CARNAP ON EXTREMAL AXIOMS, “COMPLETENESS OF THE MODELS,” AND CATEGORICITY
- PhilosophyThe Review of Symbolic Logic
The paper surveys Carnap’s different attempts to explicate the extremal properties of a theory and puts his results in context with related metamathematical research at the time.
Fraenkel's Axiom of Restriction: axiom choice, intended models, and categoricity
A recent debate has focused on different methodological principles underlying the practice of axiom choice in mathematics (cf. Feferman et al., 2000; Maddy, 1997; Easwaran, 2008). The general aim of…
Fermat’s last theorem proved in Hilbert arithmetic. II. Its proof in Hilbert arithmetic by the Kochen-Specker theorem with or without induction
Axioms of Set Theory
We shall introduce and discuss in this chapter the axioms of Zermelo–Fraenkel Set Theory with the Axiom of Choice.
The Axioms of Zermelo–Fraenkel Set Theory
In the middle and late 19th century, members of the then small mathematical community began to look for a rigorous foundation of Mathematics, but at the time it was not clear what assumptions should be made and what operations should be allowed in mathematical reasoning.
Early history of the Generalized Continuum Hypothesis: 1878 - 1938
- PhilosophyBull. Symb. Log.
This paper explores how the Generalized Continuum Hypothesis (GCH) arose from Cantor's Continuum Hypothesis in the work of Peirce, Jourdain, Hausdorff, Tarski, and how GCH was used up to Godel's…
Der axiomatische Aufbau der Mengenlehre. Die axiomatische Methode
Wir gehen nun zur ausfuhrlichen Darstellung einer der modernen Begrundungen der Mengenlehre uber, namlich der, die von E. Zermelo stammt. Sie hat neben manchem anderen auch den wesentlichen Vorzug…
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A system of axioms for geometry
CHAPTER I. The axioms and their independence. Introductory statement of the axioms . . .1 Categorical and disjunctive systems . . . ? 2 Independence proofs and historical remarks on axioms IX-XII. ?…