Abstract
We use model-theoretic ideas to present a perspicuous and versatile method of constructing full satisfaction classes on models of Peano arithmetic. We also comment on the ramifications of our work on issues related to conservativity and interpretability.
This research was partially supported by a grant from the Descartes Center of Utrecht University, which supported the first author’s visit to Utrecht to work closely with the second author.
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Notes
- 1.
- 2.
A recent noteworthy paper in this connection is McGee’s (2003).
- 3.
All of the formulae in the list can be arranged to be1-formulae in the sense of Definition 2.4.
- 4.
Throughout the paper we use the convention of using M, \(M_{0},\) N, etc. to denote the universes of discourse of structures \(\mathcal{M}\), \(\mathcal{M}_{0},\) \(\mathcal{N},\) etc.
- 5.
Note that the closure of F under direct subformulae does not guarantee that F should also contain ‘infinitely deep’ subformulae of a nonstandard formula in F.
- 6.
As shown in (Enayat and Visser) this feature can be exploited to construct ‘pathological’ satisfaction classes, such as the one mentioned at the end of Sect. 6 of this paper.
- 7.
\( PA^{ FT}\) is the relational analogue of the theory of \(CT\hspace{-0.05in}\) in Halbach’s monograph (Halbach 2011). The base theory of \(CT \hspace{-0.05in}\) is \(PA\) formulated in a functional language. The conservativity of \(CT\hspace{-0.05in}\) over the functional language version of \(PA\) can also be established using the techniques of this paper (see Sect. 16.6).
- 8.
Note that an extensional satisfaction predicate need not be closed under re-naming of bound variables.
- 9.
Here $\alpha [t:e]$ is the assignment obtained by redefining the value of $ \alpha $ at the variable $t$ to be $e$ if $t\in \mathrm{Dom}(\alpha )$; note that $\alpha [t:e]=\alpha $ if $t\not\in \mathrm{Dom}(\alpha ).$
- 10.
More specifically, first define ° on C by \(d^{\prime }\vartriangleleft ^{\circ }c\) iff \(d^{\prime }\vartriangleleft ^{*}c^{\prime }\boldsymbol{\approx} c\), for some \(c^{\prime}\in C.\) Since ° is cycle-free, C is well-founded, and therefore lends itself to a ranking function \(rank_{C}^{\circ}(c).\) Let \(n=\max \left\{ rank_{C}^{\circ}(c):c\in C\right\},\) and for \(0\leq i\leq n\) define \(D_{i}:=\{c\in C: rank_{C}^{\circ}(c)=i\}\). Next use a ‘backward’ recursion to define \(E_{n},E_{n-1},\cdot \cdot \cdot,E_{0}\) via:
-
\(E_{n}:=D_{n};\)
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\(E_{n-(i+1)}:=D_{n-(i+1)}\cup \{d:d ^{\mathcal{N} _{0}}c\) for some \(c\in E_{n-i}\}.\)
Finally, let \(\overline{C}:=E_{n}\cup \cdot \cdot \cdot \cup E_{0}.\) It is easy to see that \(\overline{C}\) is finite, extends C, and has the desired closure property.
-
- 11.
Indeed \(B^{ FS}\) turns out to be interpretable in \(B\) for all base theories Bthat have access to the full scheme of induction over their ambient ‘numbers’. In particular, \(ACA^{ FS}\) is interpretable in \(ACA.\) On the other hand, as shown in (Enayat and Visser, Sect. 8), \(ACA_{0}^{ FS}\) is not interpretable in ACA 0 (more generally, \(B^{ FS}\) is shown to be not interpretable in \(B\), if \(B\) is finitely axiomatizable).
- 12.
\( \mathrm{LL}_{1}\)-sets are a special type of ‘low’ sets.
- 13.
- 14.
Indeed, by using the technique of Friedman (1999), this conservativity result is already verifiable in the fragment \(SEFA\) (Superexponential Arithmetic) of \(PRA\).
- 15.
Halbach’s base theory in his work is the usual version of \(PA\) that is formulated in a functional language.
- 16.
We are grateful to Graham Leigh for his kind permission to quote his unpublished work here.
- 17.
As remarked in the last sentence of (Kotlarski et al.1981), this condition can also be arranged using the machinery of \(\mathcal{M}\)-logic. Note that ‘axioms of \(PA\)’ in the sense used here do not include the logical axioms.
- 18.
\( c^{\prime}\) is an alphabetic variant of c if \(c^{\prime}\) is obtainable from c by the usual rules of re-naming the bound variables of c.
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Enayat, A., Visser, A. (2015). New Constructions of Satisfaction Classes. In: Achourioti, T., Galinon, H., MartÃnez Fernández, J., Fujimoto, K. (eds) Unifying the Philosophy of Truth. Logic, Epistemology, and the Unity of Science, vol 36. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9673-6_16
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