[FOM] Elementaricity of elementaricity

[Aatu Koskensilta]

On Jul 28, 2004, at 9:37 PM, I. Natochdag wrote:

> Conservative extensions of PA are usable
> for complex-variable analysis: how can they axiomatic-methodologically
> use Logarithms and e? (I ask this out of idiotic ignorance of any such
> works).

I am not sure I understand all of your questions, so I might be 
answering to something you didn't
actually ask. A natural conservative extension of PA is ACA_0, which is 
a subsystem of second
order arithmetic (i.e. you have variables for natural numbers and 
variables for sets of
natural numbers) in which comprehension is restricted to arithmetical 
formulae. Weyl showed
in his das Kontinuum that you can do a lot of analysis in this system 
(which wasn't called ACA_0 then,
of course).

Real numbers can be defined as Cauchy sequences in the standard manner 
and piecewise
continous functions can be represented by noticing that they are 
completely determined by their
values at rational points, and can thus be represented as sets of 
natural numbers. In particular,
the real e and the logarithm function can be defined and proved to 
exist.

-- 
Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus