20 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) |
| 16052 | 'Superdupervenience' is supervenience that has a robustly materialistic explanation [Horgan,T] |
| Full Idea: The idea of a ontological supervenience that is robustly explainable in a materialistically explainable way I hereby dub 'superdupervenience'. | |
| From: Terence Horgan (From Supervenience to Superdupervenience [1993], §4) | |
| A reaction: [He credits William Lycan with the actual word] His assumption prior to this introduction is that mere supervenience just adds a new mystery. I take supervenience to be an observation of 'tracking', which presumably needs to be explained. |
| 16053 | 'Global' supervenience is facts tracking varying physical facts in every possible world [Horgan,T] |
| Full Idea: The idea of 'global supervenience' is standardly expressed as 'there are no two physically possible worlds which are exactly alike in all physical respects but different in some other respect'. | |
| From: Terence Horgan (From Supervenience to Superdupervenience [1993], §5) | |
| A reaction: [Jaegwon Kim is the source of this concept] The 'local' view will be that they do indeed track, but they could, in principle, come apart. A zombie might be a case of them possibly coming apart. Zombies are silly. |
| 16056 | Don't just observe supervenience - explain it! [Horgan,T] |
| Full Idea: Although the task of explaining supervenience has been little appreciated and little discussed in the philosophical literature, it is time for that to change. | |
| From: Terence Horgan (From Supervenience to Superdupervenience [1993], §8) | |
| A reaction: I would offer a strong addition to this: be absolutely sure that you are dealing with two distinct things in the supervenience relationship, before you waste time trying to explain how they relate to one another. |
| 16054 | Physicalism needs more than global supervenience on the physical [Horgan,T] |
| Full Idea: Global supervenience seems too weak to capture the physical facts determining all the facts. …There could be two spatio-temporal regions alike in all physical respects, but different in some intrinsic non-physical respect. | |
| From: Terence Horgan (From Supervenience to Superdupervenience [1993], §5) | |
| A reaction: I.e. there might be two physically identical regions, but one contains angels and the other doesn't (so the extra fact isn't tracking the physical facts). Physicalism I take to be the simple denial of the angels. Supervenience is an explanandum. |
| 16055 | Materialism requires that physics be causally complete [Horgan,T] |
| Full Idea: Any broadly materialistic metaphysical position needs to claim that physics is causally complete. | |
| From: Terence Horgan (From Supervenience to Superdupervenience [1993], §6) | |
| A reaction: Since 'physics' is a human creation, I presume he means that physical reality is causally complete. The interaction problem that faced Descartes seems crucial - how could something utterly non-physical effect a physical change? |
| 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) |
| 16057 | Instrumentalism normally says some discourse is useful, but not genuinely true [Horgan,T] |
| Full Idea: Instrumentalist views typically attribute utility to the given body of discourse, but deny that it expresses genuine truths. | |
| From: Terence Horgan (From Supervenience to Superdupervenience [1993], §8) | |
| A reaction: To me it is obvious to ask why anything could have a high level of utility (especially in accounts of the external physical world) without being true. Falsehoods may sometimes (though I doubt it) be handy in human life, but useful in chemistry…? |
| 5845 | Niceratus learnt the whole of Homer by heart, as a guide to goodness [Xenophon] |
| Full Idea: Niceratus said that his father, because he was concerned to make him a good man, made him learn the whole works of Homer, and he could still repeat by heart the entire 'Iliad' and 'Odyssey'. | |
| From: Xenophon (Symposium [c.391 BCE], 3.5) | |
| A reaction: This clearly shows the status which Homer had in the teaching of morality in the time of Socrates, and it is precisely this acceptance of authority which he was challenging, in his attempts to analyse the true basis of virtue |
| 5833 | Education is the greatest of human goods [Xenophon] |
| Full Idea: Education is the greatest of human goods. | |
| From: Xenophon (Apology of Socrates [c.392 BCE], 22) | |
| A reaction: Of course, one might ask what education is for, and arrive at a greater good. If you ask what is the greatest good which a society can provide for you, or which you can give to your children, this seems to me a good answer. |