[FOM] inconsistency of P

[Arnold Neumaier]

On 10/04/2011 12:56 AM, Timothy Y. Chow wrote:
> On Mon, 3 Oct 2011, Aatu Koskensilta wrote:
>> When we say that (arbitrarily large) naturals don't really exist, just
>> what are we denying?
>
> I'd say that the way to think about it is to regard mathematical
> statements as physical theories that make predictions about our physical
> experience. To doubt that arbitrarily large naturals exist is to doubt
> that we are justified in believing that our everyday experience with
> physical quantities of small finite cardinality extrapolates in the
> "obvious" way to physical quantities with arbitrarily large finite
> cardinality.

If not all naturals exist in this sense, there is a largest existing 
natural number, which would be an invariant of the universe (or perhaps 
a discontinuous function of time).

In particular, the set of existing natural numbers would violate Peano's 
axiom that every natural number has a successor.

It this really what you have in mind?

Or is there a doubt mapping that assigns to each natural number a doubt 
value, such that small natural numbers have doubt value zero, and the 
doubt increases to large values if numbers get really huge? Would this 
doubt function depend on who doubts, and thus be subjective? Or is it a 
property of the universe only? Neither option looks very convincing....


Arnold Neumaier