Title: Theory / Unimportant / Math / Analysis / Non-standard / Goldblatt+15 / Other
Tags: récupéré

Name: An infinitely small number
Definition: A number \(r\) such that [$]\abs{r}<\frac 1n[/$] for all \(n\in\mathbb N\)
Page: 3, 27
Chapter: 1, 3
Section: 1, 8
Definition2: The inverse of an infinite number
Example: \([\epsilon]\) with \(\epsilon=\langle\frac 1n\mid n\in\mathbb N\rangle\)
Name2: An infinitesimal number
Denoted by: \(\epsilon\)

Name: An infinitely large number
Definition: A number \(r\) such that [$]\abs{r}>n[/$] for all \(n\in\mathbb N\)
Page: 3, 27
Chapter: 1, 3
Section: 1, 8
Definition2: The inverse of an infinitesimal
Example: \([\omega]\) with \(\omega=\langle n\mid n\in\mathbb N\rangle\)
Name2: Infinite number
Denoted by: \(\omega\)
Name3: An unlimited number

Smaller0: \(S\) is appropriately formulated
Greater0: \(S\) is true of \(^*\mathbb R\) if and only if it is true of \(\mathbb R\)
Page: 11
Chapter: 1
Section: 3
Variables: \(S\) a statement
Name: Transfer principle

Name: A Dedekind complete set
Definition: An ordered set whose every non-empty subset with an upper bound admits a least upper bound
Page: 12,
https://proofwiki.org/wiki/Definition:Dedekind_Complete_Set
Chapter: 1
Section: 3
Index: (2)
Example: \(\mathbb R\)
Name2: A set with the supremum property
Counterexample: \(\mathbb Q\)
Name3: A set with the least upper bound property
Counterexample2: \(^*\mathbb Q\)

Name: A Dedekind cut
Definition: with \(l<u\) for \(l\in L\) and \(u\in U\)
Page: 11
Chapter: 1
Section: 3
Definition2: no largest member in \(L\)
Conjdef: A partition of \(\mathbb Q\) into \(\langle L,U\rangle\)

Name: A complete ordered field
Definition: An ordered field which is Dedekind complete
Page: 11
Chapter: 1
Section: 3
Index: (2)

Name: A Cauchy sequence
Definition: A sequence \((r_i)_{i\in\mathbb N}\) such that, [$]\lim_{n,m\to\infty}\abs{r_n-r_m}=0[/$]
Page: 11
Chapter: 1
Section: 3
Index: (4)

Name: Equivalenty Cauchy sequence
Definition: Two sequences \((r_i)_{i\in\mathbb N}, (s_i)_{i\in\mathbb N}\) such that, [$]\lim_{n\to\infty}\abs{r_n-s_n}=0[/$]
Page: 11
Chapter: 1
Section: 3
Index: (4)

Name: A real number
Definition: A Dedekind cut
Page: 11
Chapter: 1
Section: 3
Definition2: Cauchy sequence of rationals, up to equivalences
Representation: \(r_n\dots r_1r_0.r_{-1}\dots \) with \(r_i\) digits, representing \(\sum r_i 10^i\)

Name: The set of reals
Definition: The only complete ordered field up to isomorphism
Page: 11
Chapter: 1
Section: 3
Notation: \(\mathbb R\)
Definition2: The set of real numbers, for some explicit definition of them

Name: The agreement set of \((r_i)\) and \((s_i)\)
Definition: The set of \(n\) such that \(r_n=s_n\)
Page: 16, 24
Chapter: 2, 3
Section: 2, 4
Variables: \((r_i),(s_i)\) sequence of reals
Context: Non standard analysis
Notation: \(E_{rs}\)

Name: The value of \(r=s\)
Definition: The set of \(n\in\mathbb N\) such that \(P_n\) holds
Page: 24
Chapter: 3
Section: 4
Variables: \(\langle P_i\rangle\) a sequence of properties which holds or not depending on \(i\in\mathbb N\)
Notation: [$]\llbracket P\rrbracket[/$]
Example: for \(\langle r_i\rangle,\langle s_i\rangle\), [$]\llbracket r=s\rrbracket[/$] is the set of positions at which \(r_n=s_n\)
Name2: The interpretation of \(r=s\)

Name: A cofinite subset of \(X\)
Definition: A set \(Y\subseteq X\) such that \(X\setminus Y\) is finite
Page: 17
Chapter: 2
Variables: \(X\) a set

Definition: The set of \(n\) such that \(r_n<s_n\).
Page: 17
Chapter: 2
Variables: \((r_i),(s_i)\) sequence of reals
Notation: \(L_{rs}\)

Smaller0: \(\mathcal H\) is closed under taking superset
Greater0: \(\mathcal H\) is not empty
Page: 18
Chapter: 2
Section: 5
Variables: \(I\) a set,
\(\mathcal H\) a set of subsets of \(I\)
Index: (4)
Kind: Ex
Greater1: \(I\in\mathcal H\)
Connect Greater0: iff

Smaller0: each chain of \(P\) has an upper bound in \(P\)
Greater0: \(P\) contains a \(\le\)-maximal element
Page: 20
Chapter: 2
Section: 6
Variables: \((P,\le)\) partially ordered set
Smaller to greater: iff
Name: Zorn's Lemma

Name: Axiom of choice
Definition: There is a function \(f:I\to\bigcup X_i\) such that for each \(i\in I\), \(f(i)\in X_i\)
Page: 20
Chapter: 2
Section: 6
Variables: \(I\) an indexing family
\(\{X_i\}\) a family of set indexed by \(I\)

Name: Ring of real-valued sequences
Definition: Ring of functions from \(\mathbb N\) to \(\mathbb R\) with pointwise sum, product
Page: 23
Chapter: 3
Section: 1
Notation: \(\mathbb{R^n}\)
Kind: Theo
Typ: commutative ring with 0 and 1, not a field

Definition: \(\langle \frac 1n\rangle_{n\in\mathbb N}\)
Page: 27
Chapter: 3
Section: 7
Context: Non standard analysis
Notation: \(\epsilon\)

Definition: \(\langle n\rangle_{n\in\mathbb N}\)
Page: 27
Chapter: 3
Section: 7
Notation: \(\omega\)

Question: \([\omega][\epsilon]\)
Answer: equals \([1]\)
Page: 27
Chapter: 3
Section: 7

Name: Axiomatic approach to the analysis of \(\mathbb R\)
Definition: Considering the real as a model of the theory of complete ordered field
Page: 13
Chapter: 1
Section: 3
Name2: Descriptive approach to the analysis of \(\mathbb R\)

Name: Constructive approach to the analysis of \(\mathbb R\)
Definition: Considering the real as a set constructed using the rationals.
Page: 13
Chapter: 1
Section: 3
Example: Dedekind cuts
Example2: Equivalence of Cantor sequence