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Topic: some amateurish opinions on CH
Replies: 57   Last Post: Apr 16, 2013 8:12 PM

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 fom Posts: 1,968 Registered: 12/4/12
Re: some amateurish opinions on CH
Posted: Apr 9, 2013 4:24 PM

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.

Yes.

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

http://plato.stanford.edu/entries/frege-logic/

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.

Date Subject Author
4/7/13 fom
4/7/13 mueckenh@rz.fh-augsburg.de
4/7/13 Bergholt Stuttley Johnson
4/7/13 dan.ms.chaos@gmail.com
4/7/13 mueckenh@rz.fh-augsburg.de
4/7/13 dan.ms.chaos@gmail.com
4/7/13 mueckenh@rz.fh-augsburg.de
4/7/13 dan.ms.chaos@gmail.com
4/7/13 mueckenh@rz.fh-augsburg.de
4/7/13 Virgil
4/8/13 dan.ms.chaos@gmail.com
4/8/13 mueckenh@rz.fh-augsburg.de
4/8/13 dan.ms.chaos@gmail.com
4/8/13 mueckenh@rz.fh-augsburg.de
4/8/13 dan.ms.chaos@gmail.com
4/8/13 mueckenh@rz.fh-augsburg.de
4/8/13 Virgil
4/8/13 Virgil
4/9/13 apoorv
4/8/13 Virgil
4/7/13 Virgil
4/9/13 Guest
4/9/13 dan.ms.chaos@gmail.com
4/9/13 fom
4/10/13 Guest
4/10/13 dan.ms.chaos@gmail.com
4/10/13 fom
4/10/13 JT
4/11/13 apoorv
4/11/13 dan.ms.chaos@gmail.com
4/11/13 apoorv
4/11/13 fom
4/15/13 apoorv
4/15/13 fom
4/16/13 Shmuel (Seymour J.) Metz
4/16/13 fom
4/7/13 Virgil
4/7/13 William Elliot
4/7/13 fom
4/7/13 fom
4/8/13 William Elliot
4/8/13 fom
4/9/13 William Elliot
4/9/13 fom
4/9/13 William Elliot
4/9/13 fom
4/9/13 dan.ms.chaos@gmail.com
4/9/13 fom
4/9/13 dan.ms.chaos@gmail.com
4/9/13 fom
4/9/13 dan.ms.chaos@gmail.com
4/9/13 fom
4/10/13 fom
4/11/13 dan.ms.chaos@gmail.com
4/11/13 fom
4/11/13 dan.ms.chaos@gmail.com
4/11/13 fom
4/9/13 fom