39 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] |
13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
4913 | Brain lesions can erase whole categories of perception, suggesting they are hard-wired [Carter,R] |
4910 | Sense organs don't discriminate; they reduce various inputs to the same electrical pulses [Carter,R] |
4911 | The recognition sequence is: classify, name, locate, associate, feel [Carter,R, by PG] |
4919 | There seems to be no dividing line between a memory and a thought [Carter,R] |
4908 | No one knows if animals are conscious [Carter,R] |
4902 | Pain doesn't have one brain location, but is linked to attention and emotion [Carter,R] |
4904 | Proper brains appear at seven weeks, and neonates have as many neurons as adults do [Carter,R] |
4915 | In primates, brain size correlates closely with size of social group [Carter,R] |
4917 | Consciousness involves awareness, perception, self-awareness, attention and reflection [Carter,R] |
4916 | There is enormous evidence that consciousness arises in the frontal lobes of the brain [Carter,R] |
4905 | Normal babies seem to have overlapping sense experiences [Carter,R] |
4918 | In blindsight V1 (normal vision) is inactive, but V5 (movement) lights up [Carter,R] |
4912 | Out-of-body experiences may be due to temporary loss of proprioception [Carter,R] |
4903 | Scans of brains doing similar tasks produce very similar patterns of activation [Carter,R] |
4920 | Thinking takes place on the upper side of the prefrontal cortex [Carter,R] |
4906 | Babies show highly emotional brain events, but may well be unaware of them [Carter,R] |
4909 | The only way we can control our emotions is by manipulating the outside world that influences them [Carter,R] |
4914 | A frog will starve to death surrounded by dead flies [Carter,R] |
4907 | The 'locus coeruleus' is one of several candidates for the brain's 'pleasure centre' [Carter,R] |
20558 | Your representative owes you his judgement, and betrays you if he gives your opinion instead [Burke] |