Title: Theory / Unimportant / Math / Logic / Set Theory / Kunen / V1;+14
Tags: fromMath mathToDef 1ed automaticImplies récupéré FromVariables TypToProperty counterExample hasNotNewEffectReturnsMeaning hasSeenConjDef oldTypToDelete todo
On every note of this deck, unless a note says otherwise:
Chapter: 1
Chapter: 1

Name: An infinite cardinal
Definition: \(|\alpha^{<\omega}|=\alpha\)
Page: 29, 30
Chapter: 1
Section: 10
Index: 12, 19
Kind: cor, Lem, Lem
Definition2: of the form \(\omega_{\gamma}\) for \(\gamma\) an ordinal.
Conjdef: A set \(\alpha\) such that
Definition3: \(\kappa\otimes\kappa=\kappa\)
Definition4: greater than omega 0?
Typ: A limit ordinal
Assuming: ZF

Smaller0: \(cf(\beta)\)
Greater0: \(\beta\)
Page: 32
Section: 10
Variables: \(\alpha\) an ordinal
Index: 30
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)
Extra: Not in the book
Hide sides: yes

Name: Cofinal subset of \(A\)
Definition: A subset \(B\subseteq A\) such that, for every \(a\in A\), there exists \(b\in B\) such that \(aRb\).
Page: 32
Chapter: 1
Section: 10
Variables: \(\langle A,R\rangle\) an ordered set
Index: 29
Kind: Def
Extra: Not in the book

Page: 37, 33
Notation: \(\omega\)
Definition: The set of natural numbers.
Section: 7,
Chapter: 1
Kind: Def, Lem
Index: 18, 33
Regular: Yes

Name: A finite set
Definition: A set \(A\) such that \(A\preceq n\) for some natural number \(n\)
Page: 38
Chapter: 1
Section: 7
Index: 8
Kind: Def

Definition: The cardinal of \(^BA\).
Chapter: 1
Section: 10
Variables: \(A,B\) sets.
Notation: \(A^B\)
Index: 25
Kind: Def

Smaller0: \(\kappa,\lambda,\sigma\) are cardinals
Greater0: \(\kappa^{\lambda+\sigma} \)
Page: 40
Section: 8
Variables: \(\kappa,\lambda,\sigma\) sets.
Kind: Table
Hide middle: yes
Assuming: ZFC
Greater1: \(\kappa^\lambda\cdot \kappa^\sigma\)
Connect Greater0: \(=\)

Name: Set existence axiom
Definition: \(\exists x(x=x)\)
Page: xv
Chapter: Introduction
Section: 7
Index: 0
Kind: Axiom
Extra: not in v2
Intuition: the universe of discourse in non-void.
Property: Derivable form the logical axiom under most developments of formal logic

Name: \(\forall v_i(\psi)\)
Definition: \((\neg(\exists v_i(\neg(\phi))))\).
Page: 002
Chapter: 1
Section: 2

Name: Expression
Definition: a finite sequence of basis symbols.
Page: 002
Chapter: 1
Section: 2

Name: \(v_i\ne v_j\)
Definition: \(\neg(v_i=v_j)\).
Page: 002
Chapter: 1
Section: 2

Definition: \(\neg(v_i\in v_j)\).
Page: 002
Chapter: 1
Section: 2
Notation: \(v_i\not\in v_j\)

Name: A subformula of \(\phi\)
Definition: a consecutive sequence of symbols of \(\phi\) which form a formula.
Page: 004
Chapter: 1
Section: 2

Name: The scope of a quantifier
Definition: The unique subformula beginning with that quantifier.
Page: 004
Chapter: 1
Section: 2

Name: A bound occurence of \(x\) in \(f\)
Definition: An occurence of \(x\) which lies in the scope of a quantifier acting on that variable.
Page: 004
Chapter: 1
Section: 2
Variables: \(f\) a formula and \(x\) a variable.

