44 ideas
| 13520 | A 'tautology' must include connectives [Wolf,RS] |
| 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
| 13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
| 13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
| 13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
| 13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
| 13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
| 13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
| 13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
| 13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
| 13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
| 13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
| 13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS] |
| 13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
| 13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
| 13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
| 13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
| 13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
| 22886 | The modern idea of 'limit' allows infinite quantities to have a finite sum [Bardon] |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| 22914 | An equally good question would be why there was nothing instead of something [Bardon] |
| 7492 | Early societies are based on community, and modern societies on association [Tönnies, by Watson] |
| 22902 | Why does an effect require a prior event if the prior event isn't a cause? [Bardon] |
| 22905 | Becoming disordered is much easier for a system than becoming ordered [Bardon] |
| 22913 | The universe expands, so space-time is enlarging [Bardon] |
| 22889 | We should treat time as adverbial, so we don't experience time, we experience things temporally [Bardon, by Bardon] |
| 22900 | How can we question the passage of time, if the question takes time to ask? [Bardon] |
| 22898 | What is time's passage relative to, and how fast does it pass? [Bardon] |
| 22897 | The A-series says a past event is becoming more past, but how can it do that? [Bardon] |
| 22901 | The B-series needs a revised view of causes, laws and explanations [Bardon] |
| 22896 | The B-series is realist about time, but idealist about its passage [Bardon] |
| 22903 | The B-series adds directionality when it accepts 'earlier' and 'later' [Bardon] |
| 22910 | To define time's arrow by causation, we need a timeless definition of causation [Bardon] |
| 22909 | We judge memories to be of the past because the events cause the memories [Bardon] |
| 22904 | The psychological arrow of time is the direction from our memories to our anticipations [Bardon] |
| 22906 | The direction of entropy is probabilistic, not necessary, so cannot be identical to time's arrow [Bardon] |
| 22907 | It is arbitrary to reverse time in a more orderly universe, but not in a sub-system of it [Bardon] |
| 22883 | It seems hard to understand change without understanding time first [Bardon] |
| 22890 | We experience static states (while walking round a house) and observe change (ship leaving dock) [Bardon] |
| 22884 | The motion of a thing should be a fact in the present moment [Bardon] |
| 22892 | Experiences of motion may be overlapping, thus stretching out the experience [Bardon] |
| 22911 | At least eternal time gives time travellers a possible destination [Bardon] |
| 22912 | Time travel is not a paradox if we include it in the eternal continuum of events [Bardon] |
| 22882 | We use calendars for the order of events, and clocks for their passing [Bardon] |