Title: Theory / Unimportant / Math / Logic / Model theory / Chang Keisler
Tags: fromMath mathToDef mathToSee récupéré toBinaryRelation automaticImplies defToTheory
On every note of this deck, unless a note says otherwise:
Chapter: 1
Chapter: 1

Q: Equivalently:
-[...],
-[...].
A: Relation between a formal language and its interpretation; universal algebra+logic
Page: 1
Name: Model theory

Name: Classical Model Theory
Definition: Model Theory of the first-order predicate logic.
Page: 1

Name: A model for \(\mathcal S\)
Typ: A subset of \(\mathcal S\)
Page: 4
Section: 2
Chapter: 1
Variables: \(\mathcal S\) a set.

Name: The set of sentences of \(\mathcal S\)
Definition: The smallest set containing elements of \(\mathcal S\), closed under \((\neg \phi)\) and \((\phi\lor\psi)\).
Page: 4
Section: 2
Variables: \(\mathcal S\) a set.

Definition: \((\neg((\neg\phi)\land(\neg\psi)))\).
Page: 6
Section: 2
Variables: \(\phi,\psi\) be formulas
Notation: \((\phi\lor \psi)\)

Definition: \(((\neg\phi)\lor \psi)\).
Page: 6
Section: 2
Variables: \(\phi,\psi\) be formulas
Notation: \((\phi\to \psi)\)

Definition: \(((\phi\to\psi)\land(\psi\to\phi))\).
Page: 6
Section: 2
Variables: \(\phi,\psi\) be formulas
Notation: \((\phi\leftrightarrow \psi)\)

Q: [...].
.
A: either \(\phi\) is a sentence symbol belonging to \(A\),
or a conjonction on formulas having model \(A\),
or the negation of a formula not having \(A\) as model
Page: 7
Chapter: 1
Section: 2
Variables: \(\phi\) a formula and \(A\) a model
Name: \(A\) is a model of \(\phi\)
Notation: \(A\models\phi\)

Name: A sentence valid for \(\mathcal S\)
Definition: A sentence which holds for all models of \(\mathcal S\).
Page: 7
Section: 2
Variables: \(\mathcal S\) a set.

Name: An assignment for \(\mathcal S\)
Definition: A function from \(\mathcal S\) to \(\{t,f\}\).
Page: 8
Section: 2
Variables: \(\mathcal S\) a set.

Q: -if \(\phi\) is a symbol, [...],
-if \(\phi=\neg\psi\), [...] and
-if \(\phi=\psi\land\xi\), [...].
A: \(A(\phi)\); the inverse of the value of \(\psi\); \(t\) if the values of \(\psi\) and \(\xi\) are \(t\)
Page: 8
Chapter: 1
Section: 2
Variables: \(\mathcal S\) a set, \(A\) an assignment for \(\mathcal S\)and \(\phi\) a sentence for \(\mathcal S\)
Name: The value of \(\phi\)

Name: \(\phi\) is a tautology
Definition: For all assignments, the value of \(\phi\) is \(t\).
Page: 8
Section: 2
Variables: \(\mathcal S\) a set
\(\phi\) a sentence for \(\mathcal S\)
Notation: \(\vdash \phi\)

Smaller0: \(\vdash\phi\)
Greater0: \(\models\phi\)
Page: 8
Section: 2
Variables: \(\mathcal S\) a set
\(\phi\) a sentence for \(\mathcal S\)
Index: 6
Kind: Prop
Smaller to greater: iff
Hide middle: yes
Name: Completeness theorem

Name: Rule of detachment
Definition: From \(\phi\) and \((\phi\to\psi)\) infer \(\psi\).
Page: 8
Section: 2

Name: A sentence deducible from \(\Sigma\)
Definition: A sentence \(\psi_n\) such that there exists a sequence \(\psi_0,\dots,\psi_n\), and each \(\psi_i\) is either in \(\Sigma\) or inferred from two earlier sentence or a tautology.
Page: 9
Section: 2
Variables: \(\Sigma\) a set of sentence.

Smaller0: a set of sentence such that every sentences are deducible from it
Greater0: \(\Sigma\) is inconsistent \(\Sigma\vdash S\land\neg S\) for any sentence \(S\)
Page: 9, 10
Section: 2
Variables: \(\Sigma\) a set of sentence.
Kind: ,
Smaller to greater: iff
Name: Inconsistent set of sentence

Name: Maximal consistent set of sentence
Definition: A set of sentence which is consistent, but the only consistent set of sentences which includes it is itself.
Page: 9
Section: 2
Index: 8

Smaller0: \(\Sigma\) is maximal consistent
Greater0: \(\phi\in\Sigma\)
Page: 10
Section: 2
Variables: \(\Sigma\) a set ef sentences and \(\phi\) a sentence
Index: 8 (ii)
Smaller to greater: iff
Smaller1: \(\Sigma\vdash\phi\)
Connect Smaller0: and

Smaller0: \(\Sigma\cup\{\psi\}\vdash\phi\)
Greater0: \(\Sigma\vdash\psi\to\phi\)
Page: 10
Section: 2
Variables: \(\Sigma\) a set ef sentences
\(\phi,\psi\) sentences
Index: 8 (iv)
Name: Deduction Theorem

Smaller0: \(\Sigma\) is consistent
Greater0: \(\Sigma\) can be enlarged to a maximal consistent set of sentences
Page: 10
Section: 2
Variables: \(\Sigma\) a set ef sentences
Index: 9
Kind: Lem
Hide middle: yes
Name: Lindenbaum's Theorem
Extra: doublon(autre paquet)

Smaller0: \(\Sigma\) is maximal consistent
Greater0: \(\phi\in\Sigma\) or \(\neg\phi\in\Sigma\)
Page: 10
Section: 2
Variables: \(\Sigma\) a set ef sentences
\(\phi\) a sentence
Index: 10 (i)
Kind: Lem
Hide middle: yes

\(\Sigma\) is maximal consistent implies \(\phi\land\psi\in\Sigma\) iff (\(\phi\in\Sigma\) and \(\psi\in\Sigma\)).
Page: 10
Chapter: 1
Section: 2
Variables: \(\Sigma\) a set ef sentences and \(\phi,\psi\) sentences
Index: 10 (ii)
Kind: Lem

Name: A model of \(\Sigma\)
Definition: A model such that every sentence \(\phi\in\Sigma\) is true in \(A\).
Page: 10
Section: 2
Variables: \(\Sigma\) a set ef sentences and \(A\) a model

Name: Satisfiable set of sentences
Definition: A set of sentences which admits a model.
Page: 10
Section: 2

Smaller0: satisfiable
Greater0: consistent
Page: 11
Section: 2
Variables: \(\Sigma\) a set ef sentences
Kind: Theo
Smaller to greater: iff
Hide middle: yes
Prefix Greater: \(\Sigma\) is
Name: Extended Completeness Theorem

Name: Finitely satisfiable set of sentences
Definition: A set of sentences whose every finite subset of \(\Sigma\) is satisfiable.
Page: 11
Section: 2

Smaller0: \(\Sigma\) is finitely satisfiable
Greater0: \(\Sigma\) is satisfiable
Page: 11
Section: 2
Variables: \(\Sigma\) a set ef sentences
Index: 12
Kind: Cor
Hide middle: yes
Name: Compactness Theorem

Smaller0: \(\Sigma\vdash\phi\)
Greater0: \(\Sigma\models\phi\)
Page: 11
Section: 2
Variables: \(\Sigma\) a set ef sentences
\(\phi\) a sentence
Index: 13 (i)
Kind: Cor
Smaller to greater: iff

Smaller0: \(\Sigma\models\phi\)
Greater0: some finite subset \(\Sigma_0\subseteq\Sigma\)
Page: 11
Section: 2
Variables: \(\Sigma\) a set ef sentences and \(\phi\) a sentence
Index: 13 (ii)
Kind: Cor
Hide middle: yes
Prefix Greater: \(\Sigma_0\models\phi\) where \(\Sigma_0\) is

Name: A Theory
Definition: A set of sentences.
Page: 12
Section: 2

Name: A closed theory
Definition: A theory whose every consequences of \(\Sigma\) belongs to \(\Sigma\).
Page: 12
Section: 2
Extra: It is indeed closed, and not complete

Name: A set of axiom for \(\Sigma\)
Definition: A set of \(\Delta\) such that both theories have the same consequences.
Page: 12
Section: 2

Name: Finitely axiomatizable theory
Definition: A theory which admits a finite set of axiom.
Page: 12
Section: 2

Definition: The set of all consequences of \(\Sigma\).
Page: 12
Section: 2
Variables: \(\Sigma\) theories
Notation: \(\overline \Sigma\)

Smaller0: \(\Delta\) is a set of axioms for \(\Sigma\)
Greater0: \(\Sigma\) and \(\Delta\) have the same models
Page: 12
Section: 2
Variables: \(\Sigma,\Delta\) theories
Index: 14
Kind: Prop
Smaller to greater: iff

Smaller0: the set of all models of \(\Gamma_2\) is the complement of the set of all models of \(\Gamma_1\)
Greater0: the \(\Gamma_i\)'s are finitely axiomatizable
Page: 12
Section: 2
Variables: \(\Gamma_1,\Gamma_2\) theories
Index: 15
Kind: Cor

Name: A positive sentence
Definition: A sentence built without negation symbol.
Page: 13
Section: 2
Variables: \(\phi\) a sentence

Name: An increasing set of sentence
Definition: A set of sentences whose set of models is closed under taking supermodel.
Page: 13
Section: 2

Smaller0: \(A\subseteq B\)
Greater0: every positive sentence which holds in \(A\) holds in \(B\)
Page: 13
Section: 2
Variables: \(A\) and \(B\) models
Index: 16
Kind: Theo
Smaller to greater: iff

Smaller0: \(\Gamma\) is increasing
Greater0: \(\Gamma\) has a set of positive axioms
Page: 13
Section: 2
Variables: \(\Gamma\) a theory
Index: 16 (ii)
Kind: Theo
Smaller to greater: iff
Assuming: \(\Gamma\) is consistent

Smaller0: \(\phi\) is increasing
Greater0: either \(\phi\) is equivalent to a positive sentence, \(\phi\) is valid or \(\neg\phi\) is valid
Page: 13
Section: 2
Variables: \(\phi\) a sentence
Index: 16 (iii)
Kind: Theo
Smaller to greater: iff
Hide middle: yes

Name: Conditional sentence
Definition: A disjunction of any number of negative statement, and at most one positive statement.
Page: 14
Section: 2
Extra: Duplicate with horn

Name: A set of sentences preserved under (finite/arbitrary) intersections
Definition: A set of sentences whose set of (finite/arbitrary) models is closed under taking intersections of (finite/arbitrary) models.
Page: 14
Section: 2

\(\Sigma\) is preserved under finite intersections iff \(\Sigma\) is preserved under arbitrary intersections iff \(\Sigma\) has a set of conditional axioms.
Page: 14, 15
Chapter: 1
Section: 2
Variables: \(\Sigma\) a set of sentences
Index: 17,18(i)
Kind: Lem, Theo

Smaller0: \(\phi\) is preserved under intersections
Greater0: \(\phi\) is equivalent to a conditional sentence
Page: 15
Section: 2
Variables: \(\phi\) a sentence
Index: 18
Kind: Theo
Smaller to greater: iff

Name: A language
Definition: A collection of relation symbols with positive arity, function symbols with positive arity and constant symbols.
Page: 18
Section: 3

Name: Relation and function symbols
Page: 18
Section: 3
Denoted by: Capital latin letters

Name: constant symbols
Page: 18
Section: 3
Denoted by: Lower case latin letters

Name: The power of \(\mathcal L\)
Definition: \(\omega\cup|\mathcal L|\)
Page: 18
Section: 3
Variables: \(\mathcal L\) a language
Context: Model theory
Notation: \(||\mathcal L||\)
Name2: The cardinal of \(\mathcal L\)

Name: \(\mathcal L\) is a reduction of \(\mathcal L'\)
Definition: \(\mathcal L\subseteq \mathcal L'\)
Page: 18
Section: 3
Variables: \(\mathcal L\), \(\mathcal L'\) languages
Name2: \(\mathcal L'\) is an extension of \(\mathcal L\)

Name: \(\mathcal L'\) is a simple expansion of \(\mathcal L\)
Definition: \(\mathcal L'\) is an expansion of \(\mathcal L\) and each symbols of \(\mathcal L'\setminus \mathcal L\) is a constant
Page: 18
Section: 3
Variables: \(\mathcal L\), \(\mathcal L'\) languages

Name: A universe
Page: 20
Section: 3
Typ: A non empty set

Name: An interpretation
Definition: A function sendings each constants of \(L\) to an element of \(A\), each \(n\)-ary relation symbol of \(\mathcal L\) to a subset of \(\mathcal A^n\), and each \(n\)-ary function symbol of \(\mathcal L\) to a function \(A^n\to A\).
Page: 20
Section: 3
Variables: \(A\) an universe and
\(\mathcal L\) a language
Denoted by: Gothic letters

Name: A model for \(\mathcal L\)
Definition: The pair consisting of a universe and an interpretation of \(\mathcal L\) in this universe.
Page: 20
Section: 3
Variables: \(\mathcal L\) a language
Notation: \(\langle \mathcal A, R_0,\dots , G_0, \dots, c_0,\dots \rangle\) where \(A\) is a universe, \(R_i\)'s, \(G_i\)'s and \(c_i\)'s are an interpretation

Name: The corresponding relation to \(\mathfrak A(P)\) in \(\mathfrak A'\)
Definition: \(\mathfrak A'(P)\).
Page: 20
Section: 3
Variables: \(\mathcal L\) a language,
\(P\) a relation symbol of \(\mathcal L\),
\(\langle A,\mathfrak A\rangle\), \(\langle A',\mathfrak A'\rangle\) two models of \(\mathcal L\)

Name: An expansion of \(\mathfrak A\)
Definition: A model \(\langle A,\mathfrak I\cup \mathfrak J\rangle\) of \(\mathcal L'\) where \(\mathfrak J\) is an interpretation of some language \(X\) distinct from \(\mathcal L\) over \(A\).
Page: 20
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A= \langle A,\mathfrak I\rangle\) a model of \(\mathcal L\)
Typ: A model of \(\mathcal L'\)

Name: The reduct of \(\langle A,\mathfrak I\cup \mathfrak J\rangle\) to \(\mathcal L\)
Definition: The model \(\langle A,\mathfrak I\rangle\).
Page: 20
Section: 3
Variables: \(\mathcal L,X\) disjoint language,
\(\langle A,\mathfrak I\rangle\) a model of \(\mathcal L\),
\(\langle A,\mathfrak J\rangle\) a model of \(X\).
Typ: A model of \(\mathcal L\)

Name: The power of \(\mathfrak A\)
Definition: \(|A|\).
Page: 20
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\) a model of \(\mathcal L\)
Name2: The cardinal of \(\mathfrak A\)

Name: An isomorphism from \(\mathfrak A\) to \(\mathfrak A'\)
Definition: A bijective function \(f\) between \(A\) and \(A'\) such that for each symbol \(s\in\mathcal L\), \(f(\mathfrak A(s))=\mathfrak A'(s)\).
Page: 21
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\), \(\mathfrak A'=\langle A',\mathfrak I'\rangle\) models of \(\mathcal L\)

Name: Two isomorphic models
Definition: Two models with an isomorphism from one to the other.
Page: 21
Section: 3
Variables: \(\mathcal L\) a language

Page: 21
Chapter: 1
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\),
\(\mathfrak A'=\langle A',\mathfrak I'\rangle\) models
Notation: \(\mathfrak A\cong\mathfrak A'\)
Name: \(\mathfrak A\) is isomorphic to \(\mathfrak A'\)

Page: 22
Definition: There is a model \(\mathfrak A''\) and an isomorphism \(f\) such that \(f:\mathfrak A\cong \mathfrak A''\) and \(\mathfrak A''\subset \mathfrak A '\).
Chapter: 1
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\), \(\mathfrak A'=\langle A',\mathfrak I'\rangle\) models
Name: \(\mathfrak A\) is isomorphically embedded in \(\mathfrak A'\)
Extra: I'?

Name: The set of terms
Definition: The least set containing term, symbol, and which is closed under applying functions.
Page: 22
Section: 3
Variables: \(\mathcal L\) a language,

Name: The atomic formulas
Definition: The least set containing \(t_1\equiv t_2\), \(P(t_1,\dots,t_n)\) for \(t_i\) some terms and \(P\) an \(n\)-ary relation symbol.
Page: 22
Section: 3
Variables: \(\mathcal L\) a language,
Index: 2

Name: The atomic formulas
Definition: The least set containing atomic formulas, closed under conjunction, negation and universal quantification.
Page: 23
Section: 3
Variables: \(\mathcal L\) a language,

Definition: \(\neg(\forall v)\neg\phi\).
Page: 23
Section: 3
Variables: \(\mathcal L\) a language,
\(\phi\) a formula,
\(v\) a variable
Notation: \((\exists v)\phi\)

Definition: \(\phi_1\land(\phi_2\land(\dots\land\phi_n)...)\).
Page: 23
Section: 3
Variables: \(\mathcal L\) a language,
\(\phi_i\) formulas,
Notation: \(\phi_1\land\phi_2\land\dots\land\phi_n\)

Definition: \(\phi_1\lor(\phi_2\lor(\dots\lor\phi_n)\dots)\).
Page: 23
Section: 3
Variables: \(\mathcal L\) a language,
\(\phi_i\) formulas,
Notation: \(\phi_1\lor\phi_2\lor\dots\lor\phi_n\)

Definition: \((\forall v_1)\dots(\forall v_n)\phi\).
Page: 23
Section: 3
Variables: \(\mathcal L\) a language,
\(\phi\) a formula,
\(v_1,\dots,v_n\) variables
Notation: \((\forall v_1,\dots,v_n)\phi\)

Definition: \((\exists v_1)\dots(\exists v_n)\phi\).
Page: 23
Section: 3
Variables: \(\mathcal L\) a language,
\(\phi\) a formula,
\(v_1,\dots,v_n\) variables
Notation: \((\exists v_1,\dots,v_n)\phi\)

Definition: A term whose variables form a subset of \(\{v_1,\dots,v_n\}\).
Page: 23
Section: 3
Denoted by: \(t(v_1,\dots,v_n)\)

Definition: A formula whose variables form a subset of \(\{v_1,\dots,v_n\}\).
Page: 24
Section: 3
Denoted by: \(\phi(v_1,\dots,v_n)\)

Name: A sentence
Definition: A formula with no free variables.
Page: 24
Section: 3

Question: The cardinal of the set of all formulas of \(\mathcal L\) is
Answer: \(||\mathcal L||\).
Page: 24
Chapter: 1
Section: 3
Variables: \(\mathcal L\) a language
Index: 4
Kind: Prop

Name: A tautology of \(\mathcal L\)
Definition: A formula, obtained from a tautology \(\psi\) of \(\mathcal S\) by simultaneously and uniformly substituting formulas of \(\mathcal L\) for the sentences symbols of \(\psi\).
Page: 24
Section: 3
Variables: \(\mathcal L\) a language
Index: 4
Kind: Prop

Q: -[...].
-[...].
A: Rule of Detachment; Rule of generalization
Page: 24
Chapter: 1
Section: 3
Variables: \(\mathcal L\) a language
Index: 7
Name: Rules of inference

Name: Rule of Detachment
Definition: From \(\phi\) and \(\phi\to\psi\) infer \(\psi\).
Page: 24
Section: 3
Variables: \(\mathcal L\) a language
Index: 8
Name2: Modus Ponens

