[FOM] Complexity of Constructive Truth
Dmytro Taranovsky
dmytro at MIT.EDU
Sun Jun 4 12:45:42 EDT 2006
Constructive arithmetic is about natural numbers and constructions, so
it should be clear and well-defined. However, different constructivists
accepted different arithmetical principles, and there was no
satisfactory notion of constructive arithmetical truth that is
compatible with classical truth.
My paper "Constructive Mathematical Truth", available at
http://arxiv.org/abs/math.LO/0605138
defines constructive truth for arithmetic and (intuitionistic) analysis
and explores its properties. A statement is constructively true iff it
is realized by a recursive function under Kleene's function
realizability.
I prove that the set of constructively true (first order) arithmetical
statements is Pi-1-2 and Sigma-1-2 hard, and conjecture it to be
complete for second order arithmetic.
The high complexity stems from the precise meaning of constructive
implication. I have previously written on FOM that constructive
arithmetical truth is Pi-1-1 complete; however, the definition given
there was imprecise; and a variation on constructive truth, called
narrow constructive truth, is Pi-1-1 complete.
Dmytro Taranovsky
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