Definition: There is a finite sequence \(\phi_1,\dots,\phi_n\) of formulas such that \(\phi_n\) is \(\phi\) and each \(\phi_i\) belongs to \(S\), is a logical axiom or follows from the \(\phi_1,\dots,\phi_{i-1}\) by certain rules of inferences.
Page: 005
Chapter: 1
Section: 2
Variables: \(S\) a set of sentence
\(\phi\) a sentence.
Notation: \(S\vdash\phi\)

Name: \(\phi\) is logically valid
Definition: \(\emptyset\vdash\phi\).
Page: 005
Chapter: 1
Section: 2
Variables: \(\phi\) a sentence.
Notation: \(\vdash\phi\)

Name: Logically equivalent sentences
Definition: Two sentences \(\phi,\psi\) such that \(\vdash(\phi\iff\psi)\).
Page: 005
Chapter: 1
Section: 2
Variables: \(\phi,\psi\) sentences.

Smaller0: \(S\) is inconsistent
Greater0: there is a finite \(S_0\subseteq S\) such that \(S_0\) is inconsistent
Page: 006
Section: 2
Variables: \(S\) a set of sentence.
Index: 1 b
Kind: Theo
Smaller to greater: iff
Assuming: ZF

Name: A free occurence of \(x\) in \(f\)
Definition: An occurence of \(x\) which does not lie in the scope of a quantifier acting on that variable.
Page: 004
Chapter: 1
Section: 2
Variables: \(f\) a formula and \(x\) a variable.

Name: Russel's paradox
Definition: \(\exists y\forall x(x\in y\iff x\not\in x)\).
Page: 010
Chapter: 1
Section: 5

Question: The universal set
Answer: does not exists.
Page: 011
Section: 5
Index: 2
Kind: Theo
Assuming: ZF

Question: The number of isomorphism between \(\langle A,R\rangle\) and \(\langle B,S\rangle\) is
Answer: at most one.
Page: 015
Section: 6
Variables: \(\langle A,R\rangle\) and \(\langle B,S\rangle\) be well-ordered sets.
Index: 2
Kind: Lem
Assuming: ZF

Question: One of the following statement holds:
Answer: \(\langle A,R\rangle\simeq\langle B,S\rangle\) are isomorphic,
-\(\exists y\in B(\langle A,R\rangle\simeq \langle\text{pred}(B,y,S),S\rangle)\)
-\(\exists x\in B(\text{pred}(B,x,R),R\simeq \langle\langle B,S\rangle\rangle)\)
Page: 015
Section: 6
Variables: \(\langle A,R\rangle\) and \(\langle B,S\rangle\) be well-ordered sets.
Index: 3
Kind: Theo
Assuming: ZF

Smaller0: \(x\) is an ordinal
Greater0: \(y\) is an ordinal
Page: 016
Section: 7
Variables: \(x,y\) sets.
Index: 3 (1)
Kind: Theo
Hide middle: yes
Assuming: ZF
Smaller1: \(y\in x\)
Greater1: \(\text{pred}(x,y)\)
Connect Greater0: and \(y\) equals
Connect Smaller0: and

Smaller0: \(x\) and \(y\) are ordinals
Greater0: \(x=y\)
Page: 016
Section: 7
Variables: \(x,y\) sets.
Index: 3 (2)
Kind: Theo
Hide middle: yes
Assuming: ZF
Smaller1: \(x\simeq y\)
Connect Smaller0: and

Definition: \(\text{pred}(x,y,\in_x)\).
Page: 016
Chapter: 4
Section: 7
Variables: \(x\) an ordinal and \(y\in x\) a set
Notation: \(\text{pred}(x,y)\)

Definition: \(\langle x,\in_x\rangle\simeq\langle A,R\rangle\).
Page: 016
Chapter: 4
Section: 7
Variables: \(x\) an ordinal
\(A\) a set
\(R\) a relation on \(A\)
Notation: \(x\simeq\langle A,R\rangle\)

