[FOM] An intuitionistic query

[rgheck]

On 09/09/2009 10:12 AM, Arnold Neumaier wrote:
> For any of these interpretations, I have a related question.
> Is the following intuitive implication
>      A={a,b,c}  ==>    |A|=1 v |A|=2 v |A|=3      (**)
> intuitionistically valid?
>
>    
No. For a simpler case, let A = {a,b}, where we do not have a = b v a <> 
b. This last will clearly follow from |A| = 1 v |A| = 2.

> (*) is a special case of Dirichlet's pigeonhole principle.
> Are there versions of the latter that are intuitionistically
> undecidable? I found an infinite version in
>      https://eprints.kfupm.edu.sa/60600/1/60600.pdf ,
> but I am looking for a finite version.
>
>    
Each of the finite versions is certainly going to be provable by a 
version of the argument you gave, and the uniformity of those arguments 
is good intuitionistic reason to think the generalization holds. I.e, 
the obvious argument by induction should work.

Richard

-- 
-----------------------
Richard G Heck Jr
Professor of Philosophy
Brown University