Abstract
We investigate a weakening of the classical theory of Frege structures and extensions thereof which naturally interpret (predicative) theories of explicit types and names à la Jäger.
Dedicated to Gerhard Jäger on occasion of his 60th birthday.
This paper originates from the slides for the talk presented at the Jäger conference, Bern, December 12–13, 2013. We wish to thank the organizers for the nice hospitality. Thanks to an anonymous referee for comments and criticism.
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Notes
- 1.
E.g. \(\dot{\exists }f:=\dot{\lnot }(\dot{\forall }(\lambda x.\dot{\lnot }(fx)))\); \(\dot{\rightarrow }\) is assumed as primitive in Sect. 4.
- 2.
We use the standard abbreviations \(\langle t,s \rangle :=\mathsf {PAIR}ts\); \((t)_0:=\mathsf {LEFT}t \), \((t)_1:=\mathsf {RIGHT}t\). Below 1, 2, ... stand for the corresponding numerals.
- 3.
- 4.
This makes sense, since we can identify individual variables of \(\mathbf {{EET}}\) with \(\mathbf {CT}\)-variables with odd indices, and type variables of \(\mathbf {{EET}}\) with \(\mathbf {CT}\)-variables with even indices.
- 5.
Concerning (32), we are thus left only with \(\forall x P([P(x)])\).
- 6.
- 7.
- 8.
This is a sequential conjunction introduced by Aczel in [1].
- 9.
Indeed Feferman [10], noting that Aczel’s approach is based on \(\lambda \)-calculus which allows for more general interpretations, adds that “further work on systems like \(\mathbf {{DT}}\) might usefully incorporate similar features.”
- 10.
Recall footnote 2 of Sect. 2.4: think of \(\langle a,b \rangle \), \((u)_0\), \((u)_1\) as values of the terms \(\mathsf {\mathsf {PAIR}} a b \), \(\mathsf {LEFT} u\), \(\mathsf {RIGHT} u\).
- 11.
In the standard sense, see [25].
- 12.
\(\mathbf {FL}\) is reminiscent of Feferman’s logic.
- 13.
In general, if \(\Gamma :=\lbrace A_1,\ldots , A_q\rbrace \), \(\Gamma [m, n]:=\lbrace A_1[m,n],\ldots , A_q[m,n]\rbrace \).
- 14.
For a precise definition, see [11].
- 15.
- 16.
This means: the set of all \(a\in \mathcal M\) satisfying T(I(A)x in \(\mathsf {MIN_M}\) is the least fixed point of the operator defined by A in \(\mathsf {MIN_M}\).
- 17.
As for \(\mathbf {KF}\), a warning: we keep using the same label of [2] for a theory \({\mathbf {KF}}_\mu \), which is not an extension of Peano Arithmetic.
- 18.
So \(\prec \) is determinate in the sense of (2).
- 19.
We will not spell them explicitly for the sake of brevity.
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Cantini, A. (2016). About Truth and Types. In: Kahle, R., Strahm, T., Studer, T. (eds) Advances in Proof Theory. Progress in Computer Science and Applied Logic, vol 28. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-29198-7_2
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