# Models of weak theories of truth

Models of weak theories of truth In the following paper we propose a model-theoretical way of comparing the “strength” of various truth theories which are conservative over $$PA$$ P A . Let $${\mathfrak {Th}}$$ Th denote the class of models of $$PA$$ P A which admit an expansion to a model of theory $${ Th}$$ T h . We show (combining some well known results and original ideas) that \begin{aligned} {{\mathfrak {PA}}}\supset {\mathfrak {TB}}\supset {{\mathfrak {RS}}}\supset {\mathfrak {UTB}}\supseteq \mathfrak {CT^-}, \end{aligned} PA ⊃ TB ⊃ RS ⊃ UTB ⊇ CT - , where $${\mathfrak {PA}}$$ PA denotes simply the class of all models of $$PA$$ P A and $${\mathfrak {RS}}$$ RS denotes the class of recursively saturated models of $$PA$$ P A . Our main original result is that every model of $$PA$$ P A which admits an expansion to a model of $$CT ^-$$ C T - , admits also an expansion to a model of $$UTB$$ U T B . Moreover, as a corollary to one of the results (brought to us by Carlo Nicolai) we conclude that $$UTB$$ U T B is not relatively interpretable in $$TB$$ T B , thus answering the question from Fujimoto (Bull Symb Log 16:305–344, 2010). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

# Models of weak theories of truth

, Volume 56 (6) – Apr 22, 2017
22 pages

/lp/springer_journal/models-of-weak-theories-of-truth-hNPn1n5QHd
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
D.O.I.
10.1007/s00153-017-0531-1
Publisher site
See Article on Publisher Site

### Abstract

In the following paper we propose a model-theoretical way of comparing the “strength” of various truth theories which are conservative over $$PA$$ P A . Let $${\mathfrak {Th}}$$ Th denote the class of models of $$PA$$ P A which admit an expansion to a model of theory $${ Th}$$ T h . We show (combining some well known results and original ideas) that \begin{aligned} {{\mathfrak {PA}}}\supset {\mathfrak {TB}}\supset {{\mathfrak {RS}}}\supset {\mathfrak {UTB}}\supseteq \mathfrak {CT^-}, \end{aligned} PA ⊃ TB ⊃ RS ⊃ UTB ⊇ CT - , where $${\mathfrak {PA}}$$ PA denotes simply the class of all models of $$PA$$ P A and $${\mathfrak {RS}}$$ RS denotes the class of recursively saturated models of $$PA$$ P A . Our main original result is that every model of $$PA$$ P A which admits an expansion to a model of $$CT ^-$$ C T - , admits also an expansion to a model of $$UTB$$ U T B . Moreover, as a corollary to one of the results (brought to us by Carlo Nicolai) we conclude that $$UTB$$ U T B is not relatively interpretable in $$TB$$ T B , thus answering the question from Fujimoto (Bull Symb Log 16:305–344, 2010).

### Journal

Archive for Mathematical LogicSpringer Journals

Published: Apr 22, 2017

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