Ideas from 'Paradoxes of the Infinite' by Bernard Bolzano [1846], by Theme Structure

green numbers give full details    |     back to texts     |     unexpand these ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory
An aggregate in which order does not matter I call a 'set'
                        Full Idea: An aggregate whose basic conception renders the arrangement of its members a matter of indifference, and whose permutation therefore produces no essential difference, I call a 'set'.
                        From: Bernard Bolzano (Paradoxes of the Infinite [1846], 4), quoted by William W. Tait - Frege versus Cantor and Dedekind IX
                        A reaction: The idea of 'sets' was emerging before Cantor formalised it, and clarified it by thinking about infinite sets. Nowadays we also have 'ordered' sets, which rather contradicts Bolzano, and we also expect the cardinality to be determinate.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
A truly infinite quantity does not need to be a variable
                        Full Idea: A truly infinite quantity (for example, the length of a straight line, unbounded in either direction) does not by any means need to be a variable.
                        From: Bernard Bolzano (Paradoxes of the Infinite [1846]), quoted by Brian Clegg - Infinity: Quest to Think the Unthinkable 10
                        A reaction: This is an important idea, followed up by Cantor, which relegated to the sidelines the view of infinity as simply something that could increase without limit. Personally I like the old view, but there is something mathematically stable about infinity.