Name: Rule of Generalization
Definition: From \(\phi\) infer \((\forall x)\phi\).
Page: 24
Section: 3
Variables: \(\mathcal L\) a language
Index: 9

Definition: \(\phi\) is a theorem of \(\mathcal L\).
Page: 24
Section: 3
Variables: \(\mathcal L\) a language
\(\phi\) a formula
Notation: \(\vdash \phi\)

Name: \(\phi\) is deducible from \(\Sigma\)
Definition: There is a proof of \(\phi\) from the logical axioms and \(\Sigma\).
Page: 24
Section: 3
Variables: \(\mathcal L\) a language
\(\phi\) a formula,
\(\Sigma\) a set of sentences
Notation: \(\Sigma\vdash \phi\)
Name2: There is a proof of \(\phi\) from \(\Sigma\)

Definition: \(\{\sigma_1,\dots,\sigma_n\}\vdash \phi\).
Page: 24
Section: 3
Variables: \(\mathcal L\) a language
\(\phi\) a formula,
\(\sigma_1,\dots,\sigma_n\) sentences
Notation: \(\sigma_1,\dots,\sigma_n\vdash \phi\)

Name: A consistent set of sentences
Definition: A set of sentences which does not prove everything.
Page: 25
Section: 3
Variables: \(\mathcal L\) a language
\(\phi\) a formula,

Name: An inconsistent set of sentences
Definition: A set of sentences which is not consistent.
Page: 25
Section: 3
Variables: \(\mathcal L\) a language
\(\phi\) a formula,

Name: An (in)consistent sentence
Definition: A sentence \(\sigma\) such that \(\{\sigma\}\) is (in)consistent.
Page: 25
Section: 3
Variables: \(\mathcal L\) a language
\(\phi\) a formula,

Smaller0: \(\Sigma\) is consistent
Greater0: every finite subset of \(\Sigma\) is consistent
Page: 25
Section: 3
Variables: \(\mathcal L\) a language,
\(\Sigma\) a set of sentences
Index: 10 (i)
Kind: Prop
Smaller to greater: iff

Smaller0: \(\Sigma\cup\{\sigma\}\) is inconsistent
Greater0: \(\Sigma\vdash\neg\sigma\)
Page: 25
Section: 3
Variables: \(\mathcal L\) a language,
\(\Sigma\) a set of sentences,
\(\sigma\) a sentence
Index: 10 (ii)
Kind: Prop
Smaller to greater: iff

Smaller0: \(\Sigma\vdash\sigma\)
Greater0: \(\sigma\in\Sigma\)
Page: 25
Section: 3
Variables: \(\mathcal L\) a language,
\(\Sigma\) a set of sentences,
\(\sigma\) a sentence
Index: 10 (ii)
Kind: Prop
Smaller to greater: iff
Assuming: \(\Sigma\) is maximal consistent
Context: Maximal consistent set of sentence
Smallest: \(\neg\sigma\not\in\Sigma\)

Smaller0: \(\Sigma\) is maximal consistent
Greater0: \(\sigma\land\tau\in\Sigma\)
Page: 25
Section: 3
Variables: \(\mathcal L\) a language,
\(\Sigma\) a set of sentences,
\(\sigma,\tau\) sentences
Index: 10 (iii)
Kind: Prop
Greater1: \(\sigma,\tau\in\Sigma\)
Connect Greater0: iff

Name: The value of \(t(v_0\dots v_q) \) at \(x_0,\dots,x_q\)
Definition: -\(x_i\) if \(t=v_i\),
-\(\mathfrak I(c)\) if \(t=c\) a constant symbol,
-\(\mathfrak I(f)(t_1[x_0\dots x_q]\dots t_n[x_0\dots x_q])\) if \(t=f(t_1,\dots,t_n)\)..
Page: 27
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\) a model,
\(t(v_0\dots v_q)\) a term,
\(x_0,\dots,x_q\in A\)
Notation: \(t[x_0\dots x_q]\)
Index: 13

Definition: \(t_1[x_0\dots x_q]=t_2[x_0\dots x_q]\)
Page: 28
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\) a model,
\(\phi(v_0\dots v_q)\) a formula,
\(x_0,\dots,x_q\in A\)
Notation: \(\mathfrak A\models t_1\equiv t_2\)
Index: 14, 15

Definition: \((t_1[x_0\dots x_q],\dots,t_2[x_0\dots x_q])\in \mathfrak I(R)\)
Page: 28
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\) a model,
\(\phi(v_0\dots v_q)\) a formula,
\(x_0,\dots,x_q\in A\)
Notation: \(\mathfrak A\models R(t_1,\dots,t_n)\)
Index: 14, 15

Definition: \(\mathfrak A\models \psi_1\) and \(\mathfrak A\models\psi_2\)
Page: 28
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\) a model,
\(\phi(v_0\dots v_q)\) a formula,
\(x_0,\dots,x_q\in A\)
Notation: \(\mathfrak A\models \psi_1\land\psi_2\)
Index: 14, 15

Definition: \(\mathfrak A\not\models \psi\)
Page: 28
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\) a model,
\(\phi(v_0\dots v_q)\) a formula,
\(x_0,\dots,x_q\in A\)
Notation: \(\mathfrak A\models \neg\psi\)
Index: 14, 15

Definition: for every \(x\in A\), \(\mathfrak A\models \psi[x_0\dots x_{i-1}xx_{i+1}\dots x_q]\).
Page: 28
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\) a model,
\(\phi(v_0\dots v_q)\) a formula,
\(x_0,\dots,x_q\in A\)
Notation: \(\mathfrak A\models (\forall v_i)\psi\)
Index: 14, 15
Assuming: \(i\le q\)

Name: \(x_0,\dots,x_q\) satisfies \(\phi(v_0\dots v_q) \)
Page: 28
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\) a model,
\(\phi(v_0\dots v_q)\) a formula,
\(x_0,\dots,x_q\in A\)
Notation: \(\mathfrak A\models \phi[x_0\dots x_q]\)
Index: 14, 15

