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Single Idea 13474

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry ]

Full Idea

There is a familiar comparison between Euclid (unique parallel) and 'spherical' geometry (no parallel) and 'saddle' geometry (several parallels).

Gist of Idea

Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several

Source

William D. Hart (The Evolution of Logic [2010], 2)

Book Ref

Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.44

The 45 ideas from 'The Evolution of Logic'

13456 Set theory articulates the concept of order (through relations) [Hart,WD]
13443 ∈ relates across layers, while ⊆ relates within layers [Hart,WD]
13442 Without the empty set we could not form a∩b without checking that a and b meet [Hart,WD]
13462 With the Axiom of Choice every set can be well-ordered [Hart,WD]
13461 We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD]
13441 Naïve set theory has trouble with comprehension, the claim that every predicate has an extension [Hart,WD]
13460 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD]
13458 A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD]
13457 A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD]
13459 The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD]
13463 There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD]
13446 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD]
13516 If we accept that V=L, it seems to settle all the open questions of set theory [Hart,WD]
13515 To study abstract problems, some knowledge of set theory is essential [Hart,WD]
13469 Tarski showed how we could have a correspondence theory of truth, without using 'facts' [Hart,WD]
13466 We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD]
13477 The problems are the monuments of philosophy [Hart,WD]
13481 Maybe sets should be rethought in terms of the even more basic categories [Hart,WD]
13474 Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD]
13471 Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD]
13476 The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori [Hart,WD]
13475 The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD]
13488 Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD]
13490 Von Neumann defines α<β as α∈β [Hart,WD]
13491 The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD]
13492 Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD]
13484 Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that [Hart,WD]
13496 First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD]
13482 The Burali-Forti paradox is a crisis for Cantor's ordinals [Hart,WD]
13497 Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe [Hart,WD]
13494 The iterative conception may not be necessary, and may have fixed points or infinitely descending chains [Hart,WD]
13493 In the modern view, foundation is the heart of the way to do set theory [Hart,WD]
13495 Foundation Axiom: an nonempty set has a member disjoint from it [Hart,WD]
13504 Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do [Hart,WD]
13503 A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth [Hart,WD]
13500 Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent [Hart,WD]
13502 ∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...' [Hart,WD]
13507 The machinery used to solve the Liar can be rejigged to produce a new Liar [Hart,WD]
13506 The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD]
13505 Model theory studies how set theory can model sets of sentences [Hart,WD]
13509 We can establish truths about infinite numbers by means of induction [Hart,WD]
13511 Model theory is mostly confined to first-order theories [Hart,WD]
13513 Models are ways the world might be from a first-order point of view [Hart,WD]
13512 Modern model theory begins with the proof of Los's Conjecture in 1962 [Hart,WD]
13480 Fregean self-evidence is an intrinsic property of basic truths, rules and definitions [Hart,WD]