18 ideas
| 7719 | European philosophy consists of a series of footnotes to Plato [Whitehead] |
| Full Idea: The safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato. | |
| From: Alfred North Whitehead (Process and Reality [1929], p.39) | |
| A reaction: Outsiders think this is a ridiculous remark, but readers of Plato can only be struck by what a wonderful tribute Whitehead has come up with. I would say that at least 80% of this database deals with problems which were discussed at length by Plato. |
| 13030 | Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y) [Kunen] |
| Full Idea: Axiom of Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y). That is, a set is determined by its members. If every z in one set is also in the other set, then the two sets are the same. | |
| From: Kenneth Kunen (Set Theory [1980], §1.5) |
| 13032 | Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z) [Kunen] |
| Full Idea: Axiom of Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z). Any pair of entities must form a set. | |
| From: Kenneth Kunen (Set Theory [1980], §1.6) | |
| A reaction: Repeated applications of this can build the hierarchy of sets. |
| 13033 | Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A) [Kunen] |
| Full Idea: Axiom of Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A). That is, the union of a set (all the members of the members of the set) must also be a set. | |
| From: Kenneth Kunen (Set Theory [1980], §1.6) |
| 13037 | Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x) [Kunen] |
| Full Idea: Axiom of Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x). That is, there is a set which contains zero and all of its successors, hence all the natural numbers. The principal of induction rests on this axiom. | |
| From: Kenneth Kunen (Set Theory [1980], §1.7) |
| 13038 | Power Set: ∀x ∃y ∀z(z ⊂ x → z ∈ y) [Kunen] |
| Full Idea: Power Set Axiom: ∀x ∃y ∀z(z ⊂ x → z ∈ y). That is, there is a set y which contains all of the subsets of a given set. Hence we define P(x) = {z : z ⊂ x}. | |
| From: Kenneth Kunen (Set Theory [1980], §1.10) |
| 13034 | Replacement: ∀x∈A ∃!y φ(x,y) → ∃Y ∀X∈A ∃y∈Y φ(x,y) [Kunen] |
| Full Idea: Axiom of Replacement Scheme: ∀x ∈ A ∃!y φ(x,y) → ∃Y ∀X ∈ A ∃y ∈ Y φ(x,y). That is, any function from a set A will produce another set Y. | |
| From: Kenneth Kunen (Set Theory [1980], §1.6) |
| 13039 | Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ¬∃z(z∈x ∧ z∈y))) [Kunen] |
| Full Idea: Axiom of Foundation: ∀x (∃y(y ∈ x) → ∃y(y ∈ x ∧ ¬∃z(z ∈ x ∧ z ∈ y))). Aka the 'Axiom of Regularity'. Combined with Choice, it means there are no downward infinite chains. | |
| From: Kenneth Kunen (Set Theory [1980], §3.4) |
| 13036 | Choice: ∀A ∃R (R well-orders A) [Kunen] |
| Full Idea: Axiom of Choice: ∀A ∃R (R well-orders A). That is, for every set, there must exist another set which imposes a well-ordering on it. There are many equivalent versions. It is not needed in elementary parts of set theory. | |
| From: Kenneth Kunen (Set Theory [1980], §1.6) |
| 13029 | Set Existence: ∃x (x = x) [Kunen] |
| Full Idea: Axiom of Set Existence: ∃x (x = x). This says our universe is non-void. Under most developments of formal logic, this is derivable from the logical axioms and thus redundant, but we do so for emphasis. | |
| From: Kenneth Kunen (Set Theory [1980], §1.5) |
| 13031 | Comprehension: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ) [Kunen] |
| Full Idea: Comprehension Scheme: for each formula φ without y free, the universal closure of this is an axiom: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ). That is, there must be a set y if it can be defined by the formula φ. | |
| From: Kenneth Kunen (Set Theory [1980], §1.5) | |
| A reaction: Unrestricted comprehension leads to Russell's paradox, so restricting it in some way (e.g. by the Axiom of Specification) is essential. |
| 13040 | Constructibility: V = L (all sets are constructible) [Kunen] |
| Full Idea: Axiom of Constructability: this is the statement V = L (i.e. ∀x ∃α(x ∈ L(α)). That is, the universe of well-founded von Neumann sets is the same as the universe of sets which are actually constructible. A possible axiom. | |
| From: Kenneth Kunen (Set Theory [1980], §6.3) |
| 10656 | With 'extensive connection', boundary elements are not included in domains [Whitehead, by Varzi] |
| Full Idea: In Whitehead's theory of extensive connection, no boundary elements are included in the domain of quantification. ...His conception of space contains no parts of lower dimensions, such as points or boundary elements. | |
| From: report of Alfred North Whitehead (Process and Reality [1929]) by Achille Varzi - Mereology 3.1 | |
| A reaction: [Varzi says we should see B.L.Clarke 1981 for a rigorous formulation. Second half of the Idea is Varzi p.21] |
| 15389 | In Whitehead 'processes' consist of events beginning and ending [Whitehead, by Simons] |
| Full Idea: There are no items in Whitehead's ontology called 'processes'. Rather, the term 'process' refers to the way in which the basic things - which are still events - come into existence and cease to exist. Whitehead called this 'becoming'. | |
| From: report of Alfred North Whitehead (Process and Reality [1929]) by Peter Simons - Whitehead: process and cosmology 'The mature' |
| 18465 | An 'equivalence' relation is one which is reflexive, symmetric and transitive [Kunen] |
| Full Idea: R is an equivalence relation on A iff R is reflexive, symmetric and transitive on A. | |
| From: Kenneth Kunen (The Foundations of Mathematics (2nd ed) [2012], I.7.1) |
| 6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
| Full Idea: Hermarchus said that animal killing is justified by considerations of human safety and nourishment and by animals' inability to form contractual relations of justice with us. | |
| From: report of Hermarchus (fragments/reports [c.270 BCE]) by David A. Sedley - Hermarchus | |
| A reaction: Could the last argument be used to justify torturing animals? Or could we eat a human who was too brain-damaged to form contracts? |
| 15247 | Whitehead held that perception was a necessary feature of all causation [Whitehead, by Harré/Madden] |
| Full Idea: On Whitehead's view, not only is a volitional sense of 'causal power' projected on to physical events, but 'perception in the causal mode' is literally ascribed to them. | |
| From: report of Alfred North Whitehead (Process and Reality [1929]) by Harré,R./Madden,E.H. - Causal Powers 3.II | |
| A reaction: This seems to be a close relative of Leibniz's monads. 'Perception' is a daft word for it, but in some way everything is 'responsive' to the things adjacent to it. |
| 16962 | Whitehead replaced points with extended regions [Whitehead, by Quine] |
| Full Idea: Whitehead tried to avoid points, and make do with extended regions and sets of regions. | |
| From: report of Alfred North Whitehead (Process and Reality [1929]) by Willard Quine - Existence and Quantification p.93 |