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Topic: feeble compactness in syntactic formalisms - a topological analysis
of Tarski-Givant

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Registered: 12/4/12
feeble compactness in syntactic formalisms - a topological analysis
of Tarski-Givant

Posted: Dec 6, 2013 1:42 AM
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On two other occasions, I tried to talk about
the fact that a typical recursively-generated
language for a classical bivalent logic could
be structured as a minimal Hausdorff topology.

Such a topology is feebly compact.

A collection of expressions, organized to
satisfy the description of a topology, cannot
be compact since the language must also support
the representations of inconsistent sets of
expressions.

Feeble compactness is strong enough to ground
the compactness theorem with regard to
satisfiable collection. It is weak enough
to admit the use of Koenig's lemma to identify
finite sets of inconsistent formulas from an
arbitrarily given inconsistent set.

Koenig's lemma is needed -- or, possibly
just the weak Koenig's lemma -- so that
such a finite set may be obtained using a
systematic semantic tableaux as described
by Smullyan.


====

In "A Formalization of Set Theory
Without Variables" Tarski and Givant
give five axioms describing the turnstile
relation for an uninterpreted syntactic
formalism.

What follows is some thoughts with respect
to those axioms.

Tarski and Givant are careful to restrict
the notion of a formalism to the kind of
systems they are considering. But, their
general description is rather simple:

"..., but also to formalisms *G* provided
exclusively with syntactical components, and
thus construed as ordered pairs,

< *G*, |- >.

"Such formalisms are often referred to as
syntactical or uninterpreted formalisms
(as opposed to the original or interpreted
formalisms). We can thus arrive at an abstract
theory of syntactical formalisms, i.e., the
theory of all ordered pairs

< *Sigma*, |- >

(where |- is a binary relation between subsets
of *Sigma* and elements of *Sigma*."


I will be interpreting subsets of the system
as global functions on the domain. Those with
significant knowledge of the lambda calculus
may see similarities of which I am unaware. I
am considering these axioms in other contexts.
In particular, I shall interpret these global
functions as "selections".

Let X and Y be in *Sigma*. Let Phi, Psi, and
Delta be subsets of *Sigma*.


Their first axiom is

"If ( X in Phi ), then ( Phi |- X )"

Topologically, this describes a neighborhood
filter about X. That is, one has


{ X } |- X

{ X, A } |- X

{ X, A, B } |- X

{ X, A, B, C } |- X

{ X, A, B, C, ... } |- X


with respect to every pairing,


{ X } |- X

{ X, B } |- X

{ X, B, C } |- X

{ X, B, C, ..., A, ... } |- X


and so on.

For the purpose of expressing this
interpretation of the axiom in the
sense of a topological neighborhood,
consider rewriting the axiom as,


"If ( {}_f in nbhd(X) ), then ( {}_f accepts X )"


There will be some complexity associated
with this topological notion which is
comarable to what is found in the set
theory axioms. But, it is best to proceed
to the additional axioms before discussing
the issue.



Their second axiom is

"If ( Phi |- Y ) for each ( Y in Psi ), and if
( Psi |- X ), then ( Phi |- X )"


This is clearly a transitive closure operation.

In terms of finitary well-formed formulas for
which the reflexive relation of axiom 1 is
excluded, this corresponds to a directed acyclic
graph.

Now, consider this in the context of Zermelo-Fraenkel
set theory for a moment. The axiom of foundation
expresses the fact that the universe of sets is
partitioned into those with the empty set as an
element and those without that member. This is
not a standard characterization of that axiom.
But, when the empty set is an element of a set,
the axiom is trivially satisfied.

This is easy to see with the initial example
above. Consider the difference between the
presentation,


{ X } |- X

{ X, A } |- X

{ X, A, B } |- X

{ X, A, B, C } |- X

{ X, A, B, C, ... } |- X


and the presentation,


{ A } |- X

{ A, B } |- X

{ A, B, C } |- X

{ A, B, C, ... } |- X


With respect to interpreting X as the empty
set, the turnstile relation may be interpreted
as the converse of


"X is an element of the transitive closure of Y"


But, while the theory of pure sets has only
the empty set as a set without members, a
syntactic system has an antichain of incomparable
syntactic atomic well-formed parts.


