towards an independent version of tarski s system of …

Towards an independent version of Tarski’ssystem of geometry

Pierre Boutry

University of Strasbourg - ICube - CNRS

Logic Colloquium 2019Prague, August 15, 2019

IntroductionTarski’s System of Geometry

Gupta’s and Szczerba’s contributionsAn independent version of Tarski’s system of geometry?

Conclusion

Introduction

Euclid’s Elements

led to what it likely the first independenceproof.

Hilbert’s Grundlagen der Geometrie

contains a chapterdedicated to independence properties.

Tarski’s System of Geometry

and the problem of itsindependence was carefully studied by Gupta.

Pierre BoutryTowards an independent version of Tarski’s system of geometry



Conclusion

Introduction

Euclid’s Elements led to what it likely the first independenceproof.

Hilbert’s Grundlagen der Geometrie

contains a chapterdedicated to independence properties.






Conclusion

Introduction


Hilbert’s Grundlagen der Geometrie contains a chapterdedicated to independence properties.






Conclusion

Introduction


Hilbert’s Grundlagen der Geometrie contains a chapterdedicated to independence properties.

Tarski’s System of Geometry and the problem of itsindependence was carefully studied by Gupta.




Conclusion

Outline

1 Introduction

2 Tarski’s System of Geometry

3 Gupta’s and Szczerba’s contributions

4 An independent version of Tarski’s system of geometry?

5 Conclusion




Conclusion

The axiomsA model of the theory

Outline

1 Introduction




5 Conclusion




Conclusion


Tarski’s system of geometry

A single primitive type: point.

Two primitive predicates:

1 congruence AB ≡ CD;2 betweenness A B C .

11 axioms.

A parameter controls the dimension.

Good meta-theoritical properties.




Conclusion






11 axioms.



Alfred Tarski

(1901 - 1983)




Conclusion





1 congruence AB ≡ CD;

2 betweenness A B C .

11 axioms.



Alfred Tarski

(1901 - 1983)




Conclusion






11 axioms.



Alfred Tarski

(1901 - 1983)




Conclusion


Axioms about congruence

Axiom (Pseudo-transitivity for congruence)

AB ≡ CD ∧ AB ≡ EF ⇒ CD ≡ EF

Axiom (Pseudo-reflexivity for congruence)

AB ≡ BA

Axiom (Identity for congruence)

AB ≡ CC ⇒ A = B




Conclusion


Axiom about betweenness

Axiom (Identity for betweenness)

A B A⇒ A = B




Conclusion


Five-Segment Axiom

Axiom (Five-Segment)




Conclusion


Five-Segment Axiom


AB ≡ A′B ′ ∧ BC ≡ B ′C ′∧AD ≡ A′D ′ ∧ BD ≡ B ′D ′∧A B C ∧ A′ B ′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D ′

B C

D

B′ C ′

D′




Conclusion


Five-Segment Axiom


AB ≡ A′B ′ ∧ BC ≡ B ′C ′∧AD ≡ A′D ′ ∧ BD ≡ B ′D ′∧A B C ∧ A′ B ′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D ′

A B C

D

A′ B′ C ′

D′




Conclusion


Axiom of Segment Construction

Axiom (Segment Construction)

∃E ,A B E ∧ BE ≡ CD




Conclusion





A

C

B

D




Conclusion





A

C

B

D

E




Conclusion


Pasch’s axiom

Axiom (Pasch)

A P C ∧ B Q C ⇒ ∃X ,P X B ∧ Q X A




Conclusion


Pasch’s axiom

Axiom (Pasch)

A P C ∧ B Q C ⇒ ∃X ,P X B ∧ Q X A

C

B

P




Conclusion


Pasch’s axiom

Axiom (Pasch)

A P C ∧ B Q C ⇒ ∃X ,P X B ∧ Q X A

C

B

A

P




Conclusion


Pasch’s axiom

Axiom (Pasch)

A P C ∧ B Q C ⇒ ∃X ,P X B ∧ Q X A

C

B

A

P Q

X




Conclusion


2-Dimensional Axiom

Axiom (Lower 2-Dimensional)

∃ABC ,¬A B C ∧ ¬B C A ∧ ¬C A B

Axiom (Upper 2-Dimensional)

