Re: Second order arithmetic
- From: aatu.koskensilta@xxxxxxxxx
- Date: Thu, 11 Oct 2007 11:14:37 -0700
More specifically, obviously the formal system of second order
arithmetic can't give us an algorithmic procedure for deciding the
truth of arbitrary arithmetical statements. What kind of arithmetical
statement, then, is unprovable by the means of second order arithmetic?
The statement "this statement is not derivable using the given rules
of inference and axioms of arithmetic" is, among others, not provable
using the given rules of inference and axioms of arithmetic. There is
no essential difference between first-order and second-order
arithmetic in this respect. In case of second-order logic, which
axioms and rules of inference we choose to adopt as logically valid
affects which theorems of number theory are provable, but whichever
system of deduction we adopt, Gödelian arguments apply -- we can, with
some care, treat the system, in so far as their proof-theoretic
properties are concerned, as multi-sorted first-order systems, in
which the logical rules of inference and logical axioms are taken as
non-logical first-order axioms, and carry out the familiar Gödelian
arguments.
--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.
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