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Single Idea 18249

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy ]

Full Idea

A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε.

Gist of Idea

Cauchy gave a formal definition of a converging sequence.

Source

Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4)

Book Ref

Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.181

A Reaction

The sequence is 'Cauchy' if N exists.

The 142 ideas from Stewart Shapiro

15944 Second-order logic is better than set theory, since it only adds relations and operations, and nothing else [Shapiro, by Lavine]
13629 Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics? [Shapiro]
13626 Semantic consequence is ineffective in second-order logic [Shapiro]
13624 The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed [Shapiro]
13627 There is no 'correct' logic for natural languages [Shapiro]
13631 Are sets part of logic, or part of mathematics? [Shapiro]
13628 We can live well without completeness in logic [Shapiro]
13630 Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro]
13625 Mathematics and logic have no border, and logic must involve mathematics and its ontology [Shapiro]
13635 'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence [Shapiro]
13633 'Satisfaction' is a function from models, assignments, and formulas to {true,false} [Shapiro]
13634 Satisfaction is 'truth in a model', which is a model of 'truth' [Shapiro]
13632 Finding the logical form of a sentence is difficult, and there are no criteria of correctness [Shapiro]
13637 If a logic is incomplete, its semantic consequence relation is not effective [Shapiro]
13636 An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro]
13640 Russell's paradox shows that there are classes which are not iterative sets [Shapiro]
13638 Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects [Shapiro]
13641 Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals [Shapiro]
13642 Logic is the ideal for learning new propositions on the basis of others [Shapiro]
13643 Aristotelian logic is complete [Shapiro]
13644 Semantics for models uses set-theory [Shapiro]
13645 In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro]
13650 Henkin semantics has separate variables ranging over the relations and over the functions [Shapiro]
13647 Choice is essential for proving downward Löwenheim-Skolem [Shapiro]
13649 Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics [Shapiro]
13646 Compactness is derived from soundness and completeness [Shapiro]
13648 The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro]
13651 A set is 'transitive' if contains every member of each of its members [Shapiro]
13657 First-order arithmetic can't even represent basic number theory [Shapiro]
13652 The 'continuum' is the cardinality of the powerset of a denumerably infinite set [Shapiro]
13653 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro]
13654 It is central to the iterative conception that membership is well-founded, with no infinite descending chains [Shapiro]
13656 Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro]
13660 Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable [Shapiro]
13661 A language is 'semantically effective' if its logical truths are recursively enumerable [Shapiro]
13658 Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro]
13659 Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro]
13662 First-order logic was an afterthought in the development of modern logic [Shapiro]
13666 Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets [Shapiro]
13664 Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions [Shapiro]
13663 Some reject formal properties if they are not defined, or defined impredicatively [Shapiro]
13667 Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order [Shapiro]
13668 Bernays (1918) formulated and proved the completeness of propositional logic [Shapiro]
13669 Can one develop set theory first, then derive numbers, or are numbers more basic? [Shapiro]
13670 Categoricity can't be reached in a first-order language [Shapiro]
13673 The notion of finitude is actually built into first-order languages [Shapiro]
13674 We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro]
13675 Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro]
13676 Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro]
13677 Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals [Shapiro]
10294 Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro]
10590 Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
10292 Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]
10290 Second-order variables also range over properties, sets, relations or functions [Shapiro]
10588 First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro]
10591 Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro]
10296 The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro]
10299 If the aim of logic is to codify inferences, second-order logic is useless [Shapiro]
10300 Logical consequence can be defined in terms of the logical terminology [Shapiro]
10297 The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
10298 Some say that second-order logic is mathematics, not logic [Shapiro]
10301 The axiom of choice is controversial, but it could be replaced [Shapiro]
10279 Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro]
10203 We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro]
10200 We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro]
10202 Natural numbers just need an initial object, successors, and an induction principle [Shapiro]
10205 Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro]
