Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Axiomatic Systems and Their Properties

Aaron Cinzori

Fall 2012

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Axiomatic Systems

1 Any axiomatic system must contain a set of technical termsthat are deliberately chosen as undefined terms and aresubject to the interpretation of the reader.

2 All other technical terms of the system are ultimately definedby means of the undefined terms. These terms are thedefinitions of the system.

3 The axiomatic system contains a set of statements, dealingwith undefined terms and definitions, that are chosen toremain unproved. These are the axioms of the system.

4 All other statements of the system must be logicalconsequences of the axioms. These derived statements arecalled the theorems of the axiomatic system.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Axiomatic Systems

1 Any axiomatic system must contain a set of technical termsthat are deliberately chosen as undefined terms and aresubject to the interpretation of the reader.

2 All other technical terms of the system are ultimately definedby means of the undefined terms. These terms are thedefinitions of the system.

3 The axiomatic system contains a set of statements, dealingwith undefined terms and definitions, that are chosen toremain unproved. These are the axioms of the system.

4 All other statements of the system must be logicalconsequences of the axioms. These derived statements arecalled the theorems of the axiomatic system.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Axiomatic Systems

1 Any axiomatic system must contain a set of technical termsthat are deliberately chosen as undefined terms and aresubject to the interpretation of the reader.

2 All other technical terms of the system are ultimately definedby means of the undefined terms. These terms are thedefinitions of the system.

3 The axiomatic system contains a set of statements, dealingwith undefined terms and definitions, that are chosen toremain unproved. These are the axioms of the system.

4 All other statements of the system must be logicalconsequences of the axioms. These derived statements arecalled the theorems of the axiomatic system.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Axiomatic Systems

1 Any axiomatic system must contain a set of technical termsthat are deliberately chosen as undefined terms and aresubject to the interpretation of the reader.

2 All other technical terms of the system are ultimately definedby means of the undefined terms. These terms are thedefinitions of the system.

3 The axiomatic system contains a set of statements, dealingwith undefined terms and definitions, that are chosen toremain unproved. These are the axioms of the system.

4 All other statements of the system must be logicalconsequences of the axioms. These derived statements arecalled the theorems of the axiomatic system.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Example 1.2.1

Undefined Terms: Fe, Fo, belongs to.

Axiom 1: There exist exactly three distinct Fe’s in this system.

Axiom 2: Any two distinct Fe’s belong to exactly one Fo.

Axiom 3: Not all Fe’s belong to the same Fo.

Axiom 4: Any two distinct Fo’s contain at least one Fe thatbelongs to both.

Fe-Fo Theorem 1: Two distinct Fo’s contain exactly one Fe.

Fe-Fo Theorem 2: There are exactly three Fo’s.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Example 1.2.1

Undefined Terms: Fe, Fo, belongs to.

Axiom 1: There exist exactly three distinct Fe’s in this system.

Axiom 2: Any two distinct Fe’s belong to exactly one Fo.

Axiom 3: Not all Fe’s belong to the same Fo.

Axiom 4: Any two distinct Fo’s contain at least one Fe thatbelongs to both.

Fe-Fo Theorem 1: Two distinct Fo’s contain exactly one Fe.

Fe-Fo Theorem 2: There are exactly three Fo’s.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Example 1.2.1

Undefined Terms: Fe, Fo, belongs to.

Axiom 1: There exist exactly three distinct Fe’s in this system.

Axiom 2: Any two distinct Fe’s belong to exactly one Fo.

Axiom 3: Not all Fe’s belong to the same Fo.

Axiom 4: Any two distinct Fo’s contain at least one Fe thatbelongs to both.

Fe-Fo Theorem 1: Two distinct Fo’s contain exactly one Fe.

Fe-Fo Theorem 2: There are exactly three Fo’s.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Example 1.2.1

Undefined Terms: Fe, Fo, belongs to.

Axiom 1: There exist exactly three distinct Fe’s in this system.

Axiom 2: Any two distinct Fe’s belong to exactly one Fo.

Axiom 3: Not all Fe’s belong to the same Fo.

Axiom 4: Any two distinct Fo’s contain at least one Fe thatbelongs to both.

Fe-Fo Theorem 1: Two distinct Fo’s contain exactly one Fe.

Fe-Fo Theorem 2: There are exactly three Fo’s.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Example 1.2.1

Undefined Terms: Fe, Fo, belongs to.

Axiom 1: There exist exactly three distinct Fe’s in this system.

Axiom 2: Any two distinct Fe’s belong to exactly one Fo.

Axiom 3: Not all Fe’s belong to the same Fo.

Axiom 4: Any two distinct Fo’s contain at least one Fe thatbelongs to both.

Fe-Fo Theorem 1: Two distinct Fo’s contain exactly one Fe.

Fe-Fo Theorem 2: There are exactly three Fo’s.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Example 1.2.1

Undefined Terms: Fe, Fo, belongs to.

Axiom 1: There exist exactly three distinct Fe’s in this system.

Axiom 2: Any two distinct Fe’s belong to exactly one Fo.

Axiom 3: Not all Fe’s belong to the same Fo.

Axiom 4: Any two distinct Fo’s contain at least one Fe thatbelongs to both.

Fe-Fo Theorem 1: Two distinct Fo’s contain exactly one Fe.

Fe-Fo Theorem 2: There are exactly three Fo’s.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Example 1.2.1

Undefined Terms: Fe, Fo, belongs to.

Axiom 1: There exist exactly three distinct Fe’s in this system.

Axiom 2: Any two distinct Fe’s belong to exactly one Fo.

Axiom 3: Not all Fe’s belong to the same Fo.

Axiom 4: Any two distinct Fo’s contain at least one Fe thatbelongs to both.

Fe-Fo Theorem 1: Two distinct Fo’s contain exactly one Fe.

Fe-Fo Theorem 2: There are exactly three Fo’s.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Interpretations and Models

Interpret Fe’s as people (Bob, Ted, and Carol) and Fo’s ascommittees (Entertainment with Bob and Ted, Finance with Tedand Carol, and Refreshments with Bob and Carol). Then theaxioms become

Axiom 1: There exist exactly three people.

Axiom 2: Any two distinct people belong to exactly onecommittee.

Axiom 3: Not all people belong to the same committee.

Axiom 4: Any two distinct committees contain at least oneperson that belongs to both.

This is a model.

Fe-Fo Theorem 2: There are exactly three committees.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Interpretations and Models

Interpret Fe’s as people (Bob, Ted, and Carol) and Fo’s ascommittees (Entertainment with Bob and Ted, Finance with Tedand Carol, and Refreshments with Bob and Carol). Then theaxioms become

Axiom 1: There exist exactly three people.

Axiom 2: Any two distinct people belong to exactly onecommittee.

Axiom 3: Not all people belong to the same committee.

Axiom 4: Any two distinct committees contain at least oneperson that belongs to both.

This is a model.

Fe-Fo Theorem 2: There are exactly three committees.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Interpretations and Models

Interpret Fe’s as people (Bob, Ted, and Carol) and Fo’s ascommittees (Entertainment with Bob and Ted, Finance with Tedand Carol, and Refreshments with Bob and Carol). Then theaxioms become

Axiom 1: There exist exactly three people.

Axiom 2: Any two distinct people belong to exactly onecommittee.

Axiom 3: Not all people belong to the same committee.

Axiom 4: Any two distinct committees contain at least oneperson that belongs to both.

This is a model.

Fe-Fo Theorem 2: There are exactly three committees.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Interpretations and Models

Interpret Fe’s as people (Bob, Ted, and Carol) and Fo’s ascommittees (Entertainment with Bob and Ted, Finance with Tedand Carol, and Refreshments with Bob and Carol). Then theaxioms become

Axiom 1: There exist exactly three people.

Axiom 2: Any two distinct people belong to exactly onecommittee.

Axiom 3: Not all people belong to the same committee.

Axiom 4: Any two distinct committees contain at least oneperson that belongs to both.

This is a model.

Fe-Fo Theorem 2: There are exactly three committees.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Interpretations and Models

Interpret Fe’s as people (Bob, Ted, and Carol) and Fo’s ascommittees (Entertainment with Bob and Ted, Finance with Tedand Carol, and Refreshments with Bob and Carol). Then theaxioms become

Axiom 1: There exist exactly three people.

Axiom 2: Any two distinct people belong to exactly onecommittee.

Axiom 3: Not all people belong to the same committee.

Axiom 4: Any two distinct committees contain at least oneperson that belongs to both.

This is a model.

Fe-Fo Theorem 2: There are exactly three committees.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Interpretations and Models

Interpret Fe’s as people (Bob, Ted, and Carol) and Fo’s ascommittees (Entertainment with Bob and Ted, Finance with Tedand Carol, and Refreshments with Bob and Carol). Then theaxioms become

Axiom 1: There exist exactly three people.

Axiom 2: Any two distinct people belong to exactly onecommittee.

Axiom 3: Not all people belong to the same committee.

Axiom 4: Any two distinct committees contain at least oneperson that belongs to both.

This is a model.

Fe-Fo Theorem 2: There are exactly three committees.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Another Interpretation

Interpret Fe’s as books and Fo’s as shelves. Then the axiomsbecome:

Axiom 1: There exist exactly three books.

Axiom 2: Any two distinct books belong to exactly one shelf.

Axiom 3: Not all books belong to the same shelf.

Axiom 4: Any two distinct shelves contain at least one bookthat belongs to both.

