20 ideas
| 15842 | An ad hominem refutation is reasonable, if it uses the opponent's assumptions [Harte,V] |
| Full Idea: Judicious use of an opponent's assumptions is quite capable of producing a perfectly reasonable ad hominem refutation of the opponent's thesis. | |
| From: Verity Harte (Plato on Parts and Wholes [2002], 1.6) |
| 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) |
| 15841 | Mereology began as a nominalist revolt against the commitments of set theory [Harte,V] |
| Full Idea: Historically, the evolution of mereology was associated with the desire to find alternatives to set theory for those with nomimalist qualms about the commitment to abstract objects like sets. | |
| From: Verity Harte (Plato on Parts and Wholes [2002], 1.2) | |
| A reaction: Goodman, for example. It is interesting to note that the hardline nominalist Quine, pal of Goodman, eventually accepted set theory. It is difficult to account for things by merely naming their parts. |
| 15858 | Traditionally, the four elements are just what persists through change [Harte,V] |
| Full Idea: Earth, air, fire and water, viewed as elements, are, by tradition, the leading candidates for being the things that persist through change. | |
| From: Verity Harte (Plato on Parts and Wholes [2002], 4.4) | |
| A reaction: Physics still offers us things that persist through change, as conservation laws. |
| 15848 | Mereology treats constitution as a criterion of identity, as shown in the axiom of extensionality [Harte,V] |
| Full Idea: Mereologists do suppose that constitution is a criterion of identity. This view is enshrined in the Mereological axiom of extensionality; that objects with the same parts are identical. | |
| From: Verity Harte (Plato on Parts and Wholes [2002], 3.1) | |
| A reaction: A helpful explanation of why classical mereology is a very confused view of the world. It is at least obvious that a long wall and a house are different things, even if built of identical bricks. |
| 15837 | What exactly is a 'sum', and what exactly is 'composition'? [Harte,V] |
| Full Idea: The difficulty with the claim that a whole is (just) the sum of its parts is what are we to understand by 'the sum'? ...If we say wholes are 'composites' of parts, how are we to understand the relation of composition? | |
| From: Verity Harte (Plato on Parts and Wholes [2002], 1.1) |
| 15839 | If something is 'more than' the sum of its parts, is the extra thing another part, or not? [Harte,V] |
| Full Idea: Holism inherits all the difficulties associated with the term 'sum' and adds one of its own, when it says a whole is 'more than' the sum of its parts. This seems to say it has something extra? Is this something extra a part? | |
| From: Verity Harte (Plato on Parts and Wholes [2002], 1.1) | |
| A reaction: [compressed] Most people take the claim that a thing is more than the sum of its parts as metaphorical, I would think (except perhaps emergentists about the mind, and they are wrong). |
| 15838 | The problem with the term 'sum' is that it is singular [Harte,V] |
| Full Idea: For my money, the real problem with the term 'sum' is that it is singular. | |
| From: Verity Harte (Plato on Parts and Wholes [2002], 1.1) | |
| A reaction: Her point is that the surface grammar makes you accept a unity here, with no account of what unifies it, or even whether there is a unity. Does classical mereology have a concept (as the rest of us do) of 'disunity'? |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |