display all the ideas for this combination of philosophers
3 ideas
9986 | The null set was doubted, because numbering seemed to require 'units' [Tait] |
Full Idea: The conception that what can be numbered is some object (including flocks of sheep) relative to a partition - a choice of unit - survived even in the late nineteenth century in the form of the rejection of the null set (and difficulties with unit sets). | |
From: William W. Tait (Frege versus Cantor and Dedekind [1996], IX) | |
A reaction: This old view can't be entirely wrong! Frege makes the point that if asked to count a pack of cards, you must decide whether to count cards, or suits, or pips. You may not need a 'unit', but you need a concept. 'Units' name concept-extensions nicely! |
18189 | ZFC could contain a contradiction, and it can never prove its own consistency [MacLane] |
Full Idea: We have at hand no proof that the axioms of ZFC for set theory will never yield a contradiction, while Gödel's second theorem tells us that such a consistency proof cannot be conducted within ZFC. | |
From: Saunders MacLane (Mathematics: Form and Function [1986], p.406), quoted by Penelope Maddy - Naturalism in Mathematics | |
A reaction: Maddy quotes this, while defending set theory as the foundation of mathematics, but it clearly isn't the most secure foundation that could be devised. She says the benefits of set theory do not need guaranteed consistency (p.30). |
9984 | We can have a series with identical members [Tait] |
Full Idea: Why can't we have a series (as opposed to a linearly ordered set) all of whose members are identical, such as (a, a, a...,a)? | |
From: William W. Tait (Frege versus Cantor and Dedekind [1996], VII) | |
A reaction: The question is whether the items order themselves, which presumably the natural numbers are supposed to do, or whether we impose the order (and length) of the series. What decides how many a's there are? Do we order, or does nature? |