Suppose, now, that one tries to rewrite the
second axiom as discussed earlier. Then one
has something along the lines of

"If ( {}_f accepts Y ) and if ( {}_g in nbhd(Y) )
for each Y such that ( {}_f accepts Y ), then
( {}_g accepts X ) implies ( {}_f accepts X )"


For comparison, here is the original axiom:


"If ( Phi |- Y ) for each ( Y in Psi ), and if
( Psi |- X ), then ( Phi |- X )"


To get a better sense of what is intended here,
consider reading "accepts" as "selects". It is
certainly true that this reading as a "selection"
does not quite seem correct since the initial
segment of a deduction could have several
possible successor steps on the basis of
transformation rules.

But, that is the point. The characterization
given by Tarski and Givant is from subsets
to individuals. At this level of
abstraction, those dependencies are obscured
and the specification is that of a fixing a
single element from among alternatives. The
sense of "selection" corresponds with
unspecified modal alternatives. The sense
of "acceptance" corresponds with a unique
selection for any given instantiation.

The wikipedia link,


http://en.wikipedia.org/wiki/Transitive_closure#In_graph_theory


expresses the fact that the transitive closure
operation on a directed acyclic graph yields a
strict partial order. I have promoted the idea
that a single formal expression could suffice in
the construction of theories. That statement is
schematized by this expression conveying an
irreflexive transitive order,


AxAy( x irreflx y <-> ( Az( y irreflx z -> x irreflx z ) /\ Ez( x
irreflx z /\ ~( y irreflx z ) ) ) )


The next section from the link above,


http://en.wikipedia.org/wiki/Transitive_closure#In_logic_and_computational_complexity


discusses the fact that this is a fixed point
logic which is more expressive than the usual
notions of a first-order logic. If one
follows the links to the page,


http://en.wikipedia.org/wiki/NL_%28complexity%29


one finds that this logic has a complexity
class referred to as "non-deterministic
logarithmic space". As it is expressed
according to


NL=NSPACE(log n)


it is, unfortunately, not of any use to logic
programmers,


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


On the other hand, it is closely related to
certain ideas in the foundations of mathematics
toward which I am inclined to have some minor
interest. And, I am personally uninterested in
foundational perspectives based upon
metaphysically grounded distinctions.

This is a particular construction with a
well-defined complexity and appropriate
features for mathematical discourse of all
variety.

The difficulty mentioned earlier begin to
appear when we recognize that the first axiom
yields expressions of the form,


{ X } |- X


This has a great deal in common with Zermelo's
original formulation of set theory. His original
version relates singletons to denotations and
uses the sign of equality to relate denotations.
This notion had been changed as the formalisms
recommended by Skolem had been implemented.

For the present purposes, one may formulate
the expression


( {}_X=X accepts X )


Interpret this as one will. It can reflect
the "arrow only" interpretation of category
theory. It can reflect the ambiguous syntax
of point set topology wherein '{x}' is
identified with 'x' as a syntactic stipulation.
It can be a graph-theoretic notion of
self-looping. Pick a favorite belief. This
exposition is not at the level of an object
theory.

The sense by which induction is inherently
impredicative arises with the transitive closure
logic and its fixed point characteristics. The
reflexive case is an augmentation to the strict
partial order given by the transitive closure
logic.

This can be made precise by considering a single "true"
instance of the axiom above. Let '<' be used to express
the asserted relation and let '>=' be used with its usual
sense in relation to '<'. One can consider "accepted",
"undecided", and "omitted" formulas based on an iteration.

For example,

accepted: { a < b, a < c }
omitted: { b < c }
undecided: { b >= c }

accepted: { a < b, a < c, a < d }
omitted: { b < c, c < d }
undecided: { b >= c, c >= d, b >= d }

accepted: { a < b, a < c, a < d, a < e }
omitted: { b < c, c < d, d < e }
undecided: { b >= c, c >= d, b >= d, d >= e , c >= e , b >= e }

and so on.

