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Universiteit Utrecht

1. Joosten, Joost Johannes. Interpretability formalized.

Degree: 2004, Universiteit Utrecht

URL: http://dspace.library.uu.nl:8080/handle/1874/1275

The dissertation is in the first place a treatment of mathematical interpretations. Interpretations themselves will be studied, but also shall they be used to study formal theories.
Interpretations, when used in comparing theories, tell us, in a natural way, something about proof-strength of formal theories. Roughly, an interpretation j of a theory T into a theory S is a structure-preserving map, mapping axioms of T to theorems of S. Structure-preserving means that the map should com-mute with proof constructions and with logical connectives. For example, the constraints on the map should exclude the possibility that we simply map all axioms of T to some tautology of S, say 1 = 1. The notion of interpretahil- ity that is studied in the thesis, is the notion of relativized interpretability as studied by Tarski et al. in [TMR53].
In the thesis, only interpretations between first order theories of some mini-mal strength will be considered. As said before, interpretations will be used to study theories, but also shall they be the subject of our study. In the latter case, the emphasis lies on the structural behavior of interpretahility. This behavior is manifested in so-called interpretability logics. The dissertation is organised in three parts.
Part one
In the first part we introduce the notion of interpretability, and relate it to other important meta-mathematical notions. This results in the well-known Orey-Hajék characterizations of interpretability. Central notions in these characterizations are consistency statements, definable cuts and II1-conservativity.
Interpretability, being a purely syntactical notion, is formalized, and also the formalizations take place in a completely formalized setting. For every implication in the characterizations, we will carefully spell out the conditions on the meta-theory. The characterizations get an especially elegant form when formulated in terms of category theory.
At the end of the first part we will focus on interpretahility logics. In es-pecially, we shall be interested in a modal characterization of IL(All), the interpretability logic of all reasonable arithmetical theories. We present a new principle R for this logic and prove its arithmetical validity. This correctness is proved in two different ways. We present two modal systems corresponding to these different proof methods. We see that all principles in IL(All), known so far, are provable in both modal systems, again, giving rise to two different soundness proofs.
Part two
In the section part of the thesis we fully concentrate on the modal semantics for interpretability logics. A central question is the modal completeness of interpretability logics. We give completeness proofs for the logics IL, ILM, ILM0, ILW and ILW*. The completeness proofs for ILM0 and for ILW* can he seen as first proofs. We also expose some applications of completeness proofs.
We try to develop a uniform format for completeness proofs for interpretability logics. However, there still has to be done…

Subjects/Keywords: Wijsbegeerte; provability logics; interpretability; formal arithmetic; Primitive Recursive Arithmetic; PRA; Modal Semantics

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Joosten, J. J. (2004). Interpretability formalized. (Doctoral Dissertation). Universiteit Utrecht. Retrieved from http://dspace.library.uu.nl:8080/handle/1874/1275

Chicago Manual of Style (16^{th} Edition):

Joosten, Joost Johannes. “Interpretability formalized.” 2004. Doctoral Dissertation, Universiteit Utrecht. Accessed June 20, 2019. http://dspace.library.uu.nl:8080/handle/1874/1275.

MLA Handbook (7^{th} Edition):

Joosten, Joost Johannes. “Interpretability formalized.” 2004. Web. 20 Jun 2019.

Vancouver:

Joosten JJ. Interpretability formalized. [Internet] [Doctoral dissertation]. Universiteit Utrecht; 2004. [cited 2019 Jun 20]. Available from: http://dspace.library.uu.nl:8080/handle/1874/1275.

Council of Science Editors:

Joosten JJ. Interpretability formalized. [Doctoral Dissertation]. Universiteit Utrecht; 2004. Available from: http://dspace.library.uu.nl:8080/handle/1874/1275

2. Joosten, Joost Johannes. Interpretability formalized.

Degree: 2004, University Utrecht

URL: http://dspace.library.uu.nl/handle/1874/1275 ; URN:NBN:NL:UI:10-1874-1275 ; urn:isbn:90-393-3869-8 ; URN:NBN:NL:UI:10-1874-1275 ; http://dspace.library.uu.nl/handle/1874/1275

