67 ideas
5988 | Anaximander produced the first philosophy book (and maybe the first book) [Anaximander, by Bodnár] |
1496 | The earth is stationary, because it is in the centre, and has no more reason to move one way than another [Anaximander, by Aristotle] |
13634 | Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro] |
13643 | Aristotelian logic is complete [Shapiro] |
13651 | A set is 'transitive' if contains every member of each of its members [Shapiro] |
13647 | Choice is essential for proving downward Löwenheim-Skolem [Shapiro] |
13631 | Are sets part of logic, or part of mathematics? [Shapiro] |
13640 | Russell's paradox shows that there are classes which are not iterative sets [Shapiro] |
13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro] |
13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro] |
13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro] |
13642 | Logic is the ideal for learning new propositions on the basis of others [Shapiro] |
13627 | There is no 'correct' logic for natural languages [Shapiro] |
13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro] |
13668 | Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro] |
13669 | Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro] |
13662 | First-order logic was an afterthought in the development of modern logic [Shapiro] |
13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro] |
13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro] |
13673 | The notion of finitude is actually built into first-order languages [Shapiro] |
15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine] |
13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro] |
13650 | Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro] |
13645 | In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro] |
13649 | Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro] |
13637 | If a logic is incomplete, its semantic consequence relation is not effective [Shapiro] |
13626 | Semantic consequence is ineffective in second-order logic [Shapiro] |
13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro] |
13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro] |
13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro] |
13644 | Semantics for models uses set-theory [Shapiro] |
13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
13670 | Categoricity can't be reached in a first-order language [Shapiro] |
13658 | Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro] |
13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro] |
13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro] |
13675 | Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro] |
13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro] |
13628 | We can live well without completeness in logic [Shapiro] |
13630 | Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro] |
13646 | Compactness is derived from soundness and completeness [Shapiro] |
13661 | A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro] |
13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro] |
13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro] |
13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro] |
13657 | First-order arithmetic can't even represent basic number theory [Shapiro] |
13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro] |
14085 | 'Deductivist' structuralism is just theories, with no commitment to objects, or modality [Linnebo] |
14084 | Non-eliminative structuralism treats mathematical objects as positions in real abstract structures [Linnebo] |
14086 | 'Modal' structuralism studies all possible concrete models for various mathematical theories [Linnebo] |
14087 | 'Set-theoretic' structuralism treats mathematics as various structures realised among the sets [Linnebo] |
14089 | Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure [Linnebo] |
14083 | Structuralism is right about algebra, but wrong about sets [Linnebo] |
14090 | In mathematical structuralism the small depends on the large, which is the opposite of physical structures [Linnebo] |
13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro] |
13625 | Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro] |
13663 | Some reject formal properties if they are not defined, or defined impredicatively [Shapiro] |
14874 | Anaximander saw the contradiction in the world - that its own qualities destroy it [Anaximander, by Nietzsche] |
14091 | There may be a one-way direction of dependence among sets, and among natural numbers [Linnebo] |
14088 | An 'intrinsic' property is either found in every duplicate, or exists independent of all externals [Linnebo] |
13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro] |
405 | The essential nature, whatever it is, of the non-limited is everlasting and ageless [Anaximander] |
13222 | The Boundless cannot exist on its own, and must have something contrary to it [Aristotle on Anaximander] |
1495 | Anaximander introduced the idea that the first principle and element of things was the Boundless [Anaximander, by Simplicius] |
404 | Things begin and end in the Unlimited, and are balanced over time according to justice [Anaximander] |
1746 | The parts of all things are susceptible to change, but the whole is unchangeable [Anaximander, by Diog. Laertius] |