16 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) |
12270 | Being is one [Melissus, by Aristotle] |
Full Idea: Being is one. | |
From: report of Melissus (fragments/reports [c.443 BCE]) by Aristotle - Topics 104b23 | |
A reaction: I can only really understand this in terms of physics, as the belief that ultimately there is one simple theory which explains everything. That project doesn't look terribly promising, despite the lovely simplifications of modern physics. |
19087 | The meaning or purport of a symbol is all the rational conduct it would lead to [Peirce] |
Full Idea: The entire intellectual purport of any symbol consists in the total of all modes of rational conduct which, conditionally upon all the possible different circumstances and desires, would ensue upon the acceptance of the symbol. | |
From: Charles Sanders Peirce (Issues of Pragmaticism [1905], EP ii.246), quoted by Danielle Macbeth - Pragmatism and Objective Truth p.169 n1 | |
A reaction: Macbeth says pragmatism is founded on this theory of meaning, rather than on a theory of truth. I don't see why the causes of a symbol shouldn't be as much a part of its meaning as the consequences are. |
3059 | There is no real motion, only the appearance of it [Melissus, by Diog. Laertius] |
Full Idea: There is no such thing as real motion, but there only appears to be such. | |
From: report of Melissus (fragments/reports [c.443 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.4.3 |
5100 | The void is not required for change, because a plenum can alter in quality [Aristotle on Melissus] |
Full Idea: There is no need for void to be the cause of all change, because it is perfectly possible for a plenum to alter qualitatively (which is something Melissus overlooked). | |
From: comment on Melissus (fragments/reports [c.443 BCE]) by Aristotle - Physics 214a27 | |
A reaction: In modern physics this presumably gives us fluctuations in a force field. Motion is like a cat being digested by a python. The atomist claim that emptiness is needed if anything is to move still has intuitive appeal. |
456 | Nothing could come out of nothing [Melissus] |
Full Idea: If Nothing existed, in no way could anything come into being out of nothing. | |
From: Melissus (fragments/reports [c.443 BCE], B1), quoted by (who?) - where? |