Concatenation as a basis for Q and the Intuitionistic variant of Nelson's Classic Result
Rachel Sterken
Abstract:
Visser shows that a first-order theory is sequential (has ‘global
coding’) iff (roughly) it directly interprets a weak set theory,
AS. The programme behind Visser’s result is to find coordinate free
ways of thinking about notions of coding. In this thesis, we add some
results to Visser’s programme for the case of ‘local coding’.
Since Robinson’s arithmetic, Q, is mutually interpretable (but not
directly) with AS and Q is in a sense the minimal arithmetical theory
that yields enough coding to prove G¨odel’s Second Incompleteness
Theorem, we propose whether a theory interprets Q as a
characterization of the notion of ‘local coding’.
We also investigate other candidates in place of Q. We show that a
basic theory of strings, TC_Q interprets Q.
We, in addition, verify that the classical result of Nelson works in
the constructive case by showing that iQ interprets iI\Delta_0+\Omega_1.
This result entails that our characterization of ‘local coding’ also
works in the constructive case.