Click on the Idea Number for the full details | back to texts | expand these ideas

13634 | Satisfaction is 'truth in a model', which is a model of 'truth' |

13643 | Aristotelian logic is complete |

13651 | A set is 'transitive' if contains every member of each of its members |

13647 | Choice is essential for proving downward Löwenheim-Skolem |

13631 | Are sets part of logic, or part of mathematics? |

13654 | It is central to the iterative conception that membership is well-founded, with no infinite descending chains |

13640 | Russell's paradox shows that there are classes which are not iterative sets |

13666 | Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets |

13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element |

13642 | Logic is the ideal for learning new propositions on the basis of others |

13627 | There is no 'correct' logic for natural languages |

13667 | Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order |

13668 | Bernays (1918) formulated and proved the completeness of propositional logic |

13669 | Can one develop set theory first, then derive numbers, or are numbers more basic? |

13662 | First-order logic was an afterthought in the development of modern logic |

13624 | The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed |

13660 | Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable |

13673 | The notion of finitude is actually built into first-order languages |

13650 | Henkin semantics has separate variables ranging over the relations and over the functions |

13645 | In standard semantics for second-order logic, a single domain fixes the ranges for the variables |

13649 | Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics |

15944 | Second-order logic is better than set theory, since it only adds relations and operations, and nothing else |

13629 | Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? |

13626 | Semantic consequence is ineffective in second-order logic |

13637 | If a logic is incomplete, its semantic consequence relation is not effective |

13632 | Finding the logical form of a sentence is difficult, and there are no criteria of correctness |

13674 | We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models |

13633 | 'Satisfaction' is a function from models, assignments, and formulas to {true,false} |

13644 | Semantics for models uses set-theory |

13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation |

13670 | Categoricity can't be reached in a first-order language |

13648 | The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity |

13658 | Downward Löwenheim-Skolem: each satisfiable countable set always has countable models |

13659 | Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes |

13675 | Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails |

13635 | 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence |

13628 | We can live well without completeness in logic |

13630 | Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures |

13646 | Compactness is derived from soundness and completeness |

13661 | A language is 'semantically effective' if its logical truths are recursively enumerable |

13641 | Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals |

13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are |

13677 | Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals |

13652 | The 'continuum' is the cardinality of the powerset of a denumerably infinite set |

13657 | First-order arithmetic can't even represent basic number theory |

13656 | Some sets of natural numbers are definable in set-theory but not in arithmetic |

13664 | Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions |

13625 | Mathematics and logic have no border, and logic must involve mathematics and its ontology |

13663 | Some reject formal properties if they are not defined, or defined impredicatively |

13638 | Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects |