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Reflections on Skolem's Relativity of SetTheoretical Concepts!
"... From 1922 onwards Skolem maintained that settheoretical concepts are relative (in a sense of 'relative ' that we must discern). In 1958 he viewed all mat.homatlcal notions as relative. The main instrument he used in his argument for settheoretical relativity is the UiwenheimSkolem theor ..."
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From 1922 onwards Skolem maintained that settheoretical concepts are relative (in a sense of 'relative ' that we must discern). In 1958 he viewed all mat.homatlcal notions as relative. The main instrument he used in his argument for settheoretical relativity is the UiwenheimSkolem theorem, in the form that every consistent firstorder countable set of sentences has a countable model. In 1938 he said of this theorem that 'its most important application is the critique of the settheoretical concepts, and most especially that of the higher infinite powers ' (Skolem [1941], p. 460).1 My goal is to review Skolem's argument, chiefly as it appears in Skolem [1922], with the aim of gaining some insight into the ontology of set theory. Although I will often quote Skolem and try to be true to his words, what follows is not put forward as a faithful reconstruction of Skolem's actual view, but rather as an attempt to present it as a sensible one. Skolem is commonly portrayed as arguing that certain otherwise well understood concepts are suspect simply because they cannot be characterized in a firstorder language; in particular that, since all firstorder formalizations of set theory (if consistent) have countable models, the concept of uncountability is flawed. I hope to show that Skolem's position is more solid than that. I see Skolem as arguing that all the evidence that has been given for the existence of uncountable sets is inconclusive, and the reason why he insists on considering countable models is that axiomatization was put forward at the time as the only way to secure set theory, and what sets are and which sets exist was claimed to be determined by the axioms and their models (much as what Euclidean geometry is about was claimed to be determined by Hilbert's axioms and their models). In this situation, bringing countable models into play was perfectly in order, all the more so as no other models could be supplied without settheoretical means. Today
NonStandard Models of Arithmetic: a Philosophical and Historical perspective
, 2010
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JACQUES HERBRAND: LIFE, LOGIC, AND AUTOMATED DEDUCTION
"... The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1 ..."
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The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.