AP ≡ AQ ∧ BP ≡ BQ ∧ CP ≡ CQ ∧ P 6= Q ⇒A B C ∨ B C A ∨ C A B

B

A

C

Q

P

C ′




Conclusion


Euclid’s axiom

Axiom (Euclid)

A D T ∧ B D C ∧ A 6= D ⇒∃XY ,A B X ∧ A C Y ∧ X T Y




Conclusion


Euclid’s axiom

Axiom (Euclid)


A

C

B

D

T




Conclusion


Euclid’s axiom

Axiom (Euclid)


A

C

B

D

TYX




Conclusion


The axioms

Identity for betweenness A B A⇒ A = B

Transitivity for congruence AB ≡ CD ∧ AB ≡ EF ⇒ CD ≡ EF

Reflexivity for congruence AB ≡ BA

Identity for congruence AB ≡ CC ⇒ A = B

Segment Construction ∃E ,A B E ∧ BE ≡ CD

Pasch A P C ∧ B Q C ⇒ ∃X ,P X B ∧ Q X A

Five-Segment AB ≡ A′B ′ ∧ BC ≡ B ′C ′∧Transitivite de la congruence AD ≡ A′D ′ ∧ BD ≡ B ′D ′∧

A B C ∧ A′ B ′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D ′

Lower 2-Dimensional ∃ABC ,¬A B C ∧ ¬B C A ∧ ¬C A B

Upper 2-Dimensional AP ≡ AQ ∧ BP ≡ BQ ∧ CP ≡ CQ ∧ P 6= Q ⇒A B C ∨ B C A ∨ C A B

Euclid A D T ∧ B D C ∧ A 6= D ⇒∃XY ,A B X ∧ A C Y ∧ X T Y

Continuity ∀ΞΥ, (∃A, (∀XY ,ΞX ∧ΥY ⇒ A X Y ))⇒∃B, (∀XY ,ΞX ∧ΥY ⇒ X B Y )

Point equality decidability X = Y ∨ X 6= Y




Conclusion


Overview of the formalization




Conclusion


Overview of the formalization

geocoq.github.io/GeoCoq/


geocoq.github.io/GeoCoq/



Conclusion


An example of proof by computation

Our example is the nine-point circletheorem which states that the followingnine points are concyclic:

The midpoints of each side of thetriangle;The feet of each altitude;The midpoints of the line-segmentsfrom each vertex of the triangle to theorthocenter.




Conclusion





A

B

C




Conclusion




The midpoints of each side of thetriangle;

The feet of each altitude;The midpoints of the line-segmentsfrom each vertex of the triangle to theorthocenter.

A

B

C

B1

A1

C1

O




Conclusion




The midpoints of each side of thetriangle;The feet of each altitude;

The midpoints of the line-segmentsfrom each vertex of the triangle to theorthocenter.

A

B

C

B1

A1

C1

H

C2

A2

B2

O




Conclusion





A

B

C

B1

A1

C1

H

C2

A2

B2

A3

B3

C3

O




Conclusion


A model of the theory

Points: F2 where F is a real closed field.

AB ≡ CD := (xA−xB)2+(yA−yB)2 = (xC−xD)2+(yC−yD)2.

A B C := ∃k, 0 ≤ k ≤ 1 ∧ B − A = k(C − A).




Conclusion

Gupta’s contributionSzczerba’s contribution

Outline

1 Introduction




5 Conclusion




Conclusion


Gupta’s axiom system




Conclusion









Five-Segment AB ≡ A′B ′ ∧ BC ≡ B ′C ′∧Transitivity for betweenness AD ≡ A′D ′ ∧ BD ≡ B ′D ′∧

A B C ∧ A′ B ′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D ′


Upper 2-Dimensional AP ≡ AQ ∧ BP ≡ BQ ∧ CP ≡ CQ ∧ P 6= Q ⇒Transitivity for betweenness A 6= B ∧ B 6= C ∧ A 6= C ⇒

A B C ∨ B C A ∨ C A B


Continuity ∀ΞΥ, (∃A, (∀XY ,ΞX ∧ΥY ⇒ A X Y ))⇒∃B, (∀XY ,ΞX ∧ΥY ⇒ X B Y )∃B, (∀XY ,




