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Single Idea 13463

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers ]

Full Idea

Since we can map the transfinite ordinals one-one into the infinite cardinals, there are at least as many infinite cardinals as transfinite ordinals.

Gist of Idea

There are at least as many infinite cardinals as transfinite ordinals (because they will map)

Source

William D. Hart (The Evolution of Logic [2010], 1)

Book Ref

Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.27

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]