Name: \(x_0,\dots,x_q\) satisfies \(\phi(v_0\dots v_q) \)
Definition: -if \(\phi\) is \(t_1\equiv t_2\) then \(t_1[x_0\dots x_q]=t_2[x_0\dots x_q]\),
-if \(\phi\) is \(R(t_1,dots,t_n)\) then \((t_1[x_0\dots x_q],\dots,t_2[x_0\dots x_q])\in \mathfrak I(R)\),
-if \(\phi\) is \(\psi_1\land\psi_2\) then for every \(i\), \(\mathfrak A\models \psi_i[x_0\dots x_{i-1}xx_{i+1}\dots x_q]\),
-if \(\phi\) is \(\neg\psi\) then, \(\mathfrak A\not\models \psi_i[x_0\dots x_{i-1}xx_{i+1}\dots x_q]\),
-if \(\phi\) is \((\forall v_i)\psi\) with \(i\le q\) then for every \(x\in A\), \(\mathfrak A\models \psi[x_0\dots x_{i-1}xx_{i+1}\dots x_q]\).
Page: 28
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\) a model,
\(\phi(v_0\dots v_q)\) a formula,
\(x_0,\dots,x_q\in A\)
Notation: \(\mathfrak A\models \phi[x_0\dots x_q]\)
Index: 14, 15

Smaller0: \(\mathfrak A\models\phi(t_0\dots t_p)[x_0\dots x_p]\)
Greater0: \(\mathfrak A[t_0[x_0\dots x_p]\dots t_p[x_0\dots x_p]]\)
Page: 31
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\) a model,
\(\phi(v_0\dots v_p)\) a formula,
\(x_0,\dots,x_p\in A\),
\(t_0,\dots,t_p\) terms
Index: 18
Kind: Prop
Smaller to greater: iff
Assuming: No variable of the \(t_i\)'s occur bound in \(\phi\)

Name: \(\mathfrak A\) is a model of \(\Sigma\)
Definition: For each \(\sigma\in \Sigma\), \(\mathfrak A\models \sigma\).
Page: 32
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\) a model,
\(\Sigma\) a set of sentences
Notation: \(\mathfrak \models\Sigma\)

Name: A valid sentence
Definition: A sentence which holds in every models of \(\mathcal L\).
Page: 32
Section: 3
Variables: \(\mathcal L\) a language,

Name: A satisfiable sentence
Definition: A sentence which holds in a model of \(\mathcal L\).
Page: 32
Section: 3
Variables: \(\mathcal L\) a language,

Name: \(\phi\) is a consequence of \(\sigma\)
Definition: Every model of \(\sigma\) is a model of \(\phi\).
Page: 32
Section: 3
Variables: \(\mathcal L\) a language,
\(\sigma,\phi\) sentences
Notation: \(\sigma\models\phi\)

Name: \(\phi\) is a consequence of \(\Sigma\)
Definition: Every model of \(\Sigma\) is a model of \(\phi\).
Page: 32
Section: 3
Variables: \(\mathcal L\) a language,
\(\phi\) a sentences
\(\Sigma\) a set of sentences
Notation: \(\Sigma\models\phi\)

Page: 32
Definition: Two models which have the same set of true sentences.
Chapter: 1
Section: 3
Variables: \(\mathcal L\) a language
Notation: \(\equiv\) on models
Name: Elementary equivalent models

Smaller0: \(\sigma\) is a theorem
Greater0: \(\sigma\) is valid
Page: 33
Section: 3
Variables: \(\mathcal L\) a language,
\(\sigma\) a sentence
Index: 20
Kind: Prop
Smaller to greater: iff
Name: Gödel's Completeness Theorem

Smaller0: \(\Sigma\) is consistent
Greater0: \(\Sigma\) has a model
Page: 33
Section: 3
Variables: \(\mathcal L\) a language,
\(\Sigma\) a set of sentences
Index: 20
Kind: Prop
Smaller to greater: iff
Name: Extended Completeness Theorem

Name: A first-order theory
Definition: A collection of sentences of \(\mathcal L\)
Page: 36
Section: 4
Variables: \(\mathcal L\) a language
Name2: A theory
Denoted by: \(T\)

Name: A closed first-order theory
Definition: A first-order theory closed under \(\models\)
Page: 36
Section: 4
Variables: \(\mathcal L\) a language

Name: A first-order theory complete in \(\mathcal L\)
Definition: A first-order theory whose set of consequences is maximal consistent.
Page: 36
Section: 4
Variables: \(\mathcal L\) a language

Question: What can be set of \(T\) related to \(\mathcal L'\) ?
Answer: \(T\) is not a closed theory of \(\mathcal L'\).
Page: 37
Chapter: 1
Section: 4
Variables: \(\mathcal L\subsetneq\mathcal L'\) languages,
\(T\) a complete theory of \(\mathcal L\).

Name: The restriction of \(T\) to \(\mathcal L\)
Definition: The set of elements of \(T\) whose vocabulary belongs to \(\mathcal L\)
Page: 37
Section: 4
Variables: \(\mathcal L\subsetneq\mathcal L'\) languages,
\(T\) a theory of \(\mathcal L'\).
Notation: \(T\mid\mathcal L\)

Smaller0: \(T\) is a closed theory
Greater0: \(T\mid\mathcal L\) is a closed theory for \(\mathcal L'\)
Page: 37
Section: 4
Variables: \(\mathcal L\subsetneq\mathcal L'\) languages,
\(T\) a theory of \(\mathcal L'\).

Name: A subtheory of \(T\)
Definition: A subset of \(T\)
Page: 37
Section: 4
Variables: \(T\) a theory
Context: Theory

Name: An extension of \(T\)
Definition: A superset of \(T\)
Page: 37
Section: 4
Variables: \(T\) a theory

Name: A set of axioms of \(T\)
Definition: A set of sentences with the same consequences as \(T\).
Page: 37
Section: 4
Variables: \(T\) a theory
Example: \(T\)
Example2: Any set \(\Sigma\) is a set of axioms for \(\{\phi:\Sigma\models\phi\}\)

Smaller0: \(T\) is a set of valid sentences of \(\mathcal L\)
Greater0: \(\emptyset\) is a set of axioms of \(T\)
Page: 37
Section: 4
Variables: \(\mathcal L\) a language
\(T\) a theory of \(\mathcal L\),
Smaller to greater: iff

Smaller0: \(T\) is a set of valid sentences
Greater0: \(\emptyset\) is a set of axioms of \(T\)
Page: 37
Section: 4
Variables: \(\mathcal L\) a language
\(T\) a theory of \(\mathcal L\),
Smaller to greater: iff

Page: 37
Section: 4
Chapter: 1
Axiom: The set of all sentences which hold in \(\mathcal A\)
Name: The model of \(\mathcal L\)
Variables: \(\mathcal L\) a language
Let \(\mathcal A\) a model of \(\mathcal L\)
Complete: A complete theory

Question: Difference between approach by first-order theories and approach by models ?
Answer: Models have non-first-order properties, such that isomorhpism, submodels, extensions...
Page: 37
Chapter: 1
Section: 4

Question: Every countable models of the theory of dense simple order is
Answer: isomorphic to each other.
Page: 38
Chapter: 1
Section: 4
Index: 2
Kind: Prop

