14 ideas
| 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. |
| 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) |
| 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) |
| 5044 | Reality must be made of basic unities, which will be animated, substantial points [Leibniz] |
| Full Idea: A multiplicity can only be made up of true unities, ..so I had recourse to the idea of a real and animated point, or an atom of substance which must embrace some element of form or of activity in order to make a complete being. | |
| From: Gottfried Leibniz (New System and Explanation of New System [1696], p.116) | |
| A reaction: This seems to be a combination of logical atomism and panpsychism. It has a certain charm, but looks like another example of these rationalist speculators overreaching themselves. |
| 5045 | No machine or mere organised matter could have a unified self [Leibniz] |
| Full Idea: By means of the soul or form, there is a true unity which is called the 'I' in us; a thing which could not occur in artificial machines, nor in the simple mass of matter, however organised it may be. | |
| From: Gottfried Leibniz (New System and Explanation of New System [1696], p.120) | |
| A reaction: I think the unity of consciousness and the unified Self are different phenomena. A wonderful remark about artificial intelligence for 1696! Note the idea of functionalism contained in 'organised'. Personally I see the brain as a 'mass of matter'. |
| 5046 | The soul does know bodies, although they do not influence one another [Leibniz] |
| Full Idea: I do not admit that the soul does not know bodies, although this knowledge arises without their influencing one another. | |
| From: Gottfried Leibniz (New System and Explanation of New System [1696], Reply 11) | |
| A reaction: He couldn't very well admit this without moving into pure idealism. Presumably it is like "I know her - she'll be in Harrods this morning". I wonder if Satan could steal my body, but my mind continue to believe it was still there? |
| 5043 | To regard animals as mere machines may be possible, but seems improbable [Leibniz] |
| Full Idea: It seems to me that the opinion of those who transform or degrade the lower animals into mere machines, although it seems possible, is improbable, and even against the order of things. | |
| From: Gottfried Leibniz (New System and Explanation of New System [1696], p.116) | |
| A reaction: His target is Descartes. 'Against the order of things' seems to beg the question. What IS the order of things? Only a thorough-going dualist would worry about this question, and that isn't me. |
| 16713 | Philosophers are the forefathers of heretics [Tertullian] |
| Full Idea: Philosophers are the forefathers of heretics. | |
| From: Tertullian (works [c.200]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2 |
| 6610 | I believe because it is absurd [Tertullian] |
| Full Idea: I believe because it is absurd ('Credo quia absurdum est'). | |
| From: Tertullian (works [c.200]), quoted by Robert Fogelin - Walking the Tightrope of Reason n4.2 | |
| A reaction: This seems to be a rather desperate remark, in response to what must have been rather good hostile arguments. No one would abandon the support of reason if it was easy to acquire. You can't deny its engaging romantic defiance, though. |