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:
A new axiom or set of axioms not directly asserting the continuum hypothesis would be required.
Arguments to the effect that the existing theory failed to properly represent standard mathematical practice would be required
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."
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
Elimination of urelements from the domain description in favor of a theory of pure sets.
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
The axiom of foundation
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.