Smaller0: \(x\) and \(y\) are ordinals
Greater0: \(x=y\),
Page: 016
Section: 7
Variables: \(x,y\) sets.
Index: 3 (3)
Kind: Theo
Hide middle: yes
Assuming: ZF
Greater1: \(x\in y\) or \(y\in x\)
Connect Greater0: or

Smaller0: \(x\), \(y\) and \(z\) are ordinals
Greater0: \(x\in z\)
Page: 016
Section: 7
Variables: \(x,y,z\) sets.
Index: 3 (4)
Kind: Theo
Hide middle: yes
Assuming: ZF
Smaller1: \(x\in y\in z\)
Connect Smaller0: and

Smaller0: \(C\) is a non-empty set of ordinals
Greater0: \(\forall y\in C(x\in y\lor x=y)\)
Page: 016
Section: 7
Variables: \(C\) a set of sets.
Index: 3 (5)
Kind: Theo
Hide middle: yes
Prefix Greater: there exists an element \(x\in C\) such that
Assuming: ZF

Question: The set of all ordinal
Answer: does not exist.
Page: 017
Section: 7
Index: 4
Kind: Theo
Assuming: ZF

Definition: The concatenation of \(s\) and of \(t\).
Page: 021
Chapter: 1
Section: 7
Variables: \(s\) a sequence of length \(\alpha\)
\(t\) a sequence of length \(\beta\)
Notation: \(s\frown t\)
Index: 22
Kind: Def

Definition: The function \(s\) with domain \(n\) such that \(s(i)=x_i\) for all \(i< n\).
Page: 021
Chapter: 1
Section: 7
Variables: \(x_0,\dots,x_{n-1}\) sets.
Notation: \(\langle x_0,\dots,x_{n-1}\rangle\)
Index: 22
Kind: Def

Definition: \(\bigcup \{A^n:n\in\omega\}\).
Page: 021
Chapter: 1
Section: 7
Variables: \(A\) a set.
Notation: \(A^{<\omega}\)
Index: 21 b
Kind: Def

Definition: The set of functions from \(n\) into \(A\).
Page: 021
Chapter: 1
Section: 7
Variables: \(A,n\) sets.
Notation: \(A^n\)
Index: 21a
Kind: Def

Smaller0: \(\mathbb C\subseteq\mathbb{ON}\) and \(\mathbb C\ne0\)
Greater0: \(\mathbb C\) has a least element
Page: 024
Section: 9
Variables: \(\mathbb C\) be a class.
Index: 2
Kind: Theo
Assuming: ZF

Question: \(\beta-1\) is
Answer: \(\beta\) if \(\beta\) is a limit or 0. \(\gamma\) such that \(S(\gamma)=\beta\) otherwise.
Page: 026
Section: 9
Variables: \(\beta\) an ordinal.
Index: 4
Kind: Def

Question: \(\alpha^0=\)
Answer: \(1\)
Page: 026
Section: 9
Variables: \(\alpha\) an ordinal
Index: 5(1)
Kind: Def

Question: \(\alpha^{\beta+1}=\)
Answer: \(\alpha^\beta\cdot\alpha\).
Page: 026
Section: 9
Variables: \(\alpha,\beta\) ordinals
Index: 5(2)
Kind: Def

Question: \(\alpha^{\beta}\) equals
Answer: \(\text{sup}\{\alpha^\zeta:\zeta<\beta\} \).
Page: 026
Section: 9
Variables: \(\alpha,\beta\) ordinals
Index: 5(3)
Kind: Def
Assuming: \(\beta\) a limit ordinal

Definition: The least ordinal \(\alpha\) such that \(A\approx\alpha\), for some well-order of \(A\).
Page: 027
Chapter: 1
Section: 10
Variables: \(A\) a set
Context: set theory
Notation: \(|A|\)
Index: 3
Kind: Def
Assuming: \(AC\)
Property: Always defined

Smaller0: \(A\precsim B\) and \(B\precsim A\)
Greater0: \(A\approx B\)
Page: 027
Section: 10
Variables: \(A,B\) sets.
Index: 2
Kind: Theorem
Hide middle: yes
Assuming: ZF
Name: Schöder-Bernstein

