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The modal logic of arithmetic potentialism and the universal algorithm
I investigate the modal commitments of various conceptions of the philosophy
of arithmetic potentialism. Specifically, I consider the natural potentialist
systems arising from the models of arithmetic under their natural extension
concepts, such as end-extensions, arbitrary extensions, conservative extensions
and more. In these potentialist systems, I show, the propositional modal
assertions that are valid with respect to all arithmetic assertions with
parameters are exactly the assertions of S4. With respect to sentences,
however, the validities of a model lie between S4 and S5, and these bounds are
sharp in that there are models realizing both endpoints. For a model of
arithmetic to validate S5 is precisely to fulfill the arithmetic maximality
principle, which asserts that every possibly necessary statement is already
true, and these models are equivalently characterized as those satisfying a
maximal theory. The main S4 analysis makes fundamental use of the
universal algorithm, of which this article provides a simplified,
self-contained account. The paper concludes with a discussion of how the
philosophical differences of several fundamentally different potentialist
attitudes---linear inevitability, convergent potentialism and radical branching
possibility---are expressed by their corresponding potentialist modal
validities.Comment: 38 pages. Inquiries and commentary can be made at
http://jdh.hamkins.org/arithmetic-potentialism-and-the-universal-algorithm.
Version v3 has further minor revisions, including additional reference
Initial segments and end-extensions of models of arithmetic
This thesis is organized into two independent parts. In the first part, we extend the recent work on generic cuts by Kaye and the author. The focus here is the properties of the pairs (M, I) where I is a generic cut of a model M. Amongst other results, we characterize the theory of such pairs, and prove that they are existentially closed in a natural category. In the second part, we construct end-extensions of models of arithmetic that are at least as strong as ATR. Two new constructions are presented. The first one uses a variant of Fodor’s Lemma in ATR to build an internally rather classless model. The second one uses some weak versions of the Galvin–Prikry Theorem in adjoining an ideal set to a model of second-order arithmetic
On end extensions of models of subsystems of peano arithmetic
AbstractWe survey results and problems concerning subsystems of Peano Arithmetic. In particular, we deal with end extensions of models of such theories. First, we discuss the results of Paris and Kirby (Logic Colloquium ’77, North-Holland, Amsterdam, 1978, pp. 199–209) and of Clote (Fund. Math. 127 (1986) 163; Fund. Math. 158 (1998) 301), which generalize the MacDowell and Specker theorem (Proc. Symp. on Foundation of Mathematics, Warsaw, 1959, Pergamon Press, Oxford, 1961, p. 257–263) we also discuss a related problem of Kaufmann (On existence of Σn end extensions, Lecture Notes in Mathematics, Vol. 859, Springer, Berlin, 1980, pp. 92). Then we sketch an alternative proof of Clote's theorem, using the arithmetized completeness theorem in the spirit of McAloon (Trans. Amer. Math. Soc. 239 (1978) 253) and Paris (Some conservation results for fragments of arithmetic, Lecture Notes in Mathematics, Vol. 890, Springer, Berlin, 1981, p. 251)
Fragments of Arithmetic and true sentences
By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class
of the ¦n+1–sentences true in the standard model is the only (up to deductive
equivalence) consistent ¦n+1–theory which extends the scheme of induction for
parameter free ¦n+1–formulas. Motivated by this result, we present a systematic
study of extensions of bounded quantifier complexity of fragments of first–order
Peano Arithmetic. Here, we improve that result and show that this property describes
a general phenomenon valid for parameter free schemes. As a consequence,
we obtain results on the quantifier complexity, (non)finite axiomatizability and
relative strength of schemes for ¢n+1–formulas.Junta de Andalucía TIC-13
The complexity of classification problems for models of arithmetic
We observe that the classification problem for countable models of arithmetic
is Borel complete. On the other hand, the classification problems for finitely
generated models of arithmetic and for recursively saturated models of
arithmetic are Borel; we investigate the precise complexity of each of these.
Finally, we show that the classification problem for pairs of recursively
saturated models and for automorphisms of a fixed recursively saturated model
are Borel complete.Comment: 15 page
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