Combining Texts

All the ideas for 'Frege's Theory of Numbers', 'The Science of Rights' and 'An Axiomatization of Set Theory'

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4 ideas

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
15943 Limitation of Size is not self-evident, and seems too strong [Lavine on Neumann]
     Full Idea: Von Neumann's Limitation of Size axiom is not self-evident, and he himself admitted that it seemed too strong.
     From: comment on John von Neumann (An Axiomatization of Set Theory [1925]) by Shaughan Lavine - Understanding the Infinite VII.1
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
17447 Parsons says counting is tagging as first, second, third..., and converting the last to a cardinal [Parsons,C, by Heck]
     Full Idea: In Parsons's demonstrative model of counting, '1' means the first, and counting says 'the first, the second, the third', where one is supposed to 'tag' each object exactly once, and report how many by converting the last ordinal into a cardinal.
     From: report of Charles Parsons (Frege's Theory of Numbers [1965]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 3
     A reaction: This sounds good. Counting seems to rely on that fact that numbers can be both ordinals and cardinals. You don't 'convert' at the end, though, because all the way you mean 'this cardinality in this order'.
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
13672 All the axioms for mathematics presuppose set theory [Neumann]
     Full Idea: There is no axiom system for mathematics, geometry, and so forth that does not presuppose set theory.
     From: John von Neumann (An Axiomatization of Set Theory [1925]), quoted by Stewart Shapiro - Foundations without Foundationalism 8.2
     A reaction: Von Neumann was doubting whether set theory could have axioms, and hence the whole project is doomed, and we face relativism about such things. His ally was Skolem in this.
16. Persons / D. Continuity of the Self / 6. Body sustains Self
22063 Effective individuals must posit a specific material body for themselves [Fichte]
     Full Idea: Rational beings cannot posit themselves as effective individuals without ascribing to themselves a material body and determining it in doing so.
     From: Johann Fichte (The Science of Rights [1797], p.87), quoted by Ludwig Siep - Fichte
     A reaction: To be free entails a belief that one is 'effective', and a body is our only concept for that. This seems to be a transcendental proof that the body must exist, which is a neat inverted move! The Self sustains the body, for Fichte.