Name: A cardinal
Definition: an ordinal \(\alpha\) such that \(\alpha=|\alpha|\).
Page: 028
Chapter: 1
Section: 10
Index: 4
Kind: Def

Smaller0: \(\beta\in\omega\)
Greater0: \(\alpha=\beta\)
Page: 028
Section: 10
Variables: \(\alpha,\beta\) sets
Index: 6
Kind: Lemma
Hide middle: yes
Assuming: ZF
Smaller1: \(\alpha\approx \beta\)
Connect Smaller0: and

Name: An uncoutable set
Definition: A set which is not countable.
Page: 028
Chapter: 1
Section: 10
Index: 8
Kind: Def

Smaller0: \(\alpha\in\omega\)
Greater0: \(\not\approx\)
Page: 028
Section: 10
Variables: \(\alpha\) a set
Index: 6
Kind: Lemma
Hide middle: yes
Prefix Greater: \(\alpha\)
Assuming: ZF
Suffix Greater: \(\alpha+1\)

Question: \(|\alpha|\le\beta\le\alpha\) implies
Answer: \(|\beta|=|\alpha|\).
Page: 028
Section: 10
Variables: \(\alpha,\beta\) sets
Index: 5
Kind: Lemma
Assuming: ZF

Question: \(\omega\) and its elements are
Answer: cardinal.
Page: 028
Section: 10
Index: 7
Kind: Cor
Assuming: ZF

Name: A countable set
Definition: A set \(A\) such that \(|A|\le\omega\).
Page: 028
Chapter: 1
Section: 10
Index: 8
Kind: Def

Smaller0: \(x\)
Greater0: \(\mathcal P(x)\)
Page: 029
Section: 10
Variables: \(X\) a set
Index: 15
Kind: Theo
Smaller to greater: \(\prec\)
Assuming: ZF
Name: Cantor's theorem
Greater to smaller: \(\succ\)

Question: Maximal cardinal
Answer: does not exists.
Page: 030
Section: 10
Index: 16
Kind: Theo
Assuming: ZF

Page: 030, 33
Section: 10
Chapter: 1
Kind: Def
Index: 17
Variables: \(\alpha\) a set
Notation: \(\alpha^+\)
Definition: The least cardinal greater than \(\alpha\).
Regular: yes

Name: A successor cardinal
Definition: \(\alpha^+\) for some \(\alpha\).
Page: 030
Chapter: 1
Section: 10
Variables: \(\kappa\) a cardinal
Index: 17
Kind: Def

Name: A limit cardinal
Definition: A cardinal which is not a successor of a cardinal greater than \(\omega\).
Page: 030
Chapter: 1
Section: 10
Index: 17
Kind: Def
Extra: example or intuition

Page: 030
Section: 10
Chapter: 1
Kind: Def
Index: 18
Variables: \(\alpha\) an ordinal
Notation: \(\aleph_\alpha\)
Definition: \(\omega\) if \(\alpha=0\)
Definition2: \(\omega_{\alpha-1}^+\) if \(\alpha\) is a successor
Definition3: \(\text{sup}\{\omega_\alpha\mid\alpha<\gamma\}\) if \(\alpha\) is a limit ordinal

Question: Each \(\omega_{\gamma}\) is
Answer: a cardinal.
Page: 030
Section: 10
Variables: \(\gamma\) an ordinal.
Index: 19
Kind: Lem
Assuming: ZF

Smaller0: \(\alpha<\beta\)
Greater0: \(<\)
Page: 030
Section: 10
Variables: \(\alpha,\beta\) ordinals
Index: 19
Kind: Lem
Hide middle: yes
Prefix Greater: \(\omega_\alpha\)
Assuming: ZF
Suffix Greater: \(\omega_\beta\)

