[FOM] Mathematical Knowledge

[Timothy Y. Chow]

praatika at mappi.helsinki.fi wrote:
> It all depends on how exactly one understands "know" here. If one 
> requires a proof for the knowledge of consistency, I don't think we can 
> get much beyond PA. However, all ordinary mathematics can be developed 
> in ACA_0, which is a conservative extension of PA.  So one possible 
> reply would be: ACA_0 is the strongest mathematical system that 
> mathematicians regularly employ (that is, outside set theory), and we 
> know (I think) it is consistent.

I thought the graph minor theorem was unprovable in ACA_0.  That's 
certainly part of "ordinary mathematics."  It is regarded by some graph 
theorists as the greatest theorem in graph theory.

I also don't understand how the requirement of "proof" for knowledge leads 
you to PA.  Proof in what system?  Invoking the word "proof" here does not 
seem to eliminate any of the shades of gray that you are worried about.

Tim