more on this theme
|
more from this thinker
Single Idea 13459
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
]
Full Idea
We can show (using the axiom of choice) that the less-than relation, <, well-orders the ordinals, ...and that it partially orders the ordinals, ...and that it totally orders the ordinals.
Gist of Idea
The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers
Source
William D. Hart (The Evolution of Logic [2010], 1)
Book Ref
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.26
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]
|
13461
|
We can choose from finite and evident sets, but not from infinite opaque ones
[Hart,WD]
|
13462
|
With the Axiom of Choice every set can be well-ordered
[Hart,WD]
|
13441
|
Naïve set theory has trouble with comprehension, the claim that every predicate has an extension
[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]
|
13457
|
A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets
[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]
|
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]
|
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]
|
13477
|
The problems are the monuments of philosophy
[Hart,WD]
|
13466
|
We are all post-Kantians, because he set the current agenda for philosophy
[Hart,WD]
|
13481
|
Maybe sets should be rethought in terms of the even more basic categories
[Hart,WD]
|
13490
|
Von Neumann defines α<β as α∈β
[Hart,WD]
|
13488
|
Mass words do not have plurals, or numerical adjectives, or use 'fewer'
[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]
|
13496
|
First-order logic is 'compact': consequences of a set are consequences of a finite subset
[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]
|
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]
|
13500
|
Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent
[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]
|
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]
|