Smaller0: \(\omega_\alpha\) is a limit/successor ordinal
Greater0: \(\alpha\) is a limit/successor ordinal
Page: 030
Section: 10
Variables: \(\alpha\) an ordinal
Index: 19
Kind: Lem
Smaller to greater: iff
Assuming: ZF

Smaller0: there is a function \(f\) from \(X\) onto \(Y\)
Greater0: \(|Y|\le|X|\)
Page: 030
Section: 10
Variables: \(X,Y\) be sets.
Index: 20
Kind: Lem
Assuming: ZFC

Smaller0: \(\langle A,R\rangle\) is a well-ordering
Greater0: \(\not\simeq\)
Page: 014
Section: 6
Variables: \(A\) be a set
\(R\) be a binary relation on \(A\).
Kind: Lem
Hide middle: yes
Prefix Greater: for all \(x\in A\), \(\langle A,R\rangle\)
Assuming: ZF
Suffix Greater: \(\langle\text{pred}(A,x,R),R\rangle\)

Smaller0: \(\alpha\)
Greater0: \(S(\alpha)\)
Page: 018
Section: 7
Variables: \(\alpha\) an ordinal
Index: 11
Kind: Lem
Smaller to greater: \(<\)
Assuming: ZF
Greater to smaller: \(>\)

Name: Peano postulates
Definition: -\(0\in\omega\)
Page: 019
Chapter: 1
Section: 7
Index: 16
Kind: Def
Definition2: -\(\forall n\in \omega\left(S(n)\in\omega\right)\)
Conjdef: All of
Definition3: -\(n,m\in \omega\left(n\ne m\implies S(n)\ne S(m)\right)\)
Definition4: -(Induction) \(\forall X\subseteq \omega\left\{\left[0\in X\land \forall n\in X\left(S(n)\in X\right)\right]\implies X=\omega\right\}\)

Definition: \(|(\kappa\times\{0\})\cup(\lambda\times\{1\})|\)
Page: 028
Chapter: 1
Section: 10
Variables: \(\kappa,\lambda\) sets
Notation: \(\kappa\oplus\lambda\)
Index: 9
Kind: Def

Definition: \(|\kappa\times\lambda|\)
Page: 028
Chapter: 1
Section: 10
Variables: \(\kappa,\lambda\) sets
Notation: \(\kappa\otimes\lambda\)
Index: 9
Kind: Def

Question: Cardinal addition and multiplication are
Answer: commutative.
Page: 028
Section: 10
Index: 9
Kind: Def

Smaller0: \(n,m\in\omega\)
Greater0: \(n\oplus m\)
Page: 028
Section: 10
Variables: \(n,m\) sets.
Kind: Lem
Hide middle: yes
Assuming: ZF
Smaller1: n+ m
Greater1: \(\omega\)
Connect Greater0: \(<\)

Smaller0: \(\kappa\) or \(\lambda\) is an infinite cardinals
Greater0: \(\text{max}(\kappa,\lambda)\)
Page: 029
Section: 10
Variables: \(\kappa,\lambda\) sets
Index: 13
Kind: Cor
Smaller to greater: iff
Prefix Greater: \(\kappa\otimes\lambda=\)

Smaller0: \(\kappa\) or \(\lambda\) is an infinite cardinals
Greater0: \(\text{max}(\kappa,\lambda)\)
Page: 029
Section: 10
Variables: \(\kappa,\lambda\) sets
Index: 13
Kind: Cor
Smaller to greater: iff
Hide middle: yes
Prefix Greater: \(\kappa\oplus\lambda=\)

Smaller0: \(\kappa\ge\omega\)
Greater0: \(|\bigcup_{\alpha<\kappa}X_\alpha|\le\kappa\)
Page: 030
Section: 10
Variables: \(\kappa\) a cardinal
\((X_\alpha)_{\alpha<\kappa}\) a sequence of sets.
Index: 21
Kind: Lem
Hide middle: yes
Assuming: ZFC
Smaller1: \(|X_\alpha|\le \kappa\) for all \(\alpha<\kappa\)

