ARITHMETIC, SET THEORY, AND THEIR MODELS - PART ONE

99 Pages·2011·0.79 MB·English

by  Ali Enayat

ARITHMETIC, SET THEORY, AND THEIR
MODELS
PART ONE: END EXTENSIONS
Ali Enayat
YOUNGSETTHEORYWORKSHOP
KO¨NIGSWINTER, MARCH 21-25, 2011
AliEnayat ARITHMETIC,SETTHEORY,ANDTHEIRMODELS
(1) [Koepke-Koerwien] SO ≈ ZFC.
SO = Second Order Theory of Ordinals.
T−∞ := T\{Infinity}∪{¬Infinity}.
(2) [Mostowski] Z +Π1 -AC =
2 ∞
(Second Order Arithmetic + Choice Scheme) ≈
ZFC\{Power}+V = H(ℵ ) ≈
1
KMC−.
SET THEORETICAL TWINS OF ARITHMETICAL
THEORIES: THE GLOBAL VIEW
AliEnayat ARITHMETIC,SETTHEORY,ANDTHEIRMODELS
T−∞ := T\{Infinity}∪{¬Infinity}.
(2) [Mostowski] Z +Π1 -AC =
2 ∞
(Second Order Arithmetic + Choice Scheme) ≈
ZFC\{Power}+V = H(ℵ ) ≈
1
KMC−.
SET THEORETICAL TWINS OF ARITHMETICAL
THEORIES: THE GLOBAL VIEW
(1) [Koepke-Koerwien] SO ≈ ZFC.
SO = Second Order Theory of Ordinals.
AliEnayat ARITHMETIC,SETTHEORY,ANDTHEIRMODELS
(2) [Mostowski] Z +Π1 -AC =
2 ∞
(Second Order Arithmetic + Choice Scheme) ≈
ZFC\{Power}+V = H(ℵ ) ≈
1
KMC−.
SET THEORETICAL TWINS OF ARITHMETICAL
THEORIES: THE GLOBAL VIEW
(1) [Koepke-Koerwien] SO ≈ ZFC.
SO = Second Order Theory of Ordinals.
T−∞ := T\{Infinity}∪{¬Infinity}.
AliEnayat ARITHMETIC,SETTHEORY,ANDTHEIRMODELS
ZFC\{Power}+V = H(ℵ ) ≈
1
KMC−.
SET THEORETICAL TWINS OF ARITHMETICAL
THEORIES: THE GLOBAL VIEW
(1) [Koepke-Koerwien] SO ≈ ZFC.
SO = Second Order Theory of Ordinals.
T−∞ := T\{Infinity}∪{¬Infinity}.
(2) [Mostowski] Z +Π1 -AC =
2 ∞
(Second Order Arithmetic + Choice Scheme) ≈
AliEnayat ARITHMETIC,SETTHEORY,ANDTHEIRMODELS
SET THEORETICAL TWINS OF ARITHMETICAL
THEORIES: THE GLOBAL VIEW
(1) [Koepke-Koerwien] SO ≈ ZFC.
SO = Second Order Theory of Ordinals.
T−∞ := T\{Infinity}∪{¬Infinity}.
(2) [Mostowski] Z +Π1 -AC =
2 ∞
(Second Order Arithmetic + Choice Scheme) ≈
ZFC\{Power}+V = H(ℵ ) ≈
1
KMC−.
AliEnayat ARITHMETIC,SETTHEORY,ANDTHEIRMODELS
(3) [Ackermann, Mycielski, Kaye-Wong]
ACA ≈ GBC−∞+TC.
0
PA ≈ ZF−∞+TC.
(4) [Gaifman-Dimitracopoulos]
EFA (Elementary Function Arithmetic)≈ Mac−∞.
(5) [Szmielew-Tarski]
Robinson’s Q ≈ AST (Adjunctive Set Theory).
SET THEORETICAL TWINS OF ARITHMETICAL
THEORIES: THE GLOBAL VIEW (CONT’D)
AliEnayat ARITHMETIC,SETTHEORY,ANDTHEIRMODELS
(4) [Gaifman-Dimitracopoulos]
EFA (Elementary Function Arithmetic)≈ Mac−∞.
(5) [Szmielew-Tarski]
Robinson’s Q ≈ AST (Adjunctive Set Theory).
SET THEORETICAL TWINS OF ARITHMETICAL
THEORIES: THE GLOBAL VIEW (CONT’D)
(3) [Ackermann, Mycielski, Kaye-Wong]
ACA ≈ GBC−∞+TC.
0
PA ≈ ZF−∞+TC.
AliEnayat ARITHMETIC,SETTHEORY,ANDTHEIRMODELS
(5) [Szmielew-Tarski]
Robinson’s Q ≈ AST (Adjunctive Set Theory).
SET THEORETICAL TWINS OF ARITHMETICAL
THEORIES: THE GLOBAL VIEW (CONT’D)
(3) [Ackermann, Mycielski, Kaye-Wong]
ACA ≈ GBC−∞+TC.
0
PA ≈ ZF−∞+TC.
(4) [Gaifman-Dimitracopoulos]
EFA (Elementary Function Arithmetic)≈ Mac−∞.
AliEnayat ARITHMETIC,SETTHEORY,ANDTHEIRMODELS
SET THEORETICAL TWINS OF ARITHMETICAL
THEORIES: THE GLOBAL VIEW (CONT’D)
(3) [Ackermann, Mycielski, Kaye-Wong]
ACA ≈ GBC−∞+TC.
0
PA ≈ ZF−∞+TC.
(4) [Gaifman-Dimitracopoulos]
EFA (Elementary Function Arithmetic)≈ Mac−∞.
(5) [Szmielew-Tarski]
Robinson’s Q ≈ AST (Adjunctive Set Theory).
AliEnayat ARITHMETIC,SETTHEORY,ANDTHEIRMODELS