Conclusion



Transitivity for betweenness A B D ∧ B C D ⇒ A B C







A B C ∧ A′ B ′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D ′









Conclusion




Transitivity for congruence AB ≡ EF ∧ CD ≡ EF ⇒ AB ≡ CD






A B C ∧ A′ B ′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D ′









Conclusion










A B C ∧ A′ B ′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D ′


Upper 2-Dimensional AP ≡ AQ ∧ BP ≡ BQ ∧ CP ≡ CQ ∧ P 6= Q∧Transitivity for betweenness A 6= B ∧ B 6= C ∧ A 6= C ⇒







Conclusion










A B C ∧ A′ B ′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D ′





Continuity ∀ΞΥ, (∃A, (∀XY ,ΞX ∧ΥY ⇒ A X Y ))⇒∃B, (∀XY ,ΞX ∧ΥY ⇒∃B, (∀XY , X B Y ∨ X = B ∨ B = Y )




Conclusion








Pasch A X Q ∧ C Q B ⇒ ∃P,A P C ∧ B X P


A B C ∧ A′ B ′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D ′





Continuity ∀ΞΥ, (∃A, (∀XY ,ΞX ∧ΥY ⇒ A X Y ))⇒∃B, (∀XY ,ΞX ∧ΥY ⇒∃B, (∀XY , X B Y ∨ X = B ∨ B = Y )




Conclusion


Pasch’s axioms

Axiom (Inner Pasch)

A P C ∧ B Q C ⇒ ∃X ,P X B ∧ Q X A

Axiom (Outer Pasch)

A X Q ∧ C Q B ⇒ ∃P,A P C ∧ B X P




Conclusion


Pasch’s axioms

Axiom (Inner Pasch)

A P C ∧ B Q C ⇒ ∃X ,P X B ∧ Q X A

Axiom (Outer Pasch)

A X Q ∧ C Q B ⇒ ∃P,A P C ∧ B X P

C

B

A

P Q

X




Conclusion


Gupta’s independence model for Pasch

Points: F2 where F is a real closed field.

AB ≡ CD := (xA−xB)2+(yA−yB)2 = (xC−xD)2+(yC−yD)2.

A B C := ∃k, 0 ≤ k ≤ 1 ∧ B − A = k(C − A) at theexception of the cases when A = B and both A and C belongto the x-axis.




Conclusion


Szczerba’s axiom system




Conclusion


Szczerba’s axiom system

Identity for betweenness

Transitivity for congruence

Reflexivity for congruence

Identity for congruence

Segment Construction

Pasch A X Q ∧ C Q B ⇒ ∃P,A P C ∧ B X P

Five-SegmentTransitivity for betweenness

Lower 2-Dimensional

Upper 2-DimensionalTransitivity for betweenness

Euclid ¬(A B C ∨ B C A ∨ C A B)⇒∃CC ,ACC ≡ BCC ∧ ACC ≡ CCC

Continuity




Conclusion


Triangle circumscription principle

Axiom (Triangle circumscription principle)

¬(A B C ∨ B C A ∨ C A B)⇒∃CC ,ACC ≡ BCC ∧ ACC ≡ CCC

CC

A B

C




Conclusion

Parallel postulates are not equivalentHow to classify the postulates?The axioms

Outline

1 Introduction




5 Conclusion




Conclusion





Conclusion


Euclid’s axiom

Axiom (Euclid)





Conclusion


Euclid’s axiom

Axiom (Euclid)


A

C

B

D

TYX




Conclusion


Syntactic proof








A B C ∧ A′ B ′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D ′




Continuity ∀ΞΥ, (∃A, (∀XY ,X ∈ Ξ ∧ Y ∈ Υ⇒ A X Y ))⇒∃B, (∀XY ,X ∈ Ξ ∧ Y ∈ Υ⇒ X B Y )




Conclusion


Syntactic proof

A B A⇒ A = B I

AB ≡ CD ∧ AB ≡ EF ⇒ CD ≡ EF I

AB ≡ BA I

AB ≡ CC ⇒ A = BI

∃E ,A B E ∧ BE ≡ CDI

A P C ∧ B Q C ⇒ ∃X ,P X B ∧ Q X AI

AB ≡ A′B ′ ∧ BC ≡ B ′C ′∧IAD ≡ A′D ′ ∧ BD ≡ B ′D ′∧A B C ∧ A′ B ′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D ′