10207 Anti-realists reject set theory [Shapiro]
10206 Modal operators are usually treated as quantifiers [Shapiro]
10204 An 'implicit definition' gives a direct description of the relations of an entity [Shapiro]
10201 Virtually all of mathematics can be modeled in set theory [Shapiro]
10208 Axiom of Choice: some function has a value for every set in a given set [Shapiro]
10209 A function is just an arbitrary correspondence between collections [Shapiro]
10210 If mathematical objects are accepted, then a number of standard principles will follow [Shapiro]
10213 Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro]
10214 Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro]
10212 Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro]
10218 Baseball positions and chess pieces depend entirely on context [Shapiro]
10215 Platonists claim we can state the essence of a number without reference to the others [Shapiro]
10217 We can apprehend structures by focusing on or ignoring features of patterns [Shapiro]
10221 Is there is no more to structures than the systems that exemplify them? [Shapiro]
10222 Mathematical foundations may not be sets; categories are a popular rival [Shapiro]
10220 Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro]
10223 There is no 'structure of all structures', just as there is no set of all sets [Shapiro]
10224 The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro]
10228 Could infinite structures be apprehended by pattern recognition? [Shapiro]
10227 The abstract/concrete boundary now seems blurred, and would need a defence [Shapiro]
10226 Mathematicians regard arithmetic as concrete, and group theory as abstract [Shapiro]
10229 Simple types can be apprehended through their tokens, via abstraction [Shapiro]
10230 The 4-pattern is the structure common to all collections of four objects [Shapiro]
8703 Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend]
10231 Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro]
10233 Platonism must accept that the Peano Axioms could all be false [Shapiro]
10236 There is no grounding for mathematics that is more secure than mathematics [Shapiro]
10235 A sentence is 'satisfiable' if it has a model [Shapiro]
10238 The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro]
10234 Any theory with an infinite model has a model of every infinite cardinality [Shapiro]
10237 Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro]
10239 The central notion of model theory is the relation of 'satisfaction' [Shapiro]
10240 Model theory deals with relations, reference and extensions [Shapiro]
18243 Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro]
10244 Intuition is an outright hindrance to five-dimensional geometry [Shapiro]
10248 Number statements are generalizations about number sequences, and are bound variables [Shapiro]
18245 Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro]
10249 The main mathematical structures are algebraic, ordered, and topological [Shapiro]
10251 The law of excluded middle might be seen as a principle of omniscience [Shapiro]
10252 The Axiom of Choice seems to license an infinite amount of choosing [Shapiro]
10253 Either logic determines objects, or objects determine logic, or they are separate [Shapiro]
10254 Can the ideal constructor also destroy objects? [Shapiro]
10255 Presumably nothing can block a possible dynamic operation? [Shapiro]
10256 For intuitionists, proof is inherently informal [Shapiro]
10257 Intuitionism only sanctions modus ponens if all three components are proved [Shapiro]
10258 Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro]
10262 Fictionalism eschews the abstract, but it still needs the possible (without model theory) [Shapiro]
10259 The two standard explanations of consequence are semantic (in models) and deductive [Shapiro]
10268 Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro]
10266 Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro]
10270 The main versions of structuralism are all definitionally equivalent [Shapiro]
10272 The notion of 'object' is at least partially structural and mathematical [Shapiro]
10274 Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro]
10273 Some structures are exemplified by both abstract and concrete [Shapiro]
10276 Mathematical structures are defined by axioms, or in set theory [Shapiro]
10275 A blurry border is still a border [Shapiro]
10277 Structuralism blurs the distinction between mathematical and ordinary objects [Shapiro]
10280 A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro]
9554 We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro]
9626 A structure is an abstraction, focussing on relationships, and ignoring other features [Shapiro]
8725 Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro]
8730 'Impredicative' definitions refer to the thing being described [Shapiro]
8729 Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
8731 Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
8744 Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
8749 Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
8750 Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
8752 Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
8753 Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
8760 Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
8761 A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
8763 The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
8762 Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
8764 Categories are the best foundation for mathematics [Shapiro]
18249 Cauchy gave a formal definition of a converging sequence. [Shapiro]