69 research outputs found
Troubles with (the concept of) truth in mathematics
In the paper the problem of definability and undefinability of the concept of satisfaction and truth is considered. Connections between satisfaction and truth on the one hand and consistency of certain systems of omega-logic and transfinite induction on the other are indicated
Tarski's Undefinability Theorem and Diagonal Lemma
We prove the equivalence of the semantic version of Tarski's theorem on the
undefinability of truth with a semantic version of the Diagonal Lemma, and also
show the equivalence of syntactic Tarski's Undefinability Theorem with a weak
syntactic diagonal lemma. We outline two seemingly diagonal-free proofs for
these theorems from the literature, and show that syntactic Tarski's theorem
can deliver G\"odel-Rosser's Incompleteness Theorem.Comment: 8 page
Satisfaction classes in nonstandard models of first-order arithmetic
A satisfaction class is a set of nonstandard sentences respecting Tarski's
truth definition. We are mainly interested in full satisfaction classes, i.e.,
satisfaction classes which decides all nonstandard sentences. Kotlarski,
Krajewski and Lachlan proved in 1981 that a countable model of PA admits a
satisfaction class if and only if it is recursively saturated. A proof of this
fact is presented in detail in such a way that it is adaptable to a language
with function symbols. The idea that a satisfaction class can only see finitely
deep in a formula is extended to terms. The definition gives rise to new
notions of valuations of nonstandard terms; these are investigated. The notion
of a free satisfaction class is introduced, it is a satisfaction class free of
existential assumptions on nonstandard terms.
It is well known that pathologies arise in some satisfaction classes. Ideas
of how to remove those are presented in the last chapter. This is done mainly
by adding inference rules to M-logic. The consistency of many of these
extensions is left as an open question.Comment: Thesis for the degree of licentiate of philosophy, 74 pages, 4
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Curious satisfaction classes
We present two new constructions of satisfaction/truth classes over models of
PA (Peano Arithmetic) that provide a foil to the fact that the existence of a
disjunctively correct full truth class over a model M of PA implies that
Con(PA) holds in M.Comment: 12 page
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