Question: The closure of \(B\) under \(\mathcal F\) has cardinality
Answer: at most \(\kappa\).
Page: 031
Section: 10
Variables: \(\kappa\) be an infinite cardinal.
Suppose \(B\subseteq A,|B|\le \kappa\) and
\(\mathcal F\) is a set of at most \(\kappa\) finitary functions on \(A\).
Index: 23
Kind: Theorem
Assuming: ZFC

Name: An \(n\)-ary function
Definition: \(f:A^n\to A\) if \(n>0\) and an element of \(A\) if \(n=0\).
Page: 031
Chapter: 1
Section: 10
Variables: \(n\in\mathbb N\).
Index: 22
Kind: Def

Name: A finitary function
Definition: An \(n\)-ary function for some \(n\).
Page: 031
Chapter: 1
Section: 10
Variables: \(n\in\mathbb N\).
Index: 22
Kind: Def

Name: \(B\) is closed under \(f\)
Definition: \(f''B^n\subseteq B\).
Page: 031
Chapter: 1
Section: 10
Variables: \(n\in\mathbb N\),
\(A\) a set,
\(f:A^n\to A\)
\(B\subset A\).
Index: 22
Kind: Def

Name: The closure of \(B\) under \(\mathcal F\)
Definition: The least subset of \(A\), containing \(B\) and closed under all functions in \(\mathcal F\).
Page: 031
Chapter: 1
Section: 10
Variables: If \(\mathcal F\) is a set of finitary function s on \(A\) and \(B\subset A\)
Index: 22
Kind: Def

Smaller0: \((\kappa^{\lambda})^{(\sigma)}\)
Greater0: \(\kappa^{\lambda\otimes\sigma}\)
Page: 032
Section: 10
Variables: If \(\kappa,\lambda,\sigma\) are cardinals
Index: 27
Kind: Lem
Smaller to greater: =
Hide middle: yes
Assuming: ZFC

Name: The continuum hypothesis
Definition: \(2^\omega=\omega_1\).
Page: 032
Chapter: 1
Section: 10
Index: 28
Kind: Def
Abbreviation: CH

Name: The generalized continuum hypothesis
Definition: \(\forall\alpha(2^{\omega_\alpha}=\omega_{\alpha+1})\).
Page: 032, 35
Chapter: 1
Section: 10
Index: 28,44
Kind: Def
Definition2: For all \(\alpha\), \(\beth_\alpha=\omega_\alpha\)
Abbreviation: GCH

Name: A function which maps \(\alpha\) cofinally
Definition: A function \(f:\alpha\to\beta\) such that \(ran(f)\) is cofinal in \(\beta\).
Page: 032
https://en.m.wikipedia.org/wiki/Cofinal_(mathematics)
Chapter: 1
Section: 10
Index: 29
Kind: Def
Name2: A final map

Name: The cofinality of \(\beta\)
Definition: The least ordinal \(\alpha\) such that there is a map from \(\alpha\) cofinally into \(\beta\).
Page: 032
Chapter: 1
Section: 10
Variables: \(\beta\) a set
Notation: \(f(\beta)\)
Index: 30
Kind: Def

Smaller0: \(\lambda\ge\omega\)
Greater0: \(^\lambda\kappa\)
Page: 032
Section: 10
Variables: \(\kappa,\lambda\) sets
Index: 26
Kind: Lem
Hide middle: yes
Assuming: ZF
Smaller1: \(2\le\kappa\le\lambda\)
Greater1: \(\mathcal P(\lambda)\)
Connect Greater0: \(\approx\)
Greater2: \(^\lambda2\)

Question: There is a cofinal map \(f:\text{cf}(\beta)\to\beta\) which is
Answer: strictly increasing.
Page: 033
Section: 10
Variables: \(\beta\) a set
Index: 31
Kind: Lemma

