Horton graph with automorphism group,

Z/2 x S_2 x S_4

There may be other such graphs. But, the
principle being expressed here is that
the combinatorial complexity must be
accounted for. The 96 element Horton
graph with a 96 element automorphism
group expresses a minimal representation
of this complexity. But, there may
be others.

http://mathforum.org/kb/plaintext.jspa?messageID=8324328

http://mathforum.org/kb/plaintext.jspa?messageID=8334585

http://mathforum.org/kb/plaintext.jspa?messageID=8334586

describe the construction of the block
design and its relation to combinatorial
structures associated with the Mathieu
groups.

http://mathforum.org/kb/plaintext.jspa?messageID=9335214

contains a labeling of the Horton graph
on 96 elements.

The particular combinatorial structure
used in the links above is the Miracle
octad Generator and is related to the
24-dimensional binary Golay code.

Recalling that the presumed fundamental
invariant here is a 4-element collection,
the order of S_4 is 24. Moreover, when
Curtis invented the Miracle Octad Generator,
he formulated his construction to support
the representation of a 21 element
projective geometry.

The "constancy" of logical constants should
be associated with the uniqueness of the
Mathieu groups up to isomorphism. The Mathieu
groups are in the family of 3-transposition
groups. So, too, are the finite symmetric
groups.

From the standpoint of proof-theoretic
semantics, the question of grounding
the constancy of logical constants is an
issue. Evidence for this can be found

http://plato.stanford.edu/entries/proof-theoretic-semantics/#StrChaLogCon

where the duality of introduction and
elimination rules is discussed.

The account given in this posting deals
with a general duality across the spectrum
of issues that include, for example, the
association of paradoxes with a specific
equation for the mathematization of quantifiers.

Where the account in the link above perceives
duality with a preference for one class of
rule over another, the construction here
gives preference to the line elements because
that is where the order relations coincide.

That the modern compositional forms of logic
do, indeed, introduce problems concerning
logical constants is discussed in the

http://plato.stanford.edu/entries/logical-constants/

For the most part, the construction has met
its goals. There are some additional details,
however.

http://www.uwyo.edu/moorhouse/pub/planes16/

one can find projective planes of order 16 having
17 points on their lines at infinity. On lines
11 and 12 you can find planes with orbits that
include an orbit of order 96.

Finite geometries relate to one another. Many
incidence structures are recursive. So, in
some sense, the construction described here
is a "basepoint" with respect to larger
incidence structures.

In fact, it appears that there may be an
entire system of partial structures relating
to one another. If one subtracts 5 from the
number of elements in the free orthomodular
lattice on 2 generators the remainder is 91.

Now, that lattice has 96 elements and its
lattice polynomials correspond to the labeling
described above. But, the 91-point projective
planes of order 9 seem to relate to one
another as do the subdirectly irreducible
De Morgan algebras.

To see why the projective planes of order
9 might be of interest here, consider again
the system of Diophantine equations discussed
earlier. According to the Wikipedia article,
the system with 14 equations and 26 variables
can be replaced by a system having only 9
variables. This is related to Matiyasevich's
theorem. It is better described in the

http://www.scholarpedia.org/article/Matiyasevich_theorem

Now, that link expresses the fact that the
9 variable solution corresponds to positive
values. That will be the focus here. However,
when values are arbitrary integers, the number
is 11.

Consider the 11-dimensional lattice in the

http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/ANABASIC_11.html

I would be misrepresenting myself if I were
to "interpret" that lattice in the context
of this construction. However, the numbers 13
and 7 are significant with respect to the
91 point projective planes about to be discussed.
The number 96 corresponds with the work above.
I know that my own investigations have led to
tangential references to the Kirkman points

http://mathworld.wolfram.com/KirkmanPoints.html

because of the Pascal configuration,

http://en.wikipedia.org/wiki/Pascal%27s_theorem

However, what certainly does correspond to the
work above is the kissing number of 24. And,
if I recall correctly, there is a Mathieu group
M_11. But, I do not know if it is related to this
lattice.

Now, turning to the projective planes of order 9,
the failure of distributivity reappears in the
mathematics. Of the 4 planes, only one is based
upon a Galois field. The other 3 are near-fields
based on a system called mini-quaternions.

there is another construction based upon
matroids. This time it is an 11-elment
matroid formed with the 15 elments among
the line names -- that is, excluding the
NOT (absurdity) -- less the 4 names corresponding
to the truth functions that are invariant
with respect to De Morgan conjugation.

This matroid again express a structure of
order 14. And, from this structure, the
miniquaternions are developed.

The 11 elements of the matroid are segregated
into a group of 8 augmented by LEQ, XOR, and
TRU. That group of 8 has a significant role

http://mathforum.org/kb/plaintext.jspa?messageID=9347270

In particular, these 8 connectives are associated
with the threshold logic of pixelation through

indicates the simple form of directionality
involved. And, when the 8 connectives are
correlated with the remaining 8, octonion
geometry is expressed. The octonion geometry

http://mathforum.org/kb/plaintext.jspa?messageID=9334936

But, in relation to the 14 element constructions and
the 4 element De Morgan algebra involutions, the
origin of the quaternionic forms lies with the
91 point projective geometries.

On this account, it is also conjectured that the
order of these projective planes is expressed
algebraically by the Heegner numbers,

http://en.wikipedia.org/wiki/Heegner_number

To conclude these remarks, let me observe that
I have spent the last year working on constructions
based on self-similarity and based on eliminating
Boolean logic. To the extent that a bivalent
logic applies in mathematics, it applies in the
sense of a Suszko reduction to support the
presuppositions of a deductive calculus.

I have worked hard to distinguish nominal,
metamathematical uses of natural numbers
from ordinal and cardinal concepts.

My formal systems are based upon notions
of fixed points and my fundamental relations
generate tree structures.

With the elimination of a Boolean logic, Goedel's
incompleteness is better expressed in terms
of Diophantine unsolvability as follows from
the work of Matiyasevich

It should be noted that Tarski's semantics
is consistent with a constructive approach
to mathematics. It is only the mistaken
understanding that truth tables ground the
recursive definition of formal languages.
The axiomatization of truth-functionality
eliminates that interpretation. The axioms
provide the necessary mathematical form
for the recursive construction directly.

Well, I cannot be the one who decides whether
or not the arguments I have given over the
last year can be construed as justifying a
finite consisency argument. It is, however,
the best that I can manage toward that end.

I'm just a guy who swings a sledgehammer.

:-)

Date Subject Author
1/24/14 fom
1/24/14 fom
1/24/14 ross.finlayson@gmail.com