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| 10859 | A set is 'well-ordered' if every subset has a first element |
| Full Idea: For a set to be 'well-ordered' it is required that every subset of the set has a first element. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10857 | Set theory made a closer study of infinity possible |
| Full Idea: Set theory made a closer study of infinity possible. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10864 | Any set can always generate a larger set - its powerset, of subsets |
| Full Idea: The idea of the 'power set' means that it is always possible to generate a bigger one using only the elements of that set, namely the set of all its subsets. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 10872 | Extensionality: Two sets are equal if and only if they have the same elements |
| Full Idea: Axiom of Extension: Two sets are equal if and only if they have the same elements. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10875 | Pairing: For any two sets there exists a set to which they both belong |
| Full Idea: Axiom of Pairing: For any two sets there exists a set to which they both belong. So you can make a set out of two other sets. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10876 | Unions: There is a set of all the elements which belong to at least one set in a collection |
| Full Idea: Axiom of Unions: For every collection of sets there exists a set that contains all the elements that belong to at least one of the sets in the collection. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10878 | Infinity: There exists a set of the empty set and the successor of each element |
| Full Idea: Axiom of Infinity: There exists a set containing the empty set and the successor of each of its elements. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |||
| A reaction: This is rather different from the other axioms because it contains the notion of 'successor', though that can be generated by an ordering procedure. |
| 10877 | Powers: All the subsets of a given set form their own new powerset |
| Full Idea: Axiom of Powers: For each set there exists a collection of sets that contains amongst its elements all the subsets of the given set. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |||
| A reaction: Obviously this must include the whole of the base set (i.e. not just 'proper' subsets), otherwise the new set would just be a duplicate of the base set. |
| 10879 | Choice: For every set a mechanism will choose one member of any non-empty subset |
| Full Idea: Axiom of Choice: For every set we can provide a mechanism for choosing one member of any non-empty subset of the set. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |||
| A reaction: This axiom is unusual because it makes the bold claim that such a 'mechanism' can always be found. Cohen showed that this axiom is separate. The tricky bit is choosing from an infinite subset. |
| 10871 | Axiom of Existence: there exists at least one set |
| Full Idea: Axiom of Existence: there exists at least one set. This may be the empty set, but you need to start with something. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10874 | Specification: a condition applied to a set will always produce a new set |
| Full Idea: Axiom of Specification: For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true. So the concept 'number is even' produces a set from the integers. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |||
| A reaction: What if the condition won't apply to the set? 'Number is even' presumably won't produce a set if it is applied to a set of non-numbers. |
| 10880 | Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) |
| Full Idea: Three views of mathematics: 'pure' mathematics, where it doesn't matter if it could ever have any application; 'real' mathematics, where every concept must be physically grounded; and 'applied' mathematics, using the non-real if the results are real. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.17) | |||
| A reaction: Very helpful. No one can deny the activities of 'pure' mathematics, but I think it is undeniable that the origins of the subject are 'real' (rather than platonic). We do economics by pretending there are concepts like the 'average family'. |
| 10860 | An ordinal number is defined by the set that comes before it |
| Full Idea: You can think of an ordinal number as being defined by the set that comes before it, so, in the non-negative integers, ordinal 5 is defined as {0, 1, 2, 3, 4}. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10861 | Beyond infinity cardinals and ordinals can come apart |
| Full Idea: With ordinary finite numbers ordinals and cardinals are in effect the same, but beyond infinity it is possible for two sets to have the same cardinality but different ordinals. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10854 | Transcendental numbers can't be fitted to finite equations |
| Full Idea: The 'transcendental numbers' are those irrationals that can't be fitted to a suitable finite equation, of which π is far and away the best known. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10858 | By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line |
| Full Idea: The realisation that brought 'i' into the toolkit of physicists and engineers was that you could extend the 'number line' into a new dimension, with an imaginary number axis at right angles to it. ...We now have a 'number plane'. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.12) |
| 10853 | Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless |
| Full Idea: It is a chicken-and-egg problem, whether the lack of zero forced forced classical mathematicians to rely mostly on a geometric approach to mathematics, or the geometric approach made 0 a meaningless concept, but the two remain strongly tied together. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10866 | Cantor's account of infinities has the shaky foundation of irrational numbers |
| Full Idea: As far as Kronecker was concerned, Cantor had built a whole structure on the irrational numbers, and so that structure had no foundation at all. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10862 | The 'continuum hypothesis' says aleph-one is the cardinality of the reals |
| Full Idea: The 'continuum hypothesis' says that aleph-one is the cardinality of the rational and irrational numbers. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 10869 | The Continuum Hypothesis is independent of the axioms of set theory |
| Full Idea: Paul Cohen showed that the Continuum Hypothesis is independent of the axioms of set theory. | |||
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |