10237 | Coherence is a primitive, intuitive notion, not reduced to something formal |

10204 | An 'implicit definition' gives a direct description of the relations of an entity |

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

13643 | Aristotelian logic is complete |

10206 | Modal operators are usually treated as quantifiers |

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

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

10208 | Axiom of Choice: some function has a value for every set in a given set |

10252 | The Axiom of Choice seems to license an infinite amount of choosing |

10301 | The axiom of choice is controversial, but it could be replaced |

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 |

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

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

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

10207 | Anti-realists reject set theory |

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

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

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? |

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 |

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

10588 | First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems |

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? |

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 |

10299 | If the aim of logic is to codify inferences, second-order logic is useless |

10298 | Some say that second-order logic is mathematics, not logic |

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

10300 | Logical consequence can be defined in terms of the logical terminology |

10259 | The two standard explanations of consequence are semantic (in models) and deductive |

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

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

10257 | Intuitionism only sanctions modus ponens if all three components are proved |

10253 | Either logic determines objects, or objects determine logic, or they are separate |

10251 | The law of excluded middle might be seen as a principle of omniscience |

8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects |

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

10212 | Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' |

10209 | A function is just an arbitrary correspondence between collections |

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

10290 | Second-order variables also range over properties, sets, relations or functions |

10268 | Maybe plural quantifiers should be understood in terms of classes or sets |

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

10235 | A sentence is 'satisfiable' if it has a model |

13644 | Semantics for models uses set-theory |

10239 | The central notion of model theory is the relation of 'satisfaction' |

10240 | Model theory deals with relations, reference and extensions |

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 |

10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' |

10238 | The set-theoretical hierarchy contains as many isomorphism types as possible |

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

10234 | Any theory with an infinite model has a model of every infinite cardinality |

10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics |

10297 | The Löwenheim-Skolem theorem seems to be a defect of first-order logic |

10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model |

10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them |

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

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

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

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 |

10201 | Virtually all of mathematics can be modeled in set theory |

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

8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex |

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 |

10213 | Real numbers are thought of as either Cauchy sequences or Dedekind cuts |

18243 | Understanding the real-number structure is knowing usage of the axiomatic language of analysis |

18249 | Cauchy gave a formal definition of a converging sequence. |

18245 | Cuts are made by the smallest upper or largest lower number, some of them not rational |

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

8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals |

10256 | For intuitionists, proof is inherently informal |

10236 | There is no grounding for mathematics that is more secure than mathematics |

8764 | Categories are the best foundation for mathematics |

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

10202 | Natural numbers just need an initial object, successors, and an induction principle |

10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory |

10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) |

10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem |

8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 |

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

10222 | Mathematical foundations may not be sets; categories are a popular rival |

10218 | Baseball positions and chess pieces depend entirely on context |

10224 | The even numbers have the natural-number structure, with 6 playing the role of 3 |

10228 | Could infinite structures be apprehended by pattern recognition? |

10230 | The 4-pattern is the structure common to all collections of four objects |

10249 | The main mathematical structures are algebraic, ordered, and topological |

10273 | Some structures are exemplified by both abstract and concrete |

10276 | Mathematical structures are defined by axioms, or in set theory |

8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers |

8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them |

10270 | The main versions of structuralism are all definitionally equivalent |

10221 | Is there is no more to structures than the systems that exemplify them? |

10248 | Number statements are generalizations about number sequences, and are bound variables |

10220 | Because one structure exemplifies several systems, a structure is a one-over-many |

10223 | There is no 'structure of all structures', just as there is no set of all sets |

8703 | Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics |

10274 | Does someone using small numbers really need to know the infinite structure of arithmetic? |

10200 | We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) |

10210 | If mathematical objects are accepted, then a number of standard principles will follow |

10215 | Platonists claim we can state the essence of a number without reference to the others |

10233 | Platonism must accept that the Peano Axioms could all be false |

10244 | Intuition is an outright hindrance to five-dimensional geometry |

10280 | A stone is a position in some pattern, and can be viewed as an object, or as a location |

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 |

8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own |

8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names |

8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? |

8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms |

10254 | Can the ideal constructor also destroy objects? |

10255 | Presumably nothing can block a possible dynamic operation? |

8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions |

8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts |

8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities |

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

8730 | 'Impredicative' definitions refer to the thing being described |

10279 | Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? |

10227 | The abstract/concrete boundary now seems blurred, and would need a defence |

10226 | Mathematicians regard arithmetic as concrete, and group theory as abstract |

10262 | Fictionalism eschews the abstract, but it still needs the possible (without model theory) |

10277 | Structuralism blurs the distinction between mathematical and ordinary objects |

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

10591 | Logicians use 'property' and 'set' interchangeably, with little hanging on it |

10272 | The notion of 'object' is at least partially structural and mathematical |

10275 | A blurry border is still a border |

10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models |

10266 | Why does the 'myth' of possible worlds produce correct modal logic? |

8725 | Rationalism tries to apply mathematical methodology to all of knowledge |

10203 | We apprehend small, finite mathematical structures by abstraction from patterns |

10229 | Simple types can be apprehended through their tokens, via abstraction |

9626 | A structure is an abstraction, focussing on relationships, and ignoring other features |

10217 | We can apprehend structures by focusing on or ignoring features of patterns |

9554 | We can focus on relations between objects (like baseballers), ignoring their other features |

10231 | Abstract objects might come by abstraction over an equivalence class of base entities |