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**1 - 7**of**7**### 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

### Non-Standard Models of Arithmetic: a Philosophical and Historical perspective

, 2010

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### Characterizable Cardinals

"... Abstract. The notions of finite and infinite second-order characterizability of cardinal and ordinal numbers are developed. Several known results for the case of finite characterizability are extended to infinite characterizability, and investigations of the secondorder theory of ordinals lead to so ..."

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Abstract. The notions of finite and infinite second-order characterizability of cardinal and ordinal numbers are developed. Several known results for the case of finite characterizability are extended to infinite characterizability, and investigations of the secondorder theory of ordinals lead to some observations about the Fraenkel-Carnap question for well-orders and about the relationship between ordinal characterizability and ordinal arithmetic. The broader significance of cardinal characterizability and the relationships between different notions of characterizability are also discussed.

### 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.

### Abstract: Von Neumann's Birth Centenary Conference. Computational Power for Social Research 2003 How Do Metalogical Concepts Emerge?

"... We are going to share with the audience a few reflections about the origin (emergence) and development of some metalogical concepts, first of all those of categoricity and completeness. The idea of an unique description of some fundamental structures from arithmetics and geometry can be found in the ..."

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We are going to share with the audience a few reflections about the origin (emergence) and development of some metalogical concepts, first of all those of categoricity and completeness. The idea of an unique description of some fundamental structures from arithmetics and geometry can be found in the works of the American Postulate Theorists (e.g. Veblen, Huntington) as well as in the pioneering works of Dedekind, Peano and Hilbert. The concepts of categoricity and completeness were intertwined at the very beginning; this situation culminated in the Gabelbarkeitssatz proposed by Carnap in 1928. The problem of completeness (of a system of logic) was in the meantime approached and solved (Bernays 1918, Post 1920, Hilbert and Ackermann 1928, Gödel 1930). Incompleteness of most important deductive theories has been established (Gödel 1931), thus showing the limitations of the Hilbert's Program. First-order logic became a standard. Tarski has codified the foundations of metalogic; in particular, connections of (several versions of) categoricity and completeness with other concepts (e.g. that of a logical constant) have been clarified. The importance of the compactness property became evident. Some fifty years ago one could observe a revival of logical systems stronger than first-order logic. As a later consequence of this, discussions about which logic is the logic became of new interest (e.g. the first-order

### The Scope of Gödel’s First Incompleteness Theorem

"... Abstract. Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of ..."

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Abstract. Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of