The dissertation is in the first place a treatment of mathematical interpretations. Interpretations themselves will be studied, but also shall they be used to study formal theories.
Interpretations, when used in comparing theories, tell us, in a natural way, something about proof-strength of formal theories. Roughly, an interpretation j of a theory T into a theory S is a structure-preserving map, mapping axioms of T to theorems of S. Structure-preserving means that the map should com-mute with proof constructions and with logical connectives. For example, the constraints on the map should exclude the possibility that we simply map all axioms of T to some tautology of S, say 1 = 1. The notion of interpretahil- ity that is studied in the thesis, is the notion of relativized interpretability as studied by Tarski et al. in [TMR53].
In the thesis, only interpretations between first order theories of some mini-mal strength will be considered. As said before, interpretations will be used to study theories, but also shall they be the subject of our study. In the latter case, the emphasis lies on the structural behavior of interpretahility. This behavior is manifested in so-called interpretability logics. The dissertation is organised in three parts.
Part one
In the first part we introduce the notion of interpretability, and relate it to other important meta-mathematical notions. This results in the well-known Orey-Hajék characterizations of interpretability. Central notions in these characterizations are consistency statements, definable cuts and II1-conservativity.
Interpretability, being a purely syntactical notion, is formalized, and also the formalizations take place in a completely formalized setting. For every implication in the characterizations, we will carefully spell out the conditions on the meta-theory. The characterizations get an especially elegant form when formulated in terms of category theory.
At the end of the first part we will focus on interpretahility logics. In es-pecially, we shall be interested in a modal characterization of IL(All), the interpretability logic of all reasonable arithmetical theories. We present a new principle R for this logic and prove its arithmetical validity. This correctness is proved in two different ways. We present two modal systems corresponding to these different proof methods. We see that all principles in IL(All), known so far, are provable in both modal systems, again, giving rise to two different soundness proofs.
Part two
In the section part of the thesis we fully concentrate on the modal semantics for interpretability logics. A central question is the modal completeness of interpretability logics. We give completeness proofs for the logics IL, ILM, ILM0, ILW and ILW*. The completeness proofs for ILM0 and for ILW* can he seen as first proofs. We also expose some applications of completeness proofs.
We try to develop a uniform format for completeness proofs for interpretability logics. However, there still has to be done…

Subjects/Keywords: provability logics; interpretability; formal arithmetic; Primitive Recursive Arithmetic; PRA; Modal Semantics

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Joosten, J. J. (2004). Interpretability formalized. (Doctoral Dissertation). University Utrecht. Retrieved from http://dspace.library.uu.nl/handle/1874/1275 ; URN:NBN:NL:UI:10-1874-1275 ; urn:isbn:90-393-3869-8 ; URN:NBN:NL:UI:10-1874-1275 ; http://dspace.library.uu.nl/handle/1874/1275

Chicago Manual of Style (16^{th} Edition):

Joosten, Joost Johannes. “Interpretability formalized.” 2004. Doctoral Dissertation, University Utrecht. Accessed June 20, 2019. http://dspace.library.uu.nl/handle/1874/1275 ; URN:NBN:NL:UI:10-1874-1275 ; urn:isbn:90-393-3869-8 ; URN:NBN:NL:UI:10-1874-1275 ; http://dspace.library.uu.nl/handle/1874/1275.

MLA Handbook (7^{th} Edition):

Joosten, Joost Johannes. “Interpretability formalized.” 2004. Web. 20 Jun 2019.

Vancouver:

Joosten JJ. Interpretability formalized. [Internet] [Doctoral dissertation]. University Utrecht; 2004. [cited 2019 Jun 20]. Available from: http://dspace.library.uu.nl/handle/1874/1275 ; URN:NBN:NL:UI:10-1874-1275 ; urn:isbn:90-393-3869-8 ; URN:NBN:NL:UI:10-1874-1275 ; http://dspace.library.uu.nl/handle/1874/1275.

Council of Science Editors:

Joosten JJ. Interpretability formalized. [Doctoral Dissertation]. University Utrecht; 2004. Available from: http://dspace.library.uu.nl/handle/1874/1275 ; URN:NBN:NL:UI:10-1874-1275 ; urn:isbn:90-393-3869-8 ; URN:NBN:NL:UI:10-1874-1275 ; http://dspace.library.uu.nl/handle/1874/1275