Name: Associativity law for \(+\)
Definition: \(x+(y+z)\equiv(x+y)+z\)
Page: 38
Section: 4
Index: 3
Kind: Ex

Name: Commutative law for \(+\)
Definition: \(x+y\equiv y+x\)
Page: 38
Section: 4
Index: 3
Kind: Ex

Name: Absorption law for \(+\) and \(\cdot\)
Definition: \(x+(x\cdot y)\equiv x\)
Page: 38
Section: 4
Index: 3
Kind: Ex

Name: De Morgan law for \(+\) and \(\cdot\)
Definition: \(\overline{x+y}\equiv \bar x\cdot \bar y\)
Page: 38
Section: 4
Index: 3
Kind: Ex

Name: Laws of zero
Definition: \(x+0\equiv x\)
Page: 38
Section: 4
Index: 3
Kind: Ex
Definition2: \(x\cdot 0\equiv 0\)
Conjdef: Simultaneously
Definition3: \(x\cdot\bar x\equiv 0\)

Name: Laws of one
Definition: \(x+1\equiv 1\)
Page: 38
Section: 4
Context: Boolean ?
Index: 3
Kind: Ex
Definition2: \(x\cdot 1=x\)
Conjdef: Simultaneously
Definition3: \(x+\bar x\equiv 1\)

Name: Laws of zero and one
Definition: Laws of zero, laws and one and \(0\not\equiv 1\)
Page: 38
Section: 4
Index: 3
Kind: Ex

Name: Laws of double negation
Definition: \(\bar{\bar x}\equiv x\)
Page: 39
Section: 4
Index: 3
Kind: Ex

Name: A Boolean algebra
Definition: A model of the theory of Boolean algebra
Page: 39
Section: 4

Definition: \(\exists z. x+z\equiv y\)
Page: 39
Section: 4
Variables: \(\mathfrak A=\langle A,+,\cdot,\bar{\ },0,1\rangle\) a Boolean Algebra
Notation: \(x\le y\)

Question: The greatest element is
Answer: 1
Page: 39
Section: 4
Variables: \(\mathfrak A=\langle A,+,\cdot,\bar{\ },0,1\rangle\) a Boolean Algebra,
\(x,y\in A\)

Question: The least element is
Answer: 0
Page: 39
Chapter: 1
Section: 4
Variables: \(\mathfrak A=\langle A,+,\cdot,\bar{\ },0,1\rangle\) a Boolean Algebra,

Question: The least upper bound of \(x\) and \(y\) and their greatest lower bound
Answer: exists
Page: 39
Chapter: 1
Section: 4
Variables: \(\mathfrak A=\langle A,+,\cdot,\bar{\ },0,1\rangle\) a Boolean Algebra,
\(x,y\in A\)

Name: A field of sets
Definition: A collection of subsets of a nonempty set \(X\) such that both \(\emptyset, X\in \mathcal S\),
Closed under \(\cup,\cap\) and \(s\mapsto\bar{s}\)
Page: 39
Section: 4
Denoted by: \(\mathcal S\)

Question: Every Boolean algebra is
Answer: isomorphic to a field of sets.
Page: 39
Chapter: 1
Section: 4
Kind: Prop
Name: Representation Theorem for Boolean algebras

Name: An atom
Definition: A minimal non-zero element
Page: 39
Section: 4
Variables: \(\mathfrak A=\langle A,+,\cdot,\bar{\ },0,1\rangle\) a Boolean Algebra,

Name: An atomic BA
Definition: A BA such that every nonzero elements includes an atom.
Page: 39
Section: 4

Name: An atomless BA
Definition: A BA without atoms
Page: 39
Section: 4

Smaller0: \(\mathfrak A_1\) and \(\mathfrak A_2\) are countable and atomless
Greater0: \(\mathfrak A_1\) and \(\mathfrak A_2\) are isomorphic
Page: 39
Section: 4
Variables: \(\mathfrak A_i=\langle A_i,+_i,\cdot_i,\bar{\ }^i,0_i,1_i\rangle\) a Boolean Algebra,
Index: 5
Kind: Prop

Name: Inverse axiom
Definition: \(\exists y. x+y\equiv 0\land y+x\equiv 0\)
Page: 39
Section: 4
Index: 6
Kind: Ex

Name: The order of \(x\)
Definition: The least \(n\) such that \(nx\equiv 0\)
Page: 40
Section: 4
Variables: \(\langle G,+,0\rangle\) a group,
\(x\in G\)
Index: 6
Kind: Ex

Definition: \(0\) if \(n=0\) and \((n-1)x+x\) if \(n>0\).
Page: 40
Section: 4
Variables: \(\langle G,+,0\rangle\) a group,
\(x\) a term in this theory
\(n\in\mathbb N\)
Notation: \(nx\)
Index: 6
Kind: Ex

Smaller0: the \(\mathcal G_i\) are uncountable divisible torsion-free Abelian groups of the same power
Greater0: the \(\mathcal G_i\) are isomorphic
Page: 40
Section: 4
Variables: \(\mathcal G_i=\langle G_i,0_i,+_i\rangle\)
Index: 8
Kind: Prop

Definition: \(1\) if \(n=0\), \(x\cdot (x^{n-1})\) if \(n>0\)
Page: 41
Section: 4
Variables: \(\langle F,1,\cdot\rangle\) a field,
\(x\) a term of this theory,
\(n\in\mathbb N\)
Notation: \(x^n\)
Index: 9
Kind: Ex

Smaller0: the \(\mathcal F_i\) are uncountable algebraically closed fields of the same characteristic and power
Greater0: the \(\mathcal F_i\) are isomorphic
Page: 41
Section: 4
Variables: \(\mathcal F_i=\langle F_i,0_i,1_i,+_i,\cdot_i\rangle\) a field
Index: 10
Kind: Prop

Name: The schema of induction for this logic
Definition: For each formula \(\phi(v_0\dots v_n)\), with \(v_0\) does not occur bound in \(\phi\):
\(\phi(0v_1\dots v_n)\land\forall v_0(\phi(v_0v_1\dots v_n)\implies \phi(Sv_0v_1\dots v_n))\implies\forall v_0\phi(v_0\dots v_n)\)
Page: 42
Section: 4
Variables: \(T\) a theory whose language contains \(S\)

Name: The standard model of number theory
Definition: \(\langle\omega,+,\cdot,S,0\rangle\) with \(S\) successor, each other symbol have their usual meaning
Page: 42
Section: 4

Name: A nonstandard model of number theory
Definition: A model of number theory which is not standard.
Page: 42
Section: 4

