[FOM] Elementaricity of elementaricity
Aatu Koskensilta
aatu.koskensilta at xortec.fi
Thu Jul 29 07:33:51 EDT 2004
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
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