Smaller0: \(\alpha\) is a limit ordinal
Greater0: \(cf(\alpha)=cf(\beta)\)
Page: 033
Section: 10
Variables: \(\alpha\) an ordinal
Index: 32
Kind: Lemma
Hide middle: yes
Assuming: ZF
Smaller1: \(f:\alpha\to\beta\) is a strictly increasing cofinal map
Connect Smaller0: and

Question: \(\text{cf}(\text{cf}(\beta))=\)
Answer: \(\text{cf}(\beta)\).
Page: 033
Section: 10
Variables: \(\beta\) a set
Index: 33
Kind: Cor
Assuming: ZF

Name: A regular set
Definition: A limit ordinal \(\beta\) such that \(cf(\beta)=\beta\).
Page: 033
Chapter: 1
Section: 10
Context: In set theory
Index: 34
Kind: Def

Smaller0: \(\beta\) is regular
Greater0: \(\beta\) is a cardinal
Page: 033
Section: 10
Variables: \(\beta\) a set
Index: 35
Kind: Lem
Assuming: ZF

Smaller0: a limit ordinal
Greater0: \(\text{cf}(\alpha)\)
Page: 033
Section: 10
Variables: \(\alpha\) an ordinal
Index: 38
Kind: Lem
Hide middle: yes
Prefix Greater: \(\text{cf}(\omega_\alpha)=\)
Assuming: ZF
Prefix Smaller: \(\alpha\) is

Name: Weakly inaccessible cardinal
Definition: A regular limit cardinal.
Page: 034
Chapter: 1
Section: 10
Variables: \(\kappa\) a cardinal
Index: 39
Kind: Def

Name: Strongly inaccessible cardinal
Definition: A regular cardinal \(\kappa>\omega\) such that \(\forall \lambda<\kappa(2^\lambda<\kappa)\).
Page: 034
Chapter: 1
Section: 10
Index: 39
Kind: Def
Assuming: ZFC

Smaller0: \(\kappa\) is weakly inaccessible
Greater0: \(\kappa\) is strongly inaccessible
Page: 034
Section: 10
Variables: \(\kappa\) a cardinal
Index: 39
Kind: Def
Smaller to greater: iff
Assuming: ZFC and GCH

Smaller0: \(\kappa\) is strongly inaccessible
Greater0: \(\kappa\) is weakly inaccessible
Page: 034
Section: 10
Variables: \(\kappa\) a cardinal
Index: 39
Kind: Def
Hide middle: yes
Assuming: ZFC

Smaller0: \(cf(\kappa)\le\lambda\)
Greater0: \(>\kappa\)
Page: 034
Section: 10
Variables: \(\kappa\) a cardinal
Index: 40
Kind: Lemma
Hide middle: yes
Prefix Greater: \(\kappa^\lambda\)
Assuming: ZFC
Smaller1: \(\kappa\) is infinite
Name: König

Smaller0: \(\lambda\ge\omega\)
Greater0: \(>\lambda\)
Page: 034
Section: 10
Variables: \(\lambda \) an ordinal.
Index: 41
Kind: Cor
Prefix Greater: \(cf(2^\lambda)\)
Assuming: ZFC,
Counterexample: take \(\lambda=1\), \(cf(2^1)=cf(2)=1\)

Smaller0: \(\kappa\le\lambda\)
Greater0: \(\kappa^\lambda\)
Page: 034
Section: 10
Variables: \(\kappa,\lambda\ge2\) and at least one of them is infinite.
Index: 42
Kind: Lemma
Hide middle: yes
Assuming: ZFC+GCH
Greater1: \(\lambda^+\)
Connect Greater0: \(=\)

Smaller0: \(\kappa> \lambda\ge\text{cf}(\kappa)\)
Greater0: \(\kappa^\lambda\)
Page: 034
Section: 10
Variables: \(\kappa,\lambda\ge2\)
At least one of them is infinite.
Index: 42
Kind: Lemma
Hide middle: yes
Assuming: ZFC+GCH
Greater1: \(\kappa^+\)
Connect Greater0: \(=\)