Several things can be observed here.

First, the the accepted formulas for
each instance describe a pointed
set of language parameters. Second,
the accepted types are specifying a
partial collection of ordered pairs
satisfying the diversity relation
( ~x=y ) by means of a single ordered
pair. Hence, one can partition the
diversity relation in such a way that
the two components of the partition
separate into one with semantical content
and one without, where the two components
are also relational converses. Third, the
undecided formulas comprise a bounded
chain. However, subsequent axioms could
omit or accept formulas according to any
partial order that does not violate the
omitted types.

Augmenting this irreflexive
transitive order with a reflexive
relation introduces a very
significant problem.

Proper filters are characterized
by the finite intersection property.
Neighborhood filters only satisfy this
property when they are principal
filters with respect to a single
individual. An anti-chain of
order-theoretic atoms would have
empty intersection.

Boolean logic provides a means of
addressing this problem for formal
set theories. By analogy, the recursive
construction of a formal language
supports the same method.

In set theory, every set is associated
with a singleton and the axiom of
pairing can form finite sets. In the
typical recursive construction of a
countable language, every finite
collection of well-formed formulas can
form pairwise connected terms.

In both cases, the failure of
the finite intersection property
for the collection of neighborhood
filters is compensated by the
existence of an element that
corresponds with the
intension of finite intersection
or finite conjunction. This
element exists as an individual in
the theory.

This is not, however, an ad hoc mechanism.
In set theory, the axiom of pairing in
conjunction with the axiom of union
implements a directed set structure. This
is implicit to theories of arithmetic by
virtue of the difference relation.

It is inherently arithmetical, hence
mathematical.

To make this work in set theory, however,
one needs the axiom of pairing interpreted
in the sense of matroid circuits.

Matroid theory is robust enough to
distinguish matroids corresponding
to graphical self-loops from matroids
which correspond to a collection of
singletons interpreted as parallel
classes.

Yet, the definition of parallel classes
in matroid theory for a set of singletons
also presupposes that pairs form circuits
for such a matroid.

For set theory, the consequence of this
view is that the "mereological" nature
of the proper subset relation has no
large cardinal strength. This is explained
by Hamkins in the link,


http://mathoverflow.net/questions/144231/is-the-inclusion-version-of-kunen-inconsistency-theorem-true


For the formal language, one requires
something along the same lines.
Although Boolean logic provides binary
connectives, the machinery is not
sufficiently weak to convey the sense
of mereological weakness one finds in
set theory.

The notion of pairing which is needed
would be a single connective -- a complete
connective. Given any two formulas, there
is a formula corresponding to its pair. That
the pair is interpreted as a truth-functional
connective is secondary. However, the reason
for associating pairing with a complete
connective is so that pairing is semantically
weak until semantically strengthened through
interpretation.


As I have explained elsewhere, the system
of basic Boolean functions can be
represented in such a manner that a system
could use a complete connective in just
this manner. In my formulations, the NOR
connective has the semantic significance.
Consequently, one could formulate the
language recursively using the NAND
connective as a purely syntactic construct.

What I describe as mereological weakness for
set theory is expressed by the understanding
that the recursive formulation of the language
is supported by the 4096 axioms in the link,


https://groups.google.com/forum/#!original/sci.logic/zuLUred6O3U/cH_nH7GCKjYJ


Unfortunately, when one "knows" what
one means in a truth table, one fails
to understand that the recursive
construction of a language as a purely
syntactic matter ought to be based
upon axioms just as the transformation
rules may be formulated axiomatically.
Although this is a simple idea, it makes
it difficult for using the syntax effectively.
These constructions are intended to ground
typical use, not supplant it.

In recursive construction, there is a
unary operation for prefixing a primitive
quantification symbol. Such prefixing should
be compared with the sequence of nested
singletons in set theory as far as such
an analogy may allow. What is important
in the present discussion is the role of
pairing.

Before turning to the third axiom, there
is one additional issue to be mentioned.
The sense by which the finite intersection
property is "implemented by proxy" introduces
a problem similar to what one sees in
algorithmically random reals. The system
contains repetitive information. However,
the sense of atomic well-formed parts is
that they be able to take on truth valuations
without dependency.