∃ABC ,¬A B C ∧ ¬B C A ∧ ¬C A BI

AP ≡ AQ ∧ BP ≡ BQ ∧ CP ≡ CQ ∧ P 6= Q ⇒IA B C ∨ B C A ∨ C A B

A D T ∧ B D C ∧ A 6= D ⇒I∃XY ,A B X ∧ A C Y ∧ X T Y

∀ΞΥ, (∃A, (∀XY ,X ∈ Ξ ∧ Y ∈ Υ⇒ A X Y ))⇒∃B, (∀XY ,X ∈ Ξ ∧ Y ∈ Υ⇒ X B Y )




Conclusion


Syntactic proof

B

C

A

A B A⇒ A = B I


AB ≡ BA I












Conclusion


Syntactic proof

B

C

A

P

Q

X

A B A⇒ A = B I


AB ≡ BA I












Conclusion


Syntactic proof

C

A

E

B

D

A B A⇒ A = B I


AB ≡ BA I












Conclusion


Syntactic proof

T

D

A

B C

X Y

A B A⇒ A = B I


AB ≡ BA I












Conclusion


Syntactic proof








A B C ∧ A′ B ′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D ′




Continuity ∀ΞΥ, (∃A, (∀XY ,X ∈ Ξ ∧ Y ∈ Υ⇒ A X Y ))⇒∃B, (∀XY ,X ∈ Ξ ∧ Y ∈ Υ⇒ X B Y )




Conclusion


Parallel postulates are not equivalent




Conclusion



Bachmann’sLotschnit-

taxiom

Trianglepostulate

Playfair’spostulate

Archimedes’axiom

Aristotle’saxiom

Greenberg’saxiom




Conclusion




taxiom

Trianglepostulate


Archimedes’axiom

Aristotle’saxiom

Greenberg’saxiom

A1

A2

B1

B2

C1

C2

P




Conclusion




taxiom

Trianglepostulate


Tarski’sparallel

postulate

Archimedes’axiom

Aristotle’saxiom

Greenberg’saxiom




Conclusion




taxiom

Trianglepostulate


Tarski’sparallel

postulate

Archimedes’axiom

Aristotle’saxiom

Greenberg’saxiom

A

D CB

TX Y




Conclusion




taxiom

Trianglepostulate


Tarski’sparallel

postulate

Archimedes’axiom

Aristotle’saxiom

Greenberg’saxiom




Conclusion




taxiom

Trianglepostulate


Tarski’sparallel

postulate

Archimedes’axiom

Aristotle’saxiom

Greenberg’saxiom

Decidability ofintersection

of lines




Conclusion


How to classify the postulates?




Conclusion



10.11.

12.

7.

8. 6.

5.

2.

9.




Conclusion



18.15.

1.

17.

20. 19.

16.

14.

13.10.

11.

12.

7.

8. 6.

5.

2.

9.




Conclusion



18.15.

1.

17.

20. 19.

16.

14.

13.28.

30.

29.

26.

25.

24. 23.

22.

3.

21.

27.10.

11.

12.

7.

8. 6.

5.

2.

9.




Conclusion



18.15.

1.

17.

20. 19.

16.

14.

13.28.

30.

29.

26.

25.

24. 23.

22.

3.

21.

27.10.

11.

12.

7.

8. 6.

5.

2.

9.31.

33.

32. 4.

34.




Conclusion



Pursuing the project faithfully will require that we takethe extreme measure of shutting out the entreaties ofour intuitions and imaginations - a forced separationof mental powers that will quite understandably beconfusing and difficult to maintain [...].

(Richard J. Trudeau)




Conclusion


The axioms




Conclusion


The axiomsTransitivity for betweennessTransitivity for betweenness A B D ∧ B C D ⇒ A B CTransitivity for congruence AB ≡ EF ∧ CD ≡ EF ⇒ AB ≡ CDReflexivity for congruence AB ≡ BA

Identity for congruence AB ≡ CC ⇒ A = BSegment Construction ∃E ,A B E ∧ BE ≡ CD

Pasch A X Q ∧ C Q B ⇒ ∃P,A P C ∧ B X PB 6= Q ∧ Q 6= C ∧ ¬(A B C ∨ B C A ∨ C A B)⇒

Five-Segment AB ≡ A′B′ ∧ BC ≡ B′C ′∧Transitivity for betweenness AD ≡ A′D′ ∧ BD ≡ B′D′∧

