340 research outputs found
From Tarski to G\"odel. Or, how to derive the Second Incompleteness Theorem from the Undefinability of Truth without Self-reference
In this paper, we provide a fairly general self-reference-free proof of the
Second Incompleteness Theorem from Tarski's Theorem of the Undefinability of
Truth.Comment: 7 page
Arithmetic, Set Theory, Reduction and Explanation
Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences
On interpretations of bounded arithmetic and bounded set theory
In a recent paper, Kaye and Wong proved the following result, which they
considered to belong to the folklore of mathematical logic.
THEOREM: The first-order theories of Peano arithmetic and ZF with the axiom
of infinity negated are bi-interpretable: that is, they are mutually
interpretable with interpretations that are inverse to each other.
In this note, I describe a theory of sets that stands in the same relation to
the bounded arithmetic IDelta0 + exp. Because of the weakness of this theory of
sets, I cannot straightforwardly adapt Kaye and Wong's interpretation of
arithmetic in set theory. Instead, I am forced to produce a different
interpretation.Comment: 12 pages; section on omega-models removed due to error; references
added and typos correcte
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