Page: 37
Section: 4
Chapter: 1
Axiom: \(x\le y\land y\le x\implies x\equiv y\)
Name: The theory of partial order
Vocabulary: \(\{<\}\)
Axiom2: \(x\le x\)
Axiom3: \(x\le y\land y\le z\implies x\le z\)

Page: 38
Section: 4
Chapter: 1
Axiom: \(x\le y\lor y\le x\)
Name: The theory of simple order
Vocabulary: \(\{<\}\)
Extends: The theory of partial order

Page: 38
Section: 4
Chapter: 1
Axiom: \(x< y\implies\exists z(x<z<y)\)
Name: The theory of dense simple order
Vocabulary: \(\{<\}\)
Extends: The theory of simple order
Axiom2: \(\exists xy. x\not\equiv y\)

Page: 38, 41
Section: 4
Chapter: 1
Axiom: \(\exists y. x< y\) and \(\exists y. x>y\)
Name: The theory of dense simple order without endpoints
Vocabulary: \(\{<\}\)
Extends: The theory of dense simple order
Complete: yes

Page: 38
Section: 4
Chapter: 1
Axiom: Laws of double negation
Name: The theory of Boolean algebras
Vocabulary: \(\{+,\cdot,0,1\}\)
Extends: Theory of fields
Kind: Ex
Index: 3
Axiom2: idempotent of \(+\) and \(\cdot\)
Axiom3: Distributive of \(+\) and \(\cdot\) on each other
Axiom4: De morgan laws
Axiom5: Absorption laws
Axiom6: Laws of zero and one

Page: 39
Section: 4
Chapter: 1
Axiom: \(0\not\equiv x\implies\exists y\in (0,x]\forall z\le y. (z\equiv 0\lor z\equiv y)\)
Name: The theory of atomic Boolean algebras
Vocabulary: \(\{+,\cdot,0,1\}\)
Extends: Theory of Boolean algebra

Page: 39, 41
Section: 4
Chapter: 1
Axiom: \(y>0\implies \exists z\in(0,y)\)
Name: The theory of atomless Boolean algebras
Vocabulary: \(\{+,\cdot,0,1\}\)
Extends: Theory of Boolean algebra
Complete: yes

Page: 39
Section: 4
Chapter: 1
Axiom: \(+\) associative
Name: The theory of groups
Vocabulary: \(\{+,0\}\)
Kind: Ex
Index: 6
Axiom2: \(0\) is an identity
Axiom3: inverse axiom

Page: 27
Section: 1
Chapter: II
Axiom: \(^{-1}\) is an inverse for \(\cdot\)
Name: The theory of groups
Vocabulary: \(\{\cdot,^{-1},1\}\)
Extends: Theory of monoids
Kind: Ex

Page: 27
Section: 1
Chapter: I
Axiom: \(1\) is an identity for \(\cdot\)
Name: The theory of monoids
Vocabulary: \(\{\cdot,1\}\)
Extends: Theory of semigroups
Kind: Ex

Page: 40
Section: 4
Chapter: 1
Axiom: commutativity axiom
Name: The theory of Abelian group
Vocabulary: \(\{+,0\}\)
Extends: The theory of group
Kind: Ex
Index: 6

Page: 27
Section: 1
Chapter: I
Axiom: \(\cdot\) is associative
Name: The theory of semigroup
Vocabulary: \(\{\cdot\}\)
Kind: Ex

Page: 40, 41
Section: 4
Chapter: 1
Axiom: the axiom \(px\equiv0\)
Name: The theory of Abelian groups with all elements of order \(p\)
Vocabulary: \(\{+,0\}\)
Extends: The theory of Abelian group
Kind: Ex
Index: 6
Complete: yes

Page: 40
Section: 4
Chapter: 1
Axiom: The list of axioms \(x\not\equiv0\implies nx\not\equiv 0\) for each \(n\)
Name: The theory of Abelian groups with all elements of order \(\infty\)
Vocabulary: \(\{+,0\}\)
Extends: The theory of Abelian group
Kind: Prop
Index: 7
Categorical: The theory of Abelian groups with all elements torsion-free

Page: 40, 41
Section: 4
Chapter: 1
Axiom: The infinite list \(\exists y. ny\equiv x\) for each \(n\in\mathbb N^{>0}\)
Name: The theory of divisible torsion-free Abelian group
Vocabulary: \(\{+,0\}\)
Extends: The theory of torsion-free Abelian groups
Kind: Prop
Index: 7
Complete: yes

Page: 40
Section: 4
Chapter: 1
Axiom: \(1\) is a unit
Name: The theory of rings with units
Vocabulary: \(\{+,\cdot, 0,1\}\)
Extends: The theory of rings
Kind: Ex
Index: 9

Page: 40
Section: 4
Chapter: 1
Axiom: \(\cdot\) commutative
Name: The theory of commutative rings with units
Vocabulary: \(\{+,\cdot, 0,1\}\)
Extends: The theory with rings
Kind: Ex
Index: 9

Page: 40
Section: 4
Chapter: 1
Axiom: \(\cdot\) distributive over \(+\)
Name: The theory of rings
Vocabulary: \(\{+,\cdot,0\}\)
Extends: Theory of groups
Kind: Ex
Index: 9
Axiom2: \(\cdot\) associative
Extra: dupliquer

Page: 27
Section: 1
Chapter: I
Axiom: \(1\) is an identity for \(\cdot\)
Name: The theory of rings with identity
Vocabulary: \(\{+,\cdot,-,0,1\}\)
Extends: Theory of rings
Kind: Ex
Index: 4

Page: 27
Section: 1
Chapter: I
Name: The theory of rings
Vocabulary: \(\{+,\cdot,-,0\}\)
Extends: Theory of groups for \(\langle+,-,0\rangle\)
Kind: Ex
Index: 4
Extends2: Theory of semiring for \(\langle+,\cdot,0\rangle\)

Page: 41
Section: 4
Chapter: 1
Axiom: \(0\not\equiv 1\)
Name: The theory of fields
Vocabulary: \(\{+,\cdot, 0,1\}\)
Extends: The theory of integral domains
Kind: Ex
Index: 9
Axiom2: \(x\not\equiv 0\implies\exists y(y\cdot x\equiv 1)\)

Page: 41
Section: 4
Chapter: 1
Axiom: \(p1\equiv 0\).
Name: The theory of fields of characteristic \(p\)
Vocabulary: \(\{+,\cdot, 0,1\}\)
Extends: The theory of fields
Kind: Ex
Index: 9
Variables: \(p\in\mathbb N\)
Assuming: \(p\) prime

