horrors with and without the axiom of choice
play

Horrors with and without the Axiom of Choice Tanmay Inamdar - PowerPoint PPT Presentation

Nov 03, 2023 •252 likes •1.34k views

Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Horrors with and without the Axiom of Choice Tanmay Inamdar February 23, 2013 Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion

  1. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion A non-measurable set Let V be a selector. Assume(!) it is measurable. Let I ∆ ∆ = Q \ [ � 1 , 1]. For each q 2 I , let V q = V + q , the pointwise translation of each element of V by q . (i) For every q 2 I 6 =0 , V \ V q = ; . Also, for each q 2 I , V q ✓ [ � 1 , 2]. (ii) µ ( V ) = µ ( V q ). So Σ q 2 I µ ( V q ) = Σ q 2 I µ ( V ). (iii) But [0 , 1] ✓ U q 2 I V q ✓ [ � 1 , 2]. (iv) So µ ([0 , 1])  Σ q 2 I µ ( V q )  µ ([ � 1 , 2]). (v) Therefore, 1  Σ q 2 I µ ( V q )  3. (vi) So µ ( V ) � 0, but Σ q 2 I µ ( V q ) = Σ q 2 I µ ( V )  3. Contradiction.

  2. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Groups Definition A group is a set G together with a binary operation • which satisfies the following laws: (i) Closure: If a, b 2 G then a • b 2 G . (ii) Identity: There is an element e 2 G such that for all g 2 G , e • g = g . (iii) Associativity: If a, b, c 2 G , then ( a • b ) • c = a • ( b • c ). (iv) Inverse: If a 2 G , then there is an element b 2 G such that a • b = b • a = e .

  3. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Groups Definition A group is a set G together with a binary operation • which satisfies the following laws: (i) Closure: If a, b 2 G then a • b 2 G . (ii) Identity: There is an element e 2 G such that for all g 2 G , e • g = g . (iii) Associativity: If a, b, c 2 G , then ( a • b ) • c = a • ( b • c ). (iv) Inverse: If a 2 G , then there is an element b 2 G such that a • b = b • a = e . Example: ( Z , + , 0). Given a set S , the set of its permutations. SO 3 , the set of rotations of R 3 . F 2 ✓ SO 3 .

  4. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Group actions Definition Let G be a group and X a set. A (left) group action of G on X is a binary operator � : G ⇥ X ! X satisfying the following laws: (i) Associativity: If g, h 2 G and x 2 X , ( g • h ) � x = g � ( h � x ). (ii) Identity: If x 2 X , e � x = x .

  5. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Group actions Definition Let G be a group and X a set. A (left) group action of G on X is a binary operator � : G ⇥ X ! X satisfying the following laws: (i) Associativity: If g, h 2 G and x 2 X , ( g • h ) � x = g � ( h � x ). (ii) Identity: If x 2 X , e � x = x . Example: Any group acting on itself. Given a set S , any subgroup of its permutation group, acting on S . SO 3 , acting on R 3 .

  6. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Decomposing and Assembling Let G act on X . Let E, F subsets of X . Say E and F are equidecomposable via G with m pieces (denoted E s G F ) if : (i) There are g 1 , · · · g m in G and A 1 , · · · A m pairwise disjoint subsets of E such that: (ii) E = U i  m A i and F = U i  m g i A i .

  7. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Decomposing and Assembling Let G act on X . Let E, F subsets of X . Say E and F are equidecomposable via G with m pieces (denoted E s G F ) if : (i) There are g 1 , · · · g m in G and A 1 , · · · A m pairwise disjoint subsets of E such that: (ii) E = U i  m A i and F = U i  m g i A i . E is paradoxical if F = E ] E .

  8. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Decomposing and Assembling Let G act on X . Let E, F subsets of X . Say E and F are equidecomposable via G with m pieces (denoted E s G F ) if : (i) There are g 1 , · · · g m in G and A 1 , · · · A m pairwise disjoint subsets of E such that: (ii) E = U i  m A i and F = U i  m g i A i . E is paradoxical if F = E ] E . If X is the group G itself, call G paradoxical if G = G ] G .

