On 4/9/2013 2:32 PM, Dan wrote: > Unlike a natural, or even a real number, sets are built in a > transfinite hierarchical manner as our ability to conceptualize them. > Difficulties begin to appear around unambiguously conceiving > uncountable cardinalites.
Yes. But, does our difficulty to conceive preclude analysis?
In other words, the distinction between countable and uncountable is an Aristotelian differentiation of species.
Mathematically, especially at the time of Cantor, analysis of this would be done with respect to an arithmetical perspective. So, formulating a system of transfinite arithmetic for the purpose of analysis need not have anything to do with conceptions.
The conceptual building of sets is a slightly different sense that arises in connection with type theory, and, in view of the importance of the axiom of foundation, model theory.
> What I consider important to remember is > that sets are objects of thought,
This is where my non-standard views would really shine. I agree, but what is the standard of individuation?
> therefore we can say that a sentence > is true when we find sufficient reason for it to be true . That means, > what we know is true is what we can Prove is true, for some unfinished > yet ever-expanding notion of Proof.
This is astute. In order to analyze proof, one finds oneself in the position of needing the object language meta language distinction. Then one needs a hierarchy of stronger theories, stronger languages, and stronger meta languages.
For me, I find this situation forcing me to reject mathematical realisms in order to accommodate a mathematical pluralism. Wittgenstein made a remark or two concerning the fact that mathematics is in the proofs. In Posterior Analytics, Aristotle admits the possibility of knowledge from other sources, but narrows his scope to concern itself with knowledge as being the knowledge contained in demonstration.
> Now , the crucial point to > remember is how the 'Proof modality' works . > > P(a^b) <-> P(a)^P(b) > P(avb) <-> P(a)vP(b) > .................... > > All laws hold except the excluded middle .It isn't necessary either to > Prove a or to Prove not a . > While the logic of ontology is aristotelian , the logic of > epistemiology (what we can prove) is intuitionistic .
This is really a nice way to put it.
I just ran across an article on provability logic that looks interesting.
> In Peano arithmetic we have a 'rule of thumb' : > If you cannot prove (exists x , G x), for a simple enough sentence > G , then (for all x , not(G x)) may be considered provable . > An extension of this principle for set theory would be wonderful . > Although , while PA seems to obey a 'minimality principle' , set > theory seems more suited to a 'maximality principle' . > Given that epistemology is intuitionistic, I'm open to the > possibility that neither CH nor its negation could provide a 'final > argument' . >
No. My understanding of CH is of a completely different character. It had not always been.
I understand predicativism and I respect it. "Principia Mathematica" is very different from anything that happened in set theory in relation to it. At its heart, however, is Russell's description theory and his epistemic views in relation to knowledge by acquaintance.
Similarly, Frege's work is keenly astute. There is a wonderful account of it and how it has been modified at
It would be correct to say that the ideas of Frege and Russell arise in relation to syllogistic logic where classes may, indeed, be considered as extensions of concepts.
But, I am fairly certain that the Cantorian idea of set does not arise from the same considerations.
Leibniz contrasts his logic from the Scholastic logic. Scholastic logic is extensional. The notion of classes as extensions of concepts is compatible with the Scholastic view. Leibniz views individuation as a matter of increasing informational complexity. Thus, genera are prior to species and species are prior to individuals. Leibnizian logic is not extensional.
Cantorian set theory is based on a theory of ones that had been rejected by Frege in his foundations of arithmetic. So, in view of his topological insights, it seems reasonable to view Cantor's notion of set differently from the framework inherited by formalism and predicativism. It seems reasonable to view Cantor's ideas in terms of the geometric statements Leibniz made when describing the principle of identity of indiscernibles.
So, although I had to start out by thinking that CH would decide something, what I have learned is that pluralism on these matters is far more valuable.