This is not a model.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Another Interpretation

Interpret Fe’s as books and Fo’s as shelves. Then the axiomsbecome:

Axiom 1: There exist exactly three books.

Axiom 2: Any two distinct books belong to exactly one shelf.

Axiom 3: Not all books belong to the same shelf.

Axiom 4: Any two distinct shelves contain at least one bookthat belongs to both.

This is not a model.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Another Interpretation

Interpret Fe’s as books and Fo’s as shelves. Then the axiomsbecome:

Axiom 1: There exist exactly three books.

Axiom 2: Any two distinct books belong to exactly one shelf.

Axiom 3: Not all books belong to the same shelf.

Axiom 4: Any two distinct shelves contain at least one bookthat belongs to both.

This is not a model.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Another Interpretation

Interpret Fe’s as books and Fo’s as shelves. Then the axiomsbecome:

Axiom 1: There exist exactly three books.

Axiom 2: Any two distinct books belong to exactly one shelf.

Axiom 3: Not all books belong to the same shelf.

Axiom 4: Any two distinct shelves contain at least one bookthat belongs to both.

This is not a model.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Another Interpretation

Interpret Fe’s as books and Fo’s as shelves. Then the axiomsbecome:

Axiom 1: There exist exactly three books.

Axiom 2: Any two distinct books belong to exactly one shelf.

Axiom 3: Not all books belong to the same shelf.

Axiom 4: Any two distinct shelves contain at least one bookthat belongs to both.

This is not a model.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Isomorphic Models

Interpret Fe’s as the letters in the set S = {P,Q,R} and Fo’s astwo-element subsets of S . This is a model of the Fe-Fo axiomsthat is isomorphic to the “committees” model.

Letters correspond to people.

Subsets correspond to committees.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Isomorphic Models

Interpret Fe’s as the letters in the set S = {P,Q,R} and Fo’s astwo-element subsets of S . This is a model of the Fe-Fo axiomsthat is isomorphic to the “committees” model.

Letters correspond to people.

Subsets correspond to committees.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Non-Isomorphic Models

Interpret Fo’s as the letters in the set T = {x , y , z} and Fe’sas two-element subsets of T . This is a model of the Fe-Foaxioms that is not isomorphic to the “committees” model.

Consider the correspondence

{x , y} ↔ P, {P,Q} ↔ z

{y , z} ↔ Q, {Q,R} ↔ y

{x , z} ↔ R, {P,R} ↔ x

No such correspondence is an isomorphism. So this model isnot isomorphic to the “committee” model.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Non-Isomorphic Models

Interpret Fo’s as the letters in the set T = {x , y , z} and Fe’sas two-element subsets of T . This is a model of the Fe-Foaxioms that is not isomorphic to the “committees” model.

Consider the correspondence

{x , y} ↔ P, {P,Q} ↔ z

{y , z} ↔ Q, {Q,R} ↔ y

{x , z} ↔ R, {P,R} ↔ x

No such correspondence is an isomorphism. So this model isnot isomorphic to the “committee” model.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Non-Isomorphic Models

Interpret Fo’s as the letters in the set T = {x , y , z} and Fe’sas two-element subsets of T . This is a model of the Fe-Foaxioms that is not isomorphic to the “committees” model.

Consider the correspondence

{x , y} ↔ P, {P,Q} ↔ z

{y , z} ↔ Q, {Q,R} ↔ y

{x , z} ↔ R, {P,R} ↔ x

No such correspondence is an isomorphism. So this model isnot isomorphic to the “committee” model.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Independence

Interpret Fe’s as the letters in the set {A,B,C ,D} and Fo’sas the subsets {A,B}, {A,C}, {A,D}, {B,C ,D}. This is anot model of the Fe-Fo axioms because Axiom 1 fails.

But Axioms 2–4 hold.

So Axiom 1 is independent of Axioms 2–4.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Independence

Interpret Fe’s as the letters in the set {A,B,C ,D} and Fo’sas the subsets {A,B}, {A,C}, {A,D}, {B,C ,D}. This is anot model of the Fe-Fo axioms because Axiom 1 fails.

But Axioms 2–4 hold.

So Axiom 1 is independent of Axioms 2–4.

Aaron Cinzori Axiomatic Systems and Their Properties

Definitions Examples Interpretations and Models Another Interpretation Isomorphic Models Non-Isomorphic Models Independence

Independence

Interpret Fe’s as the letters in the set {A,B,C ,D} and Fo’sas the subsets {A,B}, {A,C}, {A,D}, {B,C ,D}. This is anot model of the Fe-Fo axioms because Axiom 1 fails.

But Axioms 2–4 hold.

So Axiom 1 is independent of Axioms 2–4.

Aaron Cinzori Axiomatic Systems and Their Properties