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) |
| 6019 | If someone squashed a horse to make a dog, something new would now exist [Mnesarchus] |
| Full Idea: If, for the sake of argument, someone were to mould a horse, squash it, then make a dog, it would be reasonable for us on seeing this to say that this previously did not exist but now does exist. | |
| From: Mnesarchus (fragments/reports [c.120 BCE]), quoted by John Stobaeus - Anthology 179.11 | |
| A reaction: Locke would say it is new, because the substance is the same, but a new life now exists. A sword could cease to exist and become a new ploughshare, I would think. Apply this to the Ship of Theseus. Is form more important than substance? |
| 19451 | When absorbed in deep reflection, is your reason in control, or is it you? [Feuerbach] |
| Full Idea: When, submerged in deep reflection, you forget both yourself and your surroundings, is it you who controls reason, or is it rather reason that controls and absorbs you? | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], I) | |
| A reaction: A delightful question, even if it looks like a false dichotomy. I'm not sure what to make of 'me', if my reason can be subtracted from it. Aquinas was one the same wavelength here. |
| 19450 | Reason, love and will are the highest perfections and essence of man - the purpose of his life [Feuerbach] |
| Full Idea: Reason, love and power of will are perfections of man; they are his highest powers, his absolute essence in so far as he is man, the purpose of his existence. Man exists in order to think, love and will. | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], I) | |
| A reaction: Feuerbach was a notable atheist, but adopts a religious style of language which modern atheists would find rather alien. Personally I love talk of ideals and perfections. Ideals have been discredited in modern times, but need a revival. |
| 19448 | Consciousness is said to distinguish man from animals - consciousness of his own species [Feuerbach] |
| Full Idea: What constitutes the essential difference between man and animal? The most simple, general, and most widely held answer to this question is consciousness. Consciousness is given only in the case of a being to whom his species ...is an object of thought. | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], I) | |
| A reaction: Rather speculative. Since other species cohabit and breed only with their fellow species members, one might have thought they were aware of them. |
| 19454 | A God needs justice, kindness and wisdom, but those concepts don't depend on the concept of God [Feuerbach] |
| Full Idea: The concept of God depends on the concepts of justice, kindness and wisdom - a God who is not kind, not just, and not wise is no God. But these concepts do not depend on the concept of God. That a quality is possessed by God does not make it divine. | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], II) | |
| A reaction: This is part of Feuerbach's argument for atheism, but if you ask for the source of our human concepts of justice, kindness and wisdom, no one, I would have thought, could cite God for the role. |
| 19452 | The nature of God is an expression of human nature [Feuerbach] |
| Full Idea: God is the manifestation of man's inner nature, his expressed self. | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], II) | |
| A reaction: Even if you are a deeply committed theist, you have to concede some of this point. The perfections attributed to God are usually of human qualities. Leibniz, though, says that God has an infinity of perfection, mostly unknown to us. |
| 19453 | If love, goodness and personality are human, the God who is their source is anthropomorphic [Feuerbach] |
| Full Idea: If love, goodness, and personality are human determinations, the being which constitutes their source and ...their presupposition is also an anthropomorphism; so is the existence of God. | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], II) | |
| A reaction: It is certainly a struggle for the imagination to grasp a being which is characterised by idealised versions of human virtues, and yet has an intrinsic nature which is utterly different from humanity. |
| 19449 | Religion is the consciousness of the infinite [Feuerbach] |
| Full Idea: Religion is the consciousness of the infinite. | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], I) |
| 19455 | Today's atheism will tomorrow become a religion [Feuerbach] |
| Full Idea: What is regarded as atheism today will be religion tomorrow. | |
| From: Ludwig Feuerbach (Introduction of 'Essence of Christianity' [1841], II) | |
| A reaction: Modern critics of atheism frequently accuse it of being a new religion. I doubt whether Feuerbach is right, but it is a nice provocative idea. |