fom
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Registered:
12/4/12
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some amateurish opinions on CH
Posted:
Apr 7, 2013 4:54 AM
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When I failed to complete my education in
mathematics, there had been little hope
of any professional position. Whatever
mathematical interests I might have chosen
to pursue were not dictated on the basis
of what would be beneficial to a
professional status.
I had some tangential interest in the
continuum hypothesis. I also believed
that the resolution to the continuum
hypothesis might be obtained by considering
how the sign of equality is used.
It would be a mistake to think that I
did not know what was involved with
such a topic. The independence results
dictate three requirements:
1)
A new axiom or set of axioms not directly
asserting the continuum hypothesis would
be required.
2)
Arguments to the effect that the existing
theory failed to properly represent
standard mathematical practice would
be required
3)
A proof
Well, I certainly satisfied the first
requirement.
As for the third requirement, I do not
have a proof in the syntactic sense.
It may be that the axioms are stringent
enough to exclude forcing models. But,
that argument would also involve certain
general criticisms of model theory.
In turn, those arguments involve the
use of the sign of equality. So, I
have not met the third requirement in
any meaningful fashion.
The second requirement is the most
disappointing. I take the development
of equational classes, varieties,
categories and Grothendieck universes
as tangible evidence that the
majority of mathematicians do not
have any commitment to the foundational
investigations directed along the
lines of either a purely arithmetical or
constructive foundation. On the other
hand, that same body of mathematicians
seems least likely to consider
foundational questions so long as
the ambiguous "any favorite set theory"
does not impact their own interests
directly. Those who are most likely
to entertain a discussion of foundations
are also the most likely to be
committed to arithmetical or constructive
positions.
For what this is worth, I would never
expected that questions about the
sign of equality would be such a
dramatic issue. However, I had been
unaware of the historical debates
that have led to the current situation
in the foundations of mathematics.
I would be misrepresenting myself if
I claimed knowledge of the current
situation beyond a bare minimum.
Without the counsel of instructors, the
literature is large, almost never
balanced, and rarely justified with
exact quotations from prior authors
with which some prior connection is
claimed. Mathematical treatments are
technical and sparse. So, what little
I do know and understand is minimal.
In addition, what is said about the
sign of equality is rarely directed
toward how mathematicians make use
of it. So, little of what is in the
literature is pertinent to the very
arguments needed.
When approaching the formulation
of new axioms, the lack of experience
and knowledge directed investigation
to topological considerations.
Although this would be incorrect in
view of standard interpretations,
it would be compatible with the
Brouwerian position that logic was
based upon mathematics in opposition
to the logicist program.
Here is the difference. Standard
interpretations can, in large part,
be traced to the influence of Russell.
To the extent that Russell and Frege
had been in agreement, the notion of
set is based upon the extension of
concepts. This notion does not
concern itself with the individuation
of objects over which the class
predicates may be applied. This is
comparable to the fact that Aristotelian
logic does not address individuals
satisfactorily. Indeed, the search for
simple substance in the work of Bolzano
is testament to this deficiency in
the Aristotelian logic.
In contrast, Cantor's ideas are
based on a theory of ones as described
by Hallett. This approach had been
dismissed by Frege in his work on the
foundations of arithmetic. But, Frege
also retracted his logicist approach
to arithmetic at the end of his
career. So, it is not unreasonable
to question the standard paradigm
of set theory if those doubts are
related to how Cantor differed from
Frege and Russell.
In the paradigms of analytical
philosophy most aligned with logicism
and logical positivism, great pains
are taken to criticize anything that
has the appearance of metaphysics or
epistemology. So, it is unlikely
that a quote from Leibniz explaining
the principle of identity of
indiscernibles has any import. The
relevant quote is as follows:
"What St. Thomas affirms on this point
about angels or intelligences ('that
here every individual is a lowest
species') is true of all substances,
provided one takes the specific
difference in the way that geometers
take it with regard to their figures."
Leibniz
This quote attributes the treatment
of individuals as a subtype of
logical species to Thomas Aquinas.
Leibniz, however, generalizes it
on the basis of geometric principles.
How Cantor differs from Frege and
Russell can be compared with what
is conveyed in this statement by
Leibniz. Whereas Frege and Russell
both approached their ideas in relation
to a theory of description, Cantor's
ideas arose in conjunction with his
topological ideas. Thus, the geometric
aspect of Leibniz' remark is found
in Cantor's intersection theorem,
"If m_1, m_2, ..., m_v, ... is any
countable infinite set of elements
of [the linear point manifold] M of
such a nature that [for closed
intervals given by a positive
distance]:
lim [m_(v+u), m_v] = 0 for v=oo
then there is always one and only one
element m of M such that
lim [m_(v+u), m_v] = 0 for v=oo"
Cantor to Dedekind
So, when one considers the continuum
hypothesis, one must ask whether or
not the issue of individuation does,
in fact, rest with Cantor's topological
insights and Leibniz' geometric
characterization.
There is an entire historical account
discrediting geometry and an ongoing
debate criticizing the investigation
of transfinite arithmetic. That is
a difficult hurdle with which to contend
in order to justify a set theory
grounded in topological considerations.
It is not an argument to be made when
references from the historical literature
are easily dismissed as "appeals to
authority" or simply as "philosophy"
in contrast to "mathematics". Of course,
this ignores the role of those philosophies
most influential in estblishing mathematics.
That philosophy had been dominated by
logicism and logical positivism.