  9. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Decomposing and Assembling Let G act on X . Let E, F subsets of X . Say E and F are equidecomposable via G with m pieces (denoted E s G F ) if : (i) There are g 1 , · · · g m in G and A 1 , · · · A m pairwise disjoint subsets of E such that: (ii) E = U i  m A i and F = U i  m g i A i . E is paradoxical if F = E ] E . If X is the group G itself, call G paradoxical if G = G ] G . Aside: Amenable groups.

  10. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The broken circle Example: S 1 a proper circle and S 1 \ { ⇤ } a broken circle, then S 1 \ { ⇤ } s S 1 via SO 2 , the group of rotations of R 2 with 2 pieces.

  11. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The broken circle Example: S 1 a proper circle and S 1 \ { ⇤ } a broken circle, then S 1 \ { ⇤ } s S 1 via SO 2 , the group of rotations of R 2 with 2 pieces. (i) Recall the Hilbert Hotel.

  12. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The broken circle Example: S 1 a proper circle and S 1 \ { ⇤ } a broken circle, then S 1 \ { ⇤ } s S 1 via SO 2 , the group of rotations of R 2 with 2 pieces. (i) Recall the Hilbert Hotel. (ii) Want: one rotation to do all this moving around. Let the missing point be 1 = e i 0 . Then e � i does exactly that.

  13. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The broken circle Example: S 1 a proper circle and S 1 \ { ⇤ } a broken circle, then S 1 \ { ⇤ } s S 1 via SO 2 , the group of rotations of R 2 with 2 pieces. (i) Recall the Hilbert Hotel. (ii) Want: one rotation to do all this moving around. Let the missing point be 1 = e i 0 . Then e � i does exactly that. 2 ⇡ is irrational so this rotation freely generates an infinite set. That is, e im 6 = e in for m 6 = n . (iii) To be precise, our two sets are: A ∆ = { e in ; n 2 N + } and B ∆ = ( S 1 \ { ⇤ } ) \ A .

  14. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The broken circle Example: S 1 a proper circle and S 1 \ { ⇤ } a broken circle, then S 1 \ { ⇤ } s S 1 via SO 2 , the group of rotations of R 2 with 2 pieces. (i) Recall the Hilbert Hotel. (ii) Want: one rotation to do all this moving around. Let the missing point be 1 = e i 0 . Then e � i does exactly that. 2 ⇡ is irrational so this rotation freely generates an infinite set. That is, e im 6 = e in for m 6 = n . (iii) To be precise, our two sets are: A ∆ = { e in ; n 2 N + } and B ∆ = ( S 1 \ { ⇤ } ) \ A . (iv) It is easy to see that S 1 = ( e � i A ) ] B .

  15. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The broken circle Example: S 1 a proper circle and S 1 \ { ⇤ } a broken circle, then S 1 \ { ⇤ } s S 1 via SO 2 , the group of rotations of R 2 with 2 pieces. (i) Recall the Hilbert Hotel. (ii) Want: one rotation to do all this moving around. Let the missing point be 1 = e i 0 . Then e � i does exactly that. 2 ⇡ is irrational so this rotation freely generates an infinite set. That is, e im 6 = e in for m 6 = n . (iii) To be precise, our two sets are: A ∆ = { e in ; n 2 N + } and B ∆ = ( S 1 \ { ⇤ } ) \ A . (iv) It is easy to see that S 1 = ( e � i A ) ] B . e − i sends e i ( n +1) to e in .

  16. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion F 2 Definition The free group on the 2 generators { a, b } is defined as follows: (i) The base set consists of all finite strings that can be formed from the alphabet { a, a � 1 , b, b � 1 } which do not contain the substrings aa � 1 , a � 1 a, bb � 1 , b � 1 b . (ii) The identity element is the empty string, denoted by ✏ or e . (iii) The group operation is defined as follows: Let u and v be two strings. Then u • v is the string w obtained by first concatenating u and v and then replacing all occurences of the substrings aa � 1 , a � 1 a, bb � 1 , b � 1 b by the empty string.

