20 ideas
| 14025 | The weaker version of Truthmaker: 'truth supervenes on being' [Crisp,TM] |
| Full Idea: The weaker version of Truthmaker is that 'truth supervenes on being'. | |
| From: Thomas M. Crisp (Presentism [2003], 3.4) | |
| A reaction: [He cites Lewis 2001 and Bigelow 1988] This still leaves the difficulty of truths about non-existent things, and truths about possibilities (esp. those that are possible, but are never actualised). What being do mathematical truths supervene on? |
| 14023 | The Truthmaker thesis spells trouble for presentists [Crisp,TM] |
| Full Idea: The Truthmaker thesis (that 'for every truth there is a truthmaker, that is, something whose very existence entails the truth' - Fox 1987) spells trouble for the presentist about time. | |
| From: Thomas M. Crisp (Presentism [2003], 3.4) | |
| A reaction: The point is that presentists can no longer express truths about the past (never mind the future), because the truthmakers for them don't exist. This seems to neglect the power of tense - the truth of the claim that 'p was true'. |
| 14024 | Truthmaker has problems with generalisation, non-existence claims, and property instantiations [Crisp,TM] |
| Full Idea: Truthmaker is controversial: what of truths like 'all ravens are black', or 'there are no unicorns'. And 'John is tall' is not made true by John or the property of being tall, but by the fusion of the two, but what could this non-mereological fusion be? | |
| From: Thomas M. Crisp (Presentism [2003], 3.4) | |
| A reaction: A first move is to include modal facts (or possible worlds) among the truthmakers. The unicorns are tricky, and seem to need all of actuality as their truthmaker. I don't see the tallness difficulty. Predication is odd, but so what? |
| 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) |
| 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) |
| 14021 | Worm Perdurantism has a fusion of all the parts; Stage Perdurantism has one part at a time [Crisp,TM] |
| Full Idea: Worm-theoretic Perdurantism says spatio-temporal continuants are mereological fusions of instantaneous temporal parts or stages located at different times; Stage-theoretic Perdurantism says they are instantaneous temporal stages of continuants. | |
| From: Thomas M. Crisp (Presentism [2003], 2.1) | |
| A reaction: [Armstrong, Lewis and Quine defend the first; Sider the second] The Stage view seems to be the common sense view. Sider suggests that the earlier stages are counterparts, not the thing as it currently is. |
| 14020 | 'Eternalism' is the thesis that reality includes past, present and future entities [Crisp,TM] |
| Full Idea: I use the term Eternalism for the thesis that reality includes past, present and future entities. (It is sometimes used for the view that all propositions have their truth-value eternally - it is always true or never true). | |
| From: Thomas M. Crisp (Presentism [2003], Intro n.1) | |
| A reaction: 'Eternalism' strikes me as an excellent word for the former meaning, so I shall promote that, and quietly forget the second one. The idea that the future exists has always stuck in my craw, and the belief that Napoleon still exists strikes me as a weird. |
| 14026 | Presentists can talk of 'times', with no more commitment than modalists have to possible worlds [Crisp,TM] |
| Full Idea: We can talk of 'moments of time' as abstract objects. This will be attractive to the presentist. As possible worlds give an economical theory of modal talk, so 'times' gives us a theory for temporal talk. | |
| From: Thomas M. Crisp (Presentism [2003], 3.4) | |
| A reaction: Thus we can utilise 'times', while having no more commitment to them than to possible worlds. Nice. He cites Prior and Fine 1977 and Chisholm 1979. |
| 14022 | The only three theories are Presentism, Dynamic (A-series) Eternalism and Static (B-series) Eternalism [Crisp,TM] |
| Full Idea: Three theories exhaust the options on time: presentism, dynamic eternalism (eternalism with the tensed dynamic A-series view of time, and the totality of events changing over time), and static eternalism (eternalism with the B-series). | |
| From: Thomas M. Crisp (Presentism [2003], 2.4) | |
| A reaction: I think the idea that reality is Static Eternalism is just a misunderstanding, arising from our imaginative ability to take a lofty objective overview of a very fluid reality. The other two are the serious candidates. Present, or Growing-block. |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |
| Full Idea: The Egyptians were the first to claim that the soul of a human being is immortal, and that each time the body dies the soul enters another creature just as it is being born. | |
| From: Herodotus (The Histories [c.435 BCE], 2.123.2) |