Page: 41
Section: 4
Chapter: 1
Axiom: \(p1\equiv 0\) for every \(p\) prime.
Name: The theory of fields of characteristic \(0\)
Vocabulary: \(\{+,\cdot, 0,1\}\)
Extends: The theory of fields
Kind: Ex
Index: 9

Page: 41
Section: 4
Chapter: 1
Axiom: \(\exists y(\sum_{i=0}^n x_i\cdot y^i\equiv0)\lor x_n\equiv 0\) for each \(n\ge 1\)
Name: The theory of algebraically closed fields.
Vocabulary: \(\{+,\cdot, 0,1\}\)
Extends: The theory of fields
Kind: Ex
Index: 9
Complete: yes

Page: 41
Section: 4
Chapter: 1
Axiom: \(\exists y(y^2\equiv x\lor y^2+x\equiv 0)\)
Name: The theory of real closed fields
Vocabulary: \(\{+,\cdot, 0,1\}\)
Extends: The theory of fields
Index: (18)
Axiom2: \(\sum_{i=1}^n x_i^2\equiv 0\implies\bigwedge_{i=1}^nx_i=0\) for each \(n\)
Extra: equivalent definitions
Complete: yes
Intuition: A field with properties similar to the reals

Page: 41
Section: 4
Chapter: 1
Axiom: \(x\le y\implies x+z\le y+z\)
Name: The theory of ordered fields
Vocabulary: \(\{+,\cdot, 0,1,\le\}\)
Extends: the theory of fields,
Axiom2: \(x\le y\land 0\le z\implies x\cdot z\le y\cdot z\)
Extends2: the theory of linear order

Page: 42
Section: 4
Chapter: 1
Axiom: \(0\not\equiv Sx\)
Name: The theory of one successor
Vocabulary: \(\{S\}\)
Kind: Ex
Index: 11
Axiom2: \(Sx\equiv Sy\implies x\equiv y\)
Axiom3: the schema of induction for this logic
Extra: Not in the book. But simplify definition of Peano

Page: 42
Section: 4
Chapter: 1
Axiom: \(x+Sy\equiv S(x+y)\)
Name: The theory of Presburger Arithmetic
Vocabulary: \(\{S,+\}\)
Extends: The theory of one successor
Kind: Ex
Index: 11
Axiom2: \(x+0\equiv x\)
Axiom3: the schema of induction for this logic

Page: 42
Section: 4
Chapter: 1
Axiom: The set of sentences that hold in standard model.
Name: Complete number theory
Vocabulary: \(\{S,+,\cdot\}\)
Complete: yes
Categorical: Complete arithmetic

Page: 42
Section: 4
Chapter: 1
Axiom: Non-complete theory
Name: Peano arithmetic/number theory
Vocabulary: \(\{+,\cdot,S,0\}\)
Extends: the theory of Presburger Arithmetic
Kind: Ex
Index: 11
Axiom2: \(x\cdot Sy\equiv (x\cdot y)+x\)
Axiom3: \(x\cdot 0\equiv 0\)
Axiom4: The schema of induction for this logic
Axiom8: PA
Etymology: Naturals

Name: \(\mathfrak A'\) is a submodel of \(\mathfrak A\)
Definition: For each constant \(c\in \mathcal L\), \(\mathfrak I(c)=\mathfrak I'(c)\)
Page: 21
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\), \(\mathfrak A'=\langle A',\mathfrak I'\rangle\) models
Notation: \(\mathfrak A\subseteq \mathfrak A'\)
Definition2: for each \(R\in \mathcal L\) with arity \(n\), \(\mathfrak I'(R)=\mathfrak I(R)\cap A^{\prime n}\)
Definition3: for each function symbol \(G\in\mathcal L\) of arity \(n\), for each \(x_1,\dots, x_n\), \(\mathfrak I(G)(x_1,\dots,x_n)=\mathfrak I(G)(x_1,\dots,x_n)\).
Name2: \(\mathfrak A\) is an extension of \(\mathfrak A'\)

Name: A maximal consistent set of sentences
Definition: A consistent set of sentences whithout proper consistent superset
Page: 25
Section: 3
Variables: \(\mathcal L\) a language

Name: Identity axioms
Definition: \(x\equiv x\)
Page: 24
Section: 3
Variables: \(\mathcal L\) a language
Index: 7

Smaller0: \(\Sigma\) has a model
Greater0: every finite subset of \(\Sigma\) has a model
Page: 33
Section: 3
Variables: \(\mathcal L\) a language,
\(\Sigma\) a set of sentences
Index: 20
Kind: Prop
Smaller to greater: iff
Name: Compactness Theorem

Smaller0: \(\Sigma\) is consistent
Greater0: \(\Sigma\) can be extended to a maximal consistent set of sentences
Page: 25
Section: 3
Variables: \(\mathcal L\) a language,
\(\Sigma\) a set of sentences,
Index: 11
Kind: Prop
Smaller to greater: iff
Name: Lindenbaum's Theorem

Smaller0: \(\mathfrak A\cong \mathfrak B\)
Greater0: \(\mathfrak A\equiv\mathfrak B\)
Page: 32
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak{A,B}\) models
Index: 19
Kind: Prop

Name: Logical axioms
Definition: \((\forall v)(\phi\to \psi)\to (\phi\to (\forall v)\psi)\), for \(v\) a variable not free in \(\phi\).
Page: 24
Section: 3
Variables: \(\mathcal L\) a language
Index: 6
Definition2: erased by accident

Smaller0: \(\mathfrak A\cong \mathfrak A'\)
Greater0: \(|A|=|A'|\)
Page: 21
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\), \(\mathfrak A'=\langle A',\mathfrak I'\rangle\) models
Hide middle: yes

Smaller0: \(\mathfrak A\) is finite
Greater0: \(\mathfrak A\cong \mathfrak B\)
Page: 32
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak{A,B}\) models
Index: 19
Kind: Prop
Greater1: \(\mathfrak A\equiv\mathfrak B\)
Connect Greater0: if and only if

Smaller0: \(\Sigma\) is consistent
Greater0: the set of sentences deducible from \(\Sigma\) is consistent
Page: 9
Section: 2
Variables: \(\Sigma\) a set ef sentences.
Index: 8 (i)
Kind: Prop
Smaller to greater: iff

Question: \(\phi[x_0\dots x_r]\) depends
Answer: only on the variables which are free in \(\phi\).
Page: 29
Chapter: 1
Section: 3
Variables: \(\mathcal L\) a language,
\(\mathfrak A=\langle A,\mathfrak I\rangle\) a model,
\(\phi(v_0\dots v_q)\) a formula,
\(x_0,\dots,x_r\in A\)
Index: 16
Kind: Prop