Without prejudice, I view this as cause for
rejecting systems of logical atomism. It does
not seem that this problem can be solved by
syntactic stipulations. Perhaps I am wrong.

For their third axiom, Tarski and Givant
have the following:


"If ( Phi |- X ), then ( Delta |- X ) for
some finite ( Delta subset Phi )"


This clearly resembles a statement of
compactness. Compare the form of the
statement with


"Every open cover has a finite subcover"


That is an open set notion of compactness,
and, it can be associated with consistent
theories (in distinction from closed set
models which may be associated with
paraconsistent logics).

However, this is not quite the fact of the
matter.

There are two cases. The first is where
( Phi |- X ) is true for every X. In order
to handle this case, one needs at least the
weak Koenig's lemma. The discussion of
compactness in Smullyan using Hintikka sets
is what is required. A systematic tableaux
applied to an inconsistent set will eventually
yield a finite tree. However, it must be
possible for the tree to contain an infinite
branch in order to distinguish between a
satisfiable collection and an unsatisfiable
collection.

The second case relies on the fact that the
insight of Frege had been to introduce "The
True" and "The False" as objects.

The link,

http://books.google.com/books?id=DkEuGkOtSrUC&pg=PA119&lpg=PA119&dq=%22minimal+hausdorff+topology%22&source=bl&ots=3iJBPKz0j7&sig=ZPwm_43XRhgdJneeO-82fUDZvZI&hl=en&sa=X&ei=YzOgUpPlN4azrgHqu4D4CQ&ved=0CEEQ6AEwAw#v=onepage&q=%22minimal%20hausdorff%20topology%22&f=false

has a diagram of an example of the minimal
Hausdorff topology which makes this
explanation simple.

The example begins with the Cartesian product
of the positive integers Z^+ and the set,


{ -1, -2, -3, ..., w, ..., 3, 2, 1 }


To this set is added 2 arbitrary symbols. For
the present purposes, we take these symbols
to correspond with 'T' and 'F'.

In addition, we take the 'w' to correspond with
the 'NAND' connective with which the formal
language is recursively constructed.

Suppose one is given an enumeration of the
well-formed formulas for a language. Then,
for the first collection corresponding to
the set above, take each formula P which
contains no more than one atomic formula and
form the pair,


NAND( P, P )

NAND( NAND( P, P ), NAND( P, P ) )


One can now use the enumeration to form
the first set with the less complex formulas
in the postive loci and the negated formulas
in the negative loci.

For the next collection, choose only those
formulas P which contain exactly two atomic
formulas. Again form the pairs,


NAND( P, P )

NAND( NAND( P, P ), NAND( P, P ) )


and position them at their respective loci.

In this manner, one can use the complexity on
atomic formulas to form the minimal Hausdorff
topology for the language elements in the
construction.

One now has an equisatisfiable subcollection
of the originally generated formulas. Taking
this as the collection *Sigma* provides a
language with the appropriate topological
structure for implementing the third axiom.

This is very similar to the kind of conditioning
Kolmogorov implemented to formulate notions
surrounding the law of excluded middle important
to intuitionistic and constructive reasoning.
In this case, however, it is being done to support
the topological structure of a classical,
compositional logic.

There are two properties which make this
construction of a minimal Hausdorff space
significant. Such spaces are Hausdorff-closed
and semiregular.

Every Hausdorff space can be densely embedded into
an H-closed space, and, in the class of H-closed
spaces, there is a particularly important
H-closed extension called the Katetov extension.
With respect to this extension, the original
space is an open set. The Katetov extension
is defined with respect to the free open ultrafilters,
its extended elements comprise a discrete closed
set, and the Katetov extensions of semiregular
topologies can always be embedded into a minimal
Hausdorff topology.

Since every Hausdorff space can be embedded as a
nowhere dense subspace of a semiregular space, every
Hausdorff space can be embedded into a minimal Hausdorff
topology as a nowhere dense space. With respect to
forcing models, every semiregular space can be densely
embedded into a minimal Hausdorff topology.

