14 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) |
9138 | An infinite series of sentences asserting falsehood produces the paradox without self-reference [Yablo, by Sorensen] |
Full Idea: Banning self-reference is too narrow to avoid the liar paradox. With 1) all the subsequent sentences are false, 2) all the subsequent sentences are false, 3) all the subsequent... the paradox still arises. Self-reference is a special case of this. | |
From: report of Stephen Yablo (Paradox without Self-Reference [1993]) by Roy Sorensen - Vagueness and Contradiction 11.1 | |
A reaction: [Idea 9137 pointed out that the ban was too narrow. Sorensen p.168 explains why this one is paradoxical] This is a nice example of progress in philosophy, since the Greeks would have been thrilled with this idea (unless they knew it, but it was lost). |
3570 | Maybe knowledge is belief which 'tracks' the truth [Nozick, by Williams,M] |
Full Idea: Nozick suggests that knowledge is just belief which 'tracks the truth' (hence leaving out justification). | |
From: report of Robert Nozick (Philosophical Explanations [1981]) by Michael Williams - Problems of Knowledge Ch. 2 |
2748 | A true belief isn't knowledge if it would be believed even if false. It should 'track the truth' [Nozick, by Dancy,J] |
Full Idea: Nozick says Gettier cases aren't knowledge because the proposition would be believed even if false. Proper justification must be more sensitive to the truth ("track the truth"). | |
From: report of Robert Nozick (Philosophical Explanations [1981], 3.1) by Jonathan Dancy - Intro to Contemporary Epistemology 3.1 | |
A reaction: This is a bad idea. I see a genuine tree in my garden and believe it is there, so I know it. That I might have believed it if I was in virtually reality, or observing a mirror, won't alter that. |