  17. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion F 2 Definition The free group on the 2 generators { a, b } is defined as follows: (i) The base set consists of all finite strings that can be formed from the alphabet { a, a � 1 , b, b � 1 } which do not contain the substrings aa � 1 , a � 1 a, bb � 1 , b � 1 b . (ii) The identity element is the empty string, denoted by ✏ or e . (iii) The group operation is defined as follows: Let u and v be two strings. Then u • v is the string w obtained by first concatenating u and v and then replacing all occurences of the substrings aa � 1 , a � 1 a, bb � 1 , b � 1 b by the empty string. All free groups on 2 generators are isomorphic.

  18. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion F 2 Definition The free group on the 2 generators { a, b } is defined as follows: (i) The base set consists of all finite strings that can be formed from the alphabet { a, a � 1 , b, b � 1 } which do not contain the substrings aa � 1 , a � 1 a, bb � 1 , b � 1 b . (ii) The identity element is the empty string, denoted by ✏ or e . (iii) The group operation is defined as follows: Let u and v be two strings. Then u • v is the string w obtained by first concatenating u and v and then replacing all occurences of the substrings aa � 1 , a � 1 a, bb � 1 , b � 1 b by the empty string. All free groups on 2 generators are isomorphic. Aside: The free group on 1 generator is isomorphic to ( Z , + , 0).

  19. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion F 2 Definition The free group on the 2 generators { a, b } is defined as follows: (i) The base set consists of all finite strings that can be formed from the alphabet { a, a � 1 , b, b � 1 } which do not contain the substrings aa � 1 , a � 1 a, bb � 1 , b � 1 b . (ii) The identity element is the empty string, denoted by ✏ or e . (iii) The group operation is defined as follows: Let u and v be two strings. Then u • v is the string w obtained by first concatenating u and v and then replacing all occurences of the substrings aa � 1 , a � 1 a, bb � 1 , b � 1 b by the empty string. All free groups on 2 generators are isomorphic. Aside: The free group on 1 generator is isomorphic to ( Z , + , 0). But this is not the same as ( Z ⇥ Z , (+ , +) , (0 , 0)).

  20. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion A picture of F 2

  21. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Deconstructing F 2 , Reconstructing F 2 , F 2 F 2 is paradoxical with 4 pieces.

  22. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Deconstructing F 2 , Reconstructing F 2 , F 2 F 2 is paradoxical with 4 pieces. (i) Let G a be the elements of F 2 which start with a .

  23. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Deconstructing F 2 , Reconstructing F 2 , F 2 F 2 is paradoxical with 4 pieces. (i) Let G a be the elements of F 2 which start with a . (ii) Then F 2 = { e } [ G a [ G a − 1 [ G b [ G b − 1 .

  24. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Deconstructing F 2 , Reconstructing F 2 , F 2 F 2 is paradoxical with 4 pieces. (i) Let G a be the elements of F 2 which start with a . (ii) Then F 2 = { e } [ G a [ G a − 1 [ G b [ G b − 1 . (iii) Notice that F 2 = G a ] aG a − 1 and F 2 = G b ] bG b − 1 . But e is still troubling us.

  25. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Deconstructing F 2 , Reconstructing F 2 , F 2 F 2 is paradoxical with 4 pieces. (i) Let G a be the elements of F 2 which start with a . (ii) Then F 2 = { e } [ G a [ G a − 1 [ G b [ G b − 1 . (iii) Notice that F 2 = G a ] aG a − 1 and F 2 = G b ] bG b − 1 . But e is still troubling us. (iv) So consider = G a [ { e, a � 1 , a � 2 , a � 3 · · · } ∆ G 1 = G a − 1 \ { e, a � 1 , a � 2 , a � 3 · · · } ∆ G 2 ∆ G 3 = G b ∆ G 4 = G b − 1

  26. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Deconstructing F 2 , Reconstructing F 2 , F 2 F 2 is paradoxical with 4 pieces. (i) Let G a be the elements of F 2 which start with a . (ii) Then F 2 = { e } [ G a [ G a − 1 [ G b [ G b − 1 . (iii) Notice that F 2 = G a ] aG a − 1 and F 2 = G b ] bG b − 1 . But e is still troubling us. (iv) So consider = G a [ { e, a � 1 , a � 2 , a � 3 · · · } ∆ G 1 = G a − 1 \ { e, a � 1 , a � 2 , a � 3 · · · } ∆ G 2 ∆ G 3 = G b ∆ G 4 = G b − 1 (v) Easy to verify that F 2 = G 1 ] aG 2 = G 3 ] bG 4 .

