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**1 - 4**of**4**### Non-Standard Models of Arithmetic: a Philosophical and Historical perspective MSc Thesis (Afstudeerscriptie)

, 2010

"... 1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5 ..."

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1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5

### Frege, Russell and Wittgenstein on the judgment stroke

, 2011

"... Frege is highly valued as a logician by Russell and Wittgenstein, the latter nonetheless concludes in his Tractatus that one of Frege’s central notions, the judgment stroke, is “logically quite meaningless”. In order to see why Wittgenstein thinks so, we will investigate the ‘indirect interpretation ..."

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Frege is highly valued as a logician by Russell and Wittgenstein, the latter nonetheless concludes in his Tractatus that one of Frege’s central notions, the judgment stroke, is “logically quite meaningless”. In order to see why Wittgenstein thinks so, we will investigate the ‘indirect interpretation thesis’, which says that Wittgenstein’s interpretation of Frege was strongly influenced by the reading Russell gives of the Begriffsschrift in Principia Mathematica and Principles of Mathematics. This is done by analyzing the different conceptions of logic, focusing on the representations of judgment and assertion in Frege, Russell and the early Wittgenstein. Stong similarities can be found between the interpretations of Russell and Wittgenstein, this makes the indirect interpretation thesis plausible, although Russell’s influence cannot be the only reason why Wittgenstein rejected the judgment stroke as a logical symbol.

### Fraenkel–Carnap Questions for Equivalence Relations

, 2011

"... Abstract: An equivalence is a binary relational system A = (A, ρA) where ρA is an equivalence relation on A. A simple expansion of an equivalence is a system of the form (Aa1 an) were A is an equivalence and a1,..., an are members of A. It is shown that the Fraenkel–Carnap question when restricted t ..."

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Abstract: An equivalence is a binary relational system A = (A, ρA) where ρA is an equivalence relation on A. A simple expansion of an equivalence is a system of the form (Aa1 an) were A is an equivalence and a1,..., an are members of A. It is shown that the Fraenkel–Carnap question when restricted to the class of equivalences or to the class of simple expansions of equivalences has a positive answer: that the complete second-order theory of such a system is categorical, if it is finitely axiomatizable. 1 introduction In the late 1920’s Fraenkel and Carnap independently raised the question of whether or not every semantically complete, finitely axiomatizable theory is categorical. Carnap, but not Fraenkel, restricted attention to theories formulated in the simple theory of types. A positive answer to Carnap’s question implies that a finitely axiomatizable theory is semantically complete iff it is categorical. Carnap announced a positive answer. However, his proof was flawed. It appears that the question was forgotten until the beginning of this