14 ideas
13206 | A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton] |
Full Idea: A 'linear ordering' (or 'total ordering') on A is a binary relation R meeting two conditions: R is transitive (of xRy and yRz, the xRz), and R satisfies trichotomy (either xRy or x=y or yRx). | |
From: Herbert B. Enderton (Elements of Set Theory [1977], 3:62) |
13201 | ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton] |
Full Idea: To know if A ∈ B, we look at the set A as a single object, and check if it is among B's members. But if we want to know whether A ⊆ B then we must open up set A and check whether its various members are among the members of B. | |
From: Herbert B. Enderton (Elements of Set Theory [1977], 1:04) | |
A reaction: This idea is one of the key ideas to grasp if you are going to get the hang of set theory. John ∈ USA ∈ UN, but John is not a member of the UN, because he isn't a country. See Idea 12337 for a special case. |
13204 | The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton] |
Full Idea: The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}; hence it can be proved that <u,v> = <x,y> iff u = x and v = y (given by Kuratowski in 1921). ...The definition is somewhat arbitrary, and others could be used. | |
From: Herbert B. Enderton (Elements of Set Theory [1977], 3:36) | |
A reaction: This looks to me like one of those regular cases where the formal definitions capture all the logical behaviour of the concept that are required for inference, while failing to fully capture the concept for ordinary conversation. |
13200 | Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton] |
Full Idea: Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ. A man with an empty container is better off than a man with nothing. | |
From: Herbert B. Enderton (Elements of Set Theory [1977], 1.03) |
13199 | The empty set may look pointless, but many sets can be constructed from it [Enderton] |
Full Idea: It might be thought at first that the empty set would be a rather useless or even frivolous set to mention, but from the empty set by various set-theoretic operations a surprising array of sets will be constructed. | |
From: Herbert B. Enderton (Elements of Set Theory [1977], 1:02) | |
A reaction: This nicely sums up the ontological commitments of mathematics - that we will accept absolutely anything, as long as we can have some fun with it. Sets are an abstraction from reality, and the empty set is the very idea of that abstraction. |
13203 | The singleton is defined using the pairing axiom (as {x,x}) [Enderton] |
Full Idea: Given any x we have the singleton {x}, which is defined by the pairing axiom to be {x,x}. | |
From: Herbert B. Enderton (Elements of Set Theory [1977], 2:19) | |
A reaction: An interesting contrivance which is obviously aimed at keeping the axioms to a minimum. If you can do it intuitively with a new axiom, or unintuitively with an existing axiom - prefer the latter! |
13202 | Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton] |
Full Idea: It was observed by several people that for a satisfactory theory of ordinal numbers, Zermelo's axioms required strengthening. The Axiom of Replacement was proposed by Fraenkel and others, giving rise to the Zermelo-Fraenkel (ZF) axioms. | |
From: Herbert B. Enderton (Elements of Set Theory [1977], 1:15) |
13205 | We can only define functions if Choice tells us which items are involved [Enderton] |
Full Idea: For functions, we know that for any y there exists an appropriate x, but we can't yet form a function H, as we have no way of defining one particular choice of x. Hence we need the axiom of choice. | |
From: Herbert B. Enderton (Elements of Set Theory [1977], 3:48) |
22480 | Possessing the virtue of justice disposes a person to good practical rationality [Foot] |
Full Idea: If justice is a virtue it must make action good by disposing its possessor to goodness in practical rationality; the latter consisting of the right recognition of reasons, and corresponding action. | |
From: Philippa Foot (Rationality and Virtue [1994], p.174) | |
A reaction: This somewhat inverts Aristotle, who says the possessing of good practical reason is the key to acquiring the virtues. Foot suggests that possessing the virtue promotes the practical rationality. Someone can be sensible without being virtuous. |
22477 | Calling a knife or farmer or speech or root good does not involve attitudes or feelings [Foot] |
Full Idea: No one thinks that calling a knife a good knife, a farmer a good farmer, a speech a good speech, a root a good root, necessarily expresses or even involves an attitude or feeling towards it. | |
From: Philippa Foot (Rationality and Virtue [1994], p.163) | |
A reaction: This is the Aristotelian idea (which I favour) that good derives from function. In such a case it seems obvious that it has nothing to do with expressing emotions. |
22478 | The essential thing is the 'needs' of plants and animals, and their operative parts [Foot] |
Full Idea: The key notion is the concept of need, …as when we say what a plant or animal of a certain species needs to have, …and what its operative features, such roots, leaves, hearts and lungs, need to do. | |
From: Philippa Foot (Rationality and Virtue [1994], p.164) | |
A reaction: Good. That takes it away from the idea of a function, which could be possessed by an inanimate machine (even though that still entails success and failure). Strictly, we need oxygen, but the goodness resides in the lungs. |
22479 | Observing justice is necessary to humans, like hunting to wolves or dancing to bees [Foot] |
Full Idea: The teaching and observing of the rules of justice is as necessary a part of the life of human beings as hunting together in packs with a leader is a necessary part of the lives of wolves, or dancing part of the life of the dancing bee. | |
From: Philippa Foot (Rationality and Virtue [1994], p.168) | |
A reaction: So why are some men unjust? All wolves hunt, and all appropriate bees dance. A few men even thrive on injustice. |
1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
Full Idea: Archelaus was the first person to say that the universe is boundless. | |
From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |