17 ideas
| 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) |
| 6017 | Nomos is king [Pindar] |
| Full Idea: Nomos is king. | |
| From: Pindar (poems [c.478 BCE], S 169), quoted by Thomas Nagel - The Philosophical Culture | |
| A reaction: This seems to be the earliest recorded shot in the nomos-physis wars (the debate among sophists about moral relativism). It sounds as if it carries the full relativist burden - that all that matters is what has been locally decreed. |
| 18225 | If we can't reason about value, we can reason about the unconditional source of value [Korsgaard] |
| Full Idea: If you can only know what is intrinsically valuable through intuition (as Moore claims), you can still argue about what is unconditionally valuable. There must be something unconditionally valuable because there must be a source of value. | |
| From: Christine M. Korsgaard (Aristotle and Kant on the Source of Value [1986], 8 'Three') | |
| A reaction: If you only grasped the values through intuition, does that give you enough information to infer the dependence relations between values? |
| 18228 | An end can't be an ultimate value just because it is useless! [Korsgaard] |
| Full Idea: If what is final is whatever is an end but never a means, ...why should something be more valuable just because it is useless? | |
| From: Christine M. Korsgaard (Aristotle and Kant on the Source of Value [1986], 8 'Finality') | |
| A reaction: Korsgaard is offering this as a bad reading of what Aristotle intends. |
| 18224 | Goodness is given either by a psychological state, or the attribution of a property [Korsgaard] |
| Full Idea: 'Subjectivism' identifies good ends with or by reference to some psychological state. ...'Objectivism' says that something is good as an end if a property, intrinsic goodness, is attributed to it. | |
| From: Christine M. Korsgaard (Aristotle and Kant on the Source of Value [1986], 8 'Three') | |
| A reaction: Not sure about 'objectivism' here, as attributing a 'property'. A tomato is not intrinsically good. Hopeless for building walls. |
| 18233 | Contemplation is final because it is an activity which is not a process [Korsgaard] |
| Full Idea: It is because contemplation is an activity that is not also a process that Aristotle identifies it as the most final good. | |
| From: Christine M. Korsgaard (Aristotle and Kant on the Source of Value [1986], 8 'Activity') | |
| A reaction: Quite a helpful way of labelling what Aristotle has in mind. So should we not aspire to be involved in processes, except reluctantly? I take the mind itself to be a process, so that may be difficult! |
| 18226 | For Aristotle, contemplation consists purely of understanding [Korsgaard] |
| Full Idea: Contemplation, as Aristotle understand it, is not research or inquiry, but an activity that ensues on these: an activity that consists in understanding. | |
| From: Christine M. Korsgaard (Aristotle and Kant on the Source of Value [1986], 8 'Aristotle') | |
| A reaction: Fairly obvious, when you read the last part of 'Ethics', but helpful in grasping Aristotle, because understanding is the objective of 'Posterior Analytics' and 'Metaphysics', so he tells you how to achieve the ideal moral state. |