Smaller0: \(\lambda<\text{cf}(\kappa)\)
Greater0: \(\kappa^\lambda\)
Page: 034
Section: 10
Variables: \(\kappa,\lambda\ge2\) and at least one of them is infinite.
Index: 42
Kind: Lemma
Hide middle: yes
Assuming: ZFC+GCH
Greater1: \(\kappa\)
Connect Greater0: \(=\)

Definition: \(\bigcup\{^\alpha A:\alpha<\beta\}\).
Page: 034
Chapter: 1
Section: 10
Notation: \(^{<\beta}A\)
Index: 43
Kind: Def
Assuming: ZFC
Notation2: \(A^{<\beta}\)

Smaller0: \(k^{<\lambda}\)
Greater0: \(|^{<\lambda}\kappa|\)
Page: 034
Section: 10
Index: 43
Kind: Def
Smaller to greater: =
Hide middle: yes
Assuming: ZFC

Smaller0: \(\beth_0\)
Greater0: \(\omega\)
Page: 035
Section: 10
Index: 44
Kind: Def
Smaller to greater: =
Hide middle: yes
Assuming: ZFC

Question: The true statement of NBG are
Answer: exactly the true statement of ZF.
Page: 035
Section: 12

Smaller0: \(\beth_{\alpha+1} \)
Greater0: \(2^{\beth_\alpha} \)
Page: 035
Section: 10
Variables: \(\alpha\) an ordinal
Index: 44
Kind: Def
Smaller to greater: =
Hide middle: yes
Assuming: ZFC

Smaller0: \(\beth_{\gamma} \)
Greater0: \(\text{sup}\{\beth_\alpha:\alpha<\gamma\} \)
Page: 035
Section: 10
Variables: \(\gamma\) a limit ordinal
Index: 44
Kind: Def
Smaller to greater: =
Hide middle: yes
Assuming: ZFC

Name: NBG
Definition: ZF where classes are also defined. But bound variables are only set.
Page: 035
Chapter: 1
Section: 12

Name: MK
Definition: NBG without the restriction on bounded variables.
Page: 036
Chapter: 1
Section: 12

Question: \(S'\) is
Answer: a conservative extension.
Page: 037
Section: 13
Variables: \(\mathcal L\) be a set of symbols
\(S\) be a set of sentences over \(\mathcal L\)
\(S'\) in \(\mathcal L'\) be an extension by definitions of \(S\) in \(\mathcal L\).
Kind: Theorem

Name: A recursive subset of \(\mathbb N\)
Definition: There is a program accepting exactly the elements of the set.
Page: 038
Chapter: 1
Section: 14
Variables: \(N\subseteq \mathbb N\)
Name2: A decidable subset of \(\mathbb N\)

Name: The set represented by the formula \(\xi(x)\)
Definition: A set \(R\subseteq\mathbb N\) such that, for all \(n\), [$]n\in R\implies(\xi_R(\text{\textopencorner} n\text{\textcorner}))[/$]
and [$]n\not\in R\implies(\neg\xi_R(\text{\textopencorner} n\text{\textcorner}))[/$].
Page: 040
Chapter: 1
Section: 14
Variables: \(\xi(x)\) a formula.
Kind: Theorem

Question: There is
Answer: a sentence \(\psi\) such that:
[$]ZF\vdash\psi\iff\phi(\text{\textopencorner} \psi\text{\textcorner})[/$].
Page: 040
Section: 14
Variables: \(\phi(x)\) a formula in one free variable \(x\)
Index: 2
Kind: Theorem
Name: Gödel

Name: Gödel second incompleteness Theorem
Definition: \(T\not\vdash\text{Consistency of }T\).
Page: 041
Chapter: 1
Section: 14
Variables: \(T\) be a recursive consistent extension of ZF
Index: 3
Kind: Theorem

Name: A Gödel sentence
Definition: A sentence which states its non-provability.
Page: 042
Chapter: 1
Section: 14

Question: All Gödel sentence are
Answer: equivalent.
Page: 042
Section: 14