Arising out of logicism, the notion
individuation associated with modern
set theory is that which comes from
Russellian description theory. It is
heavily influenced from other factors
as well. Wittgenstein, for example,
objected to Leibniz' principle of
identity of indiscernibles as not being
a logically necessary statement.
But, the details of these positions
seems far removed from what mathematicians
focus upon. Among the formal transformation
rules in the deductive calculus for
first-order logic one finds,
Ax(P(x)) -> P(t)
P(t) -> Ex(P(x))
These axioms of the deductive calculus
reflect the considerations that had
been involved with the description
theory of Frege and Russell.
First-order logic assumes that every
domain of discourse is not empty and
that every object in any given
domain of discourse has a denotation.
Many mathematicians, however, are
unaware of this and would consider
a statement such as
Ax(P(x)) -> Ex(P(x))
to be counterintuitive. The error
here would be to accept the possibility
of an empty domain. Thus, the
antecedent is vacuously satisfied
while the consequent would be false.
This particular oddity forms part
of the objection upon which free
logics have been grounded. But,
in the current discussion, the issue
is that these presuppositions of
the first-order logic arise from
consideration of negative existential
statements and self-contradictory
descriptions. That is, they arise
from the perspectives of Frege and
Russell when considering how to
ground a logic whose singular terms
were not vacuous.
What then, would be the relation
to Leibniz and Cantor?
The answer lies in the fact that
a description theory of names relies
on a theory of identity.
There is a difference between identity
and arbitrary equivalence relations.
Identity must necessarily involve a
theory of individuation. But, this
is precisely what is rejected by the
logicist framework. Objects are
differentiated by the predicates
which they satisfy and the predicates
are differentiated by the objects
which satisfy them. Individuation
is tantamount to the thesis that
only predicates are objects.
At first, this seems a strange position
for a theory which had been grounded
on the idea of types built up from a
class of individuals. But, then one
must recall that Russell's original
theory had been a "no classes" theory.
The possibility of such a theory is
the result of his description theory.
That theory had been specifically formulated
to address the question of what is now
called presupposition failure.
Russell's original idea had been quite
ingenious. But, from his discussion of
the axiom of reducibility, his abandonment
of that axiom is tantamount to the
assumption of the existence of sets.
His original intent had been to avoid
that scenario. Having directed his
theory toward presupposition failure
rather than instantiation and toward
generality at the expense of definiteness,
there is little to justify the idea
that Russellian predicativism reflects
the position of a majority of mathematicians.
In order for predicativism to not be
vacuous, there needs to be a meaningful
theory of individuation and a meaningful
theory of reference. In contrast to
Russell, Frege considered these matters
somewhat carefully. Frege clearly
distinguished between object identity
and concept identity. The axiom of
extension in modern presentations of
set theory is not object identity in
the Fregean sense.
It is not that Zermelo's original
axiomatization of 1908 did not address
object identity. It did. But, this
disappeared when corrections to the
theory had been made. In his introduction
to the reprint of Goedel's papers on
the continuum hypothesis, Solovay describes
the modern theory as having been developed
through four changes to the original
theory. Those changes are described
as
1)
Elimination of urelements from
the domain description in favor
of a theory of pure sets.
2)
Specification of a formal language
for the theory so that the notion
of definiteness referred to in Zermelo's
axiom of separation would no longer
be vague
3)
The axiom of foundation
4)
The axiom schema of replacement
What is not mentioned in Solovay's remarks
is that Zermelo specifically mentioned a
definite sense of identity in relation
to the denotations upon which the description
of his domain had been based. So, in
changing his domain definition and by
introducing the formal language for the
purpose of formulating the axiom schema
of separation, Fregean object identity
had been eliminated from the theory.
These considerations impacted the
axioms I formulated in a somewhat
strange way. Those axioms specifcally
choose Fregean object identity as the
identity to be applied to its singular
terms. But because the original ideas
began with consideration of topological
and geometric principles, the theory
is interpreted by others as a second
order theory. So, there is this strange
relationship. The standard theory that
is taken to be a first order theory
actually uses a notion of identity
corresponding with Frege's definition
of second order identity. The non-standard
theory that is interpreted by proponents
of the standard theory as a second order
theory had been specifically formulated
to use Frege's definition of first
order identity.
Here is what is important about the
non-standard axioms that arose from
my investigations:
All sets are classes.
There exists exactly one proper class.
It is my thought that set theories
that admit reference to a set universe
also admit reference to proper classes
different from the universe. The views
for identity discussed here admit an
extension to ZFC which admits reference
to only one proper class. Any other
notion of class is grounded in the statement
of formulas just as with standard ZFC.
As this has already taken such a great
deal of writing, I will finish with the
following remarks.
I lack the skills to investigate this
theory as well as I would like. I reject
certain aspects of model theory in favor
of certain remarks by Abraham Robinson
relating the diagonal of a model to
denotations. The sense of this may be
comparable to Goedel's assumption that
every set could be given a denotation
in his formulation of the constructible
universe for ZF.
And, in relation to what had been said
earlier, the idea I have that this
formulation may be stringent enough to
exclude forcing models may not be so
outrageous. In his introduction to
Goedel's proofs, Solovay mentions that
Cohen forcing gives no information
concerning models of
ZFC + V=L
Should my formulation of a set theory
with a single proper class serving
as the universal class be comparable,
then my statement had not been
an exaggeration. To the best of my
understanding, forcing is applicable
only where partiality may be assumed.
But, the difficulty of properly
interpreting these matters without
the help of instructors or colleagues
makes it probable that I am wrong.
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