[FOM] Definability and provability

[Paul Budnik]

  On 09/07/2010 10:35 AM, hendrik at topoi.pooq.com wrote:
> When definitions of ordinals become sufficiently complicated, it
> becomes undeterminable which is greater.  Could it be that for two such
> incomparable ordinal definitions, the existence of the ordinal one
> defines would prove a conjecture, and existence of the other would prove
> its negation?

There are two separate issues here. Is there an objective ordinal 
hierarchy and, if so, to what degree can we determine it. I have argued 
that there is an objective hierarchy of logically determined statements 
that cannot be precisely defined but is limited to statements about 
countable sets. (See 
http://www.cs.nyu.edu/pipermail/fom/2010-May/014792.html ) If two 
ordinal definitions have different implications for logically determined 
mathematics then I would say at least one of them is not a correct 
definition of an ordinal although we may not be able to determine this. 
See http://cs.nyu.edu/pipermail/fom/2010-July/014851.html for more about 
when I think we will need divergent paths to fully explore objective 
mathematics.

In contrast I think the mathematics of the uncountable is about and 
relative to (Lowneheim-Skolem) countable and logically determined models 
for the axioms of the system. I am skeptical of other interpretations. 
Thus one might have two large cardinal axioms that imply the continuum 
hypothesis is true and false, but agree on all questions of objective 
mathematics. Both axioms may be helpful in exploring the objective 
ordinal hierarchy.

Paul Budnik
www.mtnmath.com