Consequently, a minimal Hausdorff topology can
embed densely into itself. This is characterized
syntactically by the methods discussed above involving
singletons, pairs, and finite intersections or finite
meets.

To the extent that the third axiom expresses
compactness, it is feeble compactness.

A space is feebly compact if every locally finite
collection of open subsets of X is finite. For
H-closed sets, this translates into the properties,

1.

For every open cover of X, there is a finite
subfamily whose union is dense in X

2.

Every open filter has nonvoid adherence

3.

Every open ultrafilter on X converges



Notice that the first statement is not compactness.
The compactness theorem in logic is stated only
with respect to satisfiable formulas whereas this
construction involves the structure of uininterpreted
syntax. But, consistent logics are associated with
topological semantics given by open topologies. So,
feeble compactness expresses the sense of compactness
in terms of open sets. But, it cannot express the
notion of a finite cover since a consistent theory
cannot include every formula.

The second statement reflects the fact that a
consistent theory cannot have an empty domain in
standard first-order semantics.

The third statement reflects the notion above
concerning fixed point logics.

For minimal Hausdorff topologies, one has,
in addition:

Every open filter with a unique adherent
point converges.

I would presume this corresponds to being
able to assume a first expression as being
true. For typical constructions based
upon "ontological" notions of identity, that
might be the formula,

Ax( x=x )

for example.

It may also correspond with the weak weak
Koenig's lemma in the sense that a finite
tree is pruned with increasing length. Hence,
there is a sense by which infinite length is
approached by asymptotically fewer branches.
In particular, the axiom above would have
a tableaux consisting of one infinite branch.

However, the stronger notion is required to
deal with the inconsistent case. One cannot
know which branches of a systematic semantic
tableaux can be pruned until they are closed.
No asymptotic condition can be given in advance.

All of these topological relations may be found
in "Extensions and Absolutes of Hausdorff Spaces"
by Porter and Woods.

To the extent that these topological notions
translate into consistent theories, I have elsewhere
proven that a consistent first-order theory is associated
with a proximity on its langauge terms. Awodey has
shown that necessary denotations are continuous
denotations in topological semantics for first order
logic. To the extent that one is speaking of the
turnstile relation in the present context, necessary
denotations may be considered to be in correspondence
with eliminable definite descriptions.

In terms of the functional notation, the
third axiom,


"If ( Phi |- X ), then ( Delta |- X ) for some finite
( Delta subset Phi )"


might be recast along the lines,


"If ( {}_f selects X ), then there exists {}_g such
that ( {}_g selects X ) and ( {}_g in nbhd(Y) ) for
only finitely many Y for which ( {}_f in nbhd(Y) )
holds"




Next, fourth and fith axioms given in Tarski
and Givant should be considered together.
These axioms are:


"*Sigma* is countable"

"There is a finite ( Delta subset *Sigma* ) such that
( Delta |- Y ) for each ( Y in *Sigma* )"


Recast these axioms as


"Given any distinct {}_i and {}_j, one has that
dom( {}_i ) = dom( {}_j ) and dom( {}_i ) is
countable"

"For each Y there exists {}_i such that
( {}_i in nbhd(X) ) for finitely many X
and such that ( {}_i accepts Y )"



Taken together, these axioms seem to assert the
existence of an omega-regular filter. A filter
F over a domain D is an omega-regular if and only
if there exists a set E such that ( E subset F ),
( |E| = |omega| ), and for each ( d in D ) there
exist only finitely many ( e in E ) such that
( d in e ).

Every omega-regular filter is countably incomplete.

Since every open filter has non-void adherence, this
cannot be an open filter.

But, such a filter is necessary. Seemingly, this
filter would correspond to the filter associated
with fallacies. By the nature of omega-filters,
to the best that I can discern, this filter would
contain every finite set.

The existence of this filter has a number of
consequences as discussed in Chapter 4 of Chang
and Keisler. I have surveyed the various statements
and found nothing obviously unsupportable with such
a construction interpreted in this way.

This concludes a somewhat difficult analysis of
the Tarski-Givant axioms.




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