30 ideas
| 10859 | A set is 'well-ordered' if every subset has a first element [Clegg] |
| 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 [Clegg] |
| 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 [Clegg] |
| 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 [Clegg] |
| 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 [Clegg] |
| 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 [Clegg] |
| 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 [Clegg] |
| 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 [Clegg] |
| 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 [Clegg] |
| 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 [Clegg] |
| 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 [Clegg] |
| 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) [Clegg] |
| 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 [Clegg] |
| 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 [Clegg] |
| 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 [Clegg] |
| 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 [Clegg] |
| 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 [Clegg] |
| 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) |
| 16014 | It is controversial whether only 'numerical identity' allows two things to be counted as one [Noonan] |
| Full Idea: 'Numerical identity' implies the controversial view that it is the only identity relation in accordance with which we can properly count (or number) things: x and y are to be properly counted as one just in case they are numerically identical. | |
| From: Harold Noonan (Identity [2009], §1) | |
| A reaction: Noonan cites Geach, presumably to remind us of relative identity, where two things may be one or two, depending on what they are relative to. The one 'guard on the gate' may actually be two men. |
| 10866 | Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg] |
| 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) |
| 10869 | The Continuum Hypothesis is independent of the axioms of set theory [Clegg] |
| 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) |
| 10862 | The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg] |
| 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) |
| 16024 | I could have died at five, but the summation of my adult stages could not [Noonan] |
| Full Idea: Persons have different modal properties from the summations of person-stages. …I might have died when I was five. But the maximal summation of person-stages which perdurantists say is me could not have had a temporal extent of a mere five years. | |
| From: Harold Noonan (Identity [2009], §5) | |
| A reaction: Thus the summation of stages seems to fail Leibniz's Law, since truths about a part are not true of the whole. But my foot might be amputated without me being amputated. The objection is the fallacy of composition? |
| 16023 | Stage theorists accept four-dimensionalism, but call each stage a whole object [Noonan] |
| Full Idea: Stage theorists, accepting the ontology of perdurance, modify the semantics to secure the result that fatness is a property of a cat. Every temporal part of a cat (such as Tabby-on-Monday) is a cat. …(but they pay a price over the counting of cats). | |
| From: Harold Noonan (Identity [2009], §5) | |
| A reaction: [Noonan cites Hawley and Sider for this view. The final parenthesis compresses Noonan] I would take the difficulty over counting cats to be fatal to the view. It produces too many cats, or too few, or denies counting altogether. |
| 16015 | Problems about identity can't even be formulated without the concept of identity [Noonan] |
| Full Idea: If identity is problematic, it is difficult to see how the problem could be resolved, since it is difficult to see how a thinker could have the conceptual resources with which to explain the concept of identity whilst lacking that concept itself. | |
| From: Harold Noonan (Identity [2009], §1) | |
| A reaction: I don't think I accept this. We can comprehend the idea of a mind that didn't think in terms of identities (at least for objects). I suppose any relation of a mind to the world has to distinguish things in some way. Does the Parmenidean One have identity? |
| 16017 | Identity is usually defined as the equivalence relation satisfying Leibniz's Law [Noonan] |
| Full Idea: Numerical identity is usually defined as the equivalence relation (or: the reflexive relation) satisfying Leibniz's Law, the indiscernibility of identicals, where everything true of x is true of y. | |
| From: Harold Noonan (Identity [2009], §2) | |
| A reaction: Noonan says this must include 'is identical to x' among the truths, and so is circular |
| 16016 | Identity definitions (such as self-identity, or the smallest equivalence relation) are usually circular [Noonan] |
| Full Idea: Identity can be circularly defined, as 'the relation everything has to itself and to nothing else', …or as 'the smallest equivalence relation'. | |
| From: Harold Noonan (Identity [2009], §2) | |
| A reaction: The first one is circular because 'nothing else' implies identity. The second is circular because it has to quantify over all equivalence relations. (So says Noonan). |
| 16020 | Identity can only be characterised in a second-order language [Noonan] |
| Full Idea: There is no condition in a first-order language for a predicate to express identity, rather than indiscernibility within the resources of the language. Leibniz's Law is statable in a second-order language, so identity can be uniquely characterised. | |
| From: Harold Noonan (Identity [2009], §2) | |
| A reaction: The point is that first-order languages only refer to all objects, but you need to refer to all properties to include Leibniz's Law. Quine's 'Identity, Ostension and Hypostasis' is the source of this idea. |
| 16018 | Indiscernibility is basic to our understanding of identity and distinctness [Noonan] |
| Full Idea: Leibniz's Law (the indiscernibility of identicals) appears to be crucial to our understanding of identity, and, more particularly, to our understanding of distinctness. | |
| From: Harold Noonan (Identity [2009], §2) | |
| A reaction: True, but indiscernibility concerns the epistemology, and identity concerns the ontology. |
| 16019 | Leibniz's Law must be kept separate from the substitutivity principle [Noonan] |
| Full Idea: Leibniz's Law must be clearly distinguished from the substitutivity principle, that if 'a' and 'b' are codesignators they are substitutable salva veritate. | |
| From: Harold Noonan (Identity [2009], §2) | |
| A reaction: He gives a bunch of well-known problem cases for substitutivity. The Morning Star, Giorgione, and the number of planets won't work. Belief contexts, or facts about spelling, may not be substitutable. |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |
| Full Idea: The Egyptians were the first to claim that the soul of a human being is immortal, and that each time the body dies the soul enters another creature just as it is being born. | |
| From: Herodotus (The Histories [c.435 BCE], 2.123.2) |