2005 | A Tour through Mathematical Logic |
Pref | p.-8 | 13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth |
Pref | p.-8 | 13518 | Modern mathematics has unified all of its objects within set theory |
1.2 | p.11 | 13520 | A 'tautology' must include connectives |
1.3 | p.20 | 13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance |
1.3 | p.20 | 13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) |
1.3 | p.20 | 13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P |
1.3 | p.31 | 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof |
1.7 | p.54 | 13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens |
2.2 | p.62 | 13526 | Comprehension Axiom: if a collection is clearly specified, it is a set |
2.2 | p.64 | 13527 | Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC |
2.2 | p.67 | 13528 | Continuum Hypothesis: there are no sets between N and P(N) |
2.3 | p.70 | 13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists |
2.4 | p.77 | 13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive |
5.1 | p.165 | 13531 | Model theory reveals the structures of mathematics |
5.2 | p.167 | 13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' |
5.3 | p.172 | 13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem |
5.3 | p.172 | 13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide |
5.3 | p.174 | 13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation |
5.4 | p.181 | 13537 | An 'isomorphism' is a bijection that preserves all structural components |
5.5 | p.191 | 13538 | If a theory is complete, only a more powerful language can strengthen it |
5.7 | p.224 | 13539 | The LST Theorem is a serious limitation of first-order logic |