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Single Idea 18101

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number ]

Full Idea

If you add to the ordinals you produce many different ordinals, each measuring the length of the sequence of ordinals less than it. They each have cardinality aleph-0. The cardinality eventually increases, but we can't say where this break comes.

Gist of Idea

Each addition changes the ordinality but not the cardinality, prior to aleph-1

Source

David Bostock (Philosophy of Mathematics [2009], 4.5)

Book Ref

Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.110

Related Idea

Idea 18106 Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock]

The 121 ideas from David Bostock

13821 Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects [Bostock]
13346 Truth is the basic notion in classical logic [Bostock]
13347 Validity is a conclusion following for premises, even if there is no proof [Bostock]
13348 It seems more natural to express |= as 'therefore', rather than 'entails' [Bostock]
13349 Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock]
13350 'Assumptions' says that a formula entails itself (φ|=φ) [Bostock]
13351 'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock]
13352 'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock]
13353 'Negation' says that Γ,¬φ|= iff Γ|=φ [Bostock]
13354 'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ [Bostock]
13355 'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [Bostock]
13356 The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [Bostock]
13421 'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope [Bostock]
13422 'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope [Bostock]
13357 Truth-functors are usually held to be defined by their truth-tables [Bostock]
13358 Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock]
13359 Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock]
13362 If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality [Bostock]
13360 In logic, a name is just any expression which refers to a particular single object [Bostock]
13361 An expression is only a name if it succeeds in referring to a real object [Bostock]
13363 A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock]
13364 Interpretation by assigning objects to names, or assigning them to variables first [Bostock, by PG]
13438 'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors [Bostock]
13439 Venn Diagrams map three predicates into eight compartments, then look for the conclusion [Bostock]
13611 Tableau proofs use reduction - seeking an impossible consequence from an assumption [Bostock]
13612 Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock]
13540 A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock]
13542 A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock]
13541 For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock]
13613 A completed open branch gives an interpretation which verifies those formulae [Bostock]
13543 A relation is not reflexive, just because it is transitive and symmetrical [Bostock]
13544 Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock]
13545 Elementary logic cannot distinguish clearly between the finite and the infinite [Bostock]
13623 The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' [Bostock]
13610 A logic with ¬ and → needs three axiom-schemas and one rule as foundation [Bostock]
13617 MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock]
13614 MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock]
13615 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ [Bostock]
13616 The Deduction Theorem greatly simplifies the search for proof [Bostock]
13618 Compactness means an infinity of sequents on the left will add nothing new [Bostock]
13619 Quantification adds two axiom-schemas and a new rule [Bostock]
13620 Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem [Bostock]
13621 The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth [Bostock]
13622 Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine... [Bostock]
13753 Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part [Bostock]
13755 Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it [Bostock]
13754 Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E) [Bostock]
13756 A tree proof becomes too broad if its only rule is Modus Ponens [Bostock]
13758 In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle [Bostock]
13757 Unlike natural deduction, semantic tableaux have recipes for proving things [Bostock]
13759 Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded [Bostock]
13760 A sequent calculus is good for comparing proof systems [Bostock]
13761 In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock]
13762 Tableau rules are all elimination rules, gradually shortening formulae [Bostock]
13801 An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English [Bostock]
13799 The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [Bostock]
13803 If we are to express that there at least two things, we need identity [Bostock]
13800 |= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock]
13802 Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock]
13812 A 'zero-place' function just has a single value, so it is a name [Bostock]
13811 A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock]
13813 Definite descriptions don't always pick out one thing, as in denials of existence, or errors [Bostock]
13814 Definite desciptions resemble names, but can't actually be names, if they don't always refer [Bostock]
13816 Because of scope problems, definite descriptions are best treated as quantifiers [Bostock]
13817 Definite descriptions are usually treated like names, and are just like them if they uniquely refer [Bostock]
13815 Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem [Bostock]
13818 If we allow empty domains, we must allow empty names [Bostock]
13820 The idea that anything which can be proved is necessary has a problem with empty names [Bostock]
13822 Fictional characters wreck elementary logic, as they have contradictions and no excluded middle [Bostock]
13846 A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock]
13847 If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock]
13848 We are only obliged to treat definite descriptions as non-names if only the former have scope [Bostock]
18095 Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock]
18093 For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock]
18097 The Peano Axioms describe a unique structure [Bostock]
18099 The number of reals is the number of subsets of the natural numbers [Bostock]
18100 ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock]
18101 Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock]
18102 A cardinal is the earliest ordinal that has that number of predecessors [Bostock]
18106 Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock]
18107 A 'proper class' cannot be a member of anything [Bostock]
18105 Replacement enforces a 'limitation of size' test for the existence of sets [Bostock]
18109 The completeness of first-order logic implies its compactness [Bostock]
18108 First-order logic is not decidable: there is no test of whether any formula is valid [Bostock]
18110 Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock]
18111 Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock]
18116 Numbers can't be positions, if nothing decides what position a given number has [Bostock]
18115 We could add axioms to make sets either as small or as large as possible [Bostock]
18114 There is no single agreed structure for set theory [Bostock]
18117 Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock]
18121 In logic a proposition means the same when it is and when it is not asserted [Bostock]
18122 Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock]
18120 The Deduction Theorem is what licenses a system of natural deduction [Bostock]
18123 Substitutional quantification is just standard if all objects in the domain have a name [Bostock]
18125 Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock]
18127 Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock]
18129 Many crucial logicist definitions are in fact impredicative [Bostock]
18131 If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock]
18134 Predicativism makes theories of huge cardinals impossible [Bostock]
18135 If mathematics rests on science, predicativism may be the best approach [Bostock]
18136 If we can only think of what we can describe, predicativism may be implied [Bostock]
18133 The usual definitions of identity and of natural numbers are impredicative [Bostock]
18132 The predicativity restriction makes a difference with the real numbers [Bostock]
18137 Impredicative definitions are wrong, because they change the set that is being defined? [Bostock]
18140 The best version of conceptualism is predicativism [Bostock]
18138 Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock]
18139 The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock]
18141 Nominalism about mathematics is either reductionist, or fictionalist [Bostock]
18158 Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock]
18159 Higher cardinalities in sets are just fairy stories [Bostock]
18144 Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock]
18147 Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock]
18146 If Hume's Principle is the whole story, that implies structuralism [Bostock]
18149 There are many criteria for the identity of numbers [Bostock]
18148 Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock]
18145 Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock]
18143 Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock]
18150 Actual measurement could never require the precision of the real numbers [Bostock]
18155 A fairy tale may give predictions, but only a true theory can give explanations [Bostock]
18156 Modern axioms of geometry do not need the real numbers [Bostock]
18157 Nominalism as based on application of numbers is no good, because there are too many applications [Bostock]