A B C ∧ A′ B′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D′

Lower 2-Dimensional ∃ABC ,¬A B C ∧ ¬B C A ∧ ¬C A BUpper 2-Dimensional AP ≡ AQ ∧ BP ≡ BQ ∧ CP ≡ CQ ∧ P 6= Q∧

Transitivity for betweenness A 6= B ∧ B 6= C ∧ A 6= C ⇒A B C ∨ B C A ∨ C A B


Continuity ∀ΞΥ, (∃A, (∀XY ,ΞX ∧ΥY ⇒ A X Y ))⇒∃B, (∀XY ,ΞX ∧ΥY ⇒ X B Y ∨ X = B ∨ B = Y )

Point equality decidability X = Y ∨ X 6= YPierre Boutry

Towards an independent version of Tarski’s system of geometry



Conclusion


The axiomsSymmetry for betweenness A B C ⇒ C B ATransitivity for betweenness A B D ∧ B C D ⇒ A B CTransitivity for congruence AB ≡ EF ∧ CD ≡ EF ⇒ AB ≡ CDReflexivity for congruence AB ≡ BA




A B C ∧ A′ B′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D′









Conclusion


The axiomsSymmetry for betweenness A B C ⇒ C B A

Transitivity for betweenness A B D ∧ B C D ⇒ A B CTransitivity for congruence AB ≡ EF ∧ CD ≡ EF ⇒ AB ≡ CDReflexivity for congruence AB ≡ BA




A B C ∧ A′ B′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D′









Conclusion





Pasch A P C ∧ B Q C ∧ A 6= P ∧ P 6= C∧B 6= Q ∧ Q 6= C ∧ ¬(A B C ∨ B C A ∨ C A B)⇒∃X ,P X B ∧ Q X A


A B C ∧ A′ B′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D′









Conclusion







A B C ∧ A′ B′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D′



Proclus






Conclusion


A few definitions

Collinearity


Coplanarity

∃X , (Col AB X ∧ Col C D X ) ∨ (Col AC X ∧ Col B D X ) ∨(Col AD X ∧ Col B C X )

Strict parallelism

A 6= B ∧ C 6= D ∧ Cp AB C D ∧ ¬∃X ,Col AB X ∧ Col C D X

Parallelism

AB ‖s CD ∨ (A 6= B ∧ C 6= D ∧ Col AC D ∧ Col B C D)




Conclusion


Proclus’ axiom

Axiom (Proclus’ axiom)

AB ‖ CD ∧ Col AB P ∧ ¬Col AB Q ⇒∃Y ,Col C D Y ∧ Col P Q Y




Conclusion


Proclus’ axiom

Axiom (Proclus’ axiom)

AB ‖ CD ∧ Col AB P ∧ ¬Col AB Q ⇒∃Y ,Col C D Y ∧ Col P Q Y

A

B

C

D

Q

Y

P




Conclusion


The axioms




Conclusion







A B C ∧ A′ B′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D′



ProclusContinuity ∀ΞΥ, (∃A, (∀XY ,ΞX ∧ΥY ⇒ A X Y ))⇒

∃B, (∀XY ,ΞX ∧ΥY ⇒ X B Y ∨ X = B ∨ B = Y )Point equality decidability X = Y ∨ X 6= Y




Conclusion


The axiomsSymmetry for betweenness A B C ⇒ C B ATransitivity for betweenness A B D ∧ B C D ⇒ A B CTransitivity for congruence AB ≡ EF ∧ CD ≡ EF ⇒ AB ≡ CDReflexivity for congruence AB ≡ BA




A B C ∧ A′ B′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D′








Conclusion



Transitivity for betweenness A B D ∧ B C D ⇒ A B CTransitivity for congruence AB ≡ EF ∧ CD ≡ EF ⇒ AB ≡ CD

Reflexivity for congruence AB ≡ BAIdentity for congruence AB ≡ CC ⇒ A = BSegment Construction ∃E ,A B E ∧ BE ≡ CD



A B C ∧ A′ B′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D′








Conclusion







A B C ∧ A′ B′ C ′ ∧ A 6= B ⇒ CD ≡ C ′D′








Conclusion

Results

An independent version of Tarski’s system of geometry.

We had to correct one of Gupta’s models.

Inner Pasch holds in the models that were given for outerPasch.




Conclusion

Thank you!


towards an independent version of tarski s system of …

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