  27. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Another picture of F 2

  28. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Hausdor ff Paradox ( AC ) Hausdor ff Paradox: There is a countable set D such that F 2 acts on S 2 \ D paradoxically with 4 pieces.

  29. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Hausdor ff Paradox ( AC ) Hausdor ff Paradox: There is a countable set D such that F 2 acts on S 2 \ D paradoxically with 4 pieces. (i) Let D ∆ = { x 2 S 2 ; 9 f 2 F 2 ( f • x = x ) } . (ii) Let G 1 , G 2 , G 3 , G 4 be as previously.

  30. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Hausdor ff Paradox ( AC ) Hausdor ff Paradox: There is a countable set D such that F 2 acts on S 2 \ D paradoxically with 4 pieces. (i) Let D ∆ = { x 2 S 2 ; 9 f 2 F 2 ( f • x = x ) } . (ii) Let G 1 , G 2 , G 3 , G 4 be as previously. (iii) The following is then an equivalence relation on S 2 \ D : x ⇠ y i ff there is a f 2 F 2 such that f � x = y .

  31. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Hausdor ff Paradox ( AC ) Hausdor ff Paradox: There is a countable set D such that F 2 acts on S 2 \ D paradoxically with 4 pieces. (i) Let D ∆ = { x 2 S 2 ; 9 f 2 F 2 ( f • x = x ) } . (ii) Let G 1 , G 2 , G 3 , G 4 be as previously. (iii) The following is then an equivalence relation on S 2 \ D : x ⇠ y i ff there is a f 2 F 2 such that f � x = y . (iv) Let X be a selector. So S 2 \ D = U x 2 X F 2 x .

  32. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Hausdor ff Paradox ( AC ) Hausdor ff Paradox: There is a countable set D such that F 2 acts on S 2 \ D paradoxically with 4 pieces. (i) Let D ∆ = { x 2 S 2 ; 9 f 2 F 2 ( f • x = x ) } . (ii) Let G 1 , G 2 , G 3 , G 4 be as previously. (iii) The following is then an equivalence relation on S 2 \ D : x ⇠ y i ff there is a f 2 F 2 such that f � x = y . (iv) Let X be a selector. So S 2 \ D = U x 2 X F 2 x . (v) But F 2 x = ( G 1 x ] aG 2 x ).

  33. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Hausdor ff Paradox ( AC ) Hausdor ff Paradox: There is a countable set D such that F 2 acts on S 2 \ D paradoxically with 4 pieces. (i) Let D ∆ = { x 2 S 2 ; 9 f 2 F 2 ( f • x = x ) } . (ii) Let G 1 , G 2 , G 3 , G 4 be as previously. (iii) The following is then an equivalence relation on S 2 \ D : x ⇠ y i ff there is a f 2 F 2 such that f � x = y . (iv) Let X be a selector. So S 2 \ D = U x 2 X F 2 x . (v) But F 2 x = ( G 1 x ] aG 2 x ). ∆ (vi) Then Ω i = U x 2 X G i x for i 2 { 1 , 2 , 3 , 4 } do the job.

  34. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Hausdor ff Paradox ( AC ) Hausdor ff Paradox: There is a countable set D such that F 2 acts on S 2 \ D paradoxically with 4 pieces. (i) Let D ∆ = { x 2 S 2 ; 9 f 2 F 2 ( f • x = x ) } . (ii) Let G 1 , G 2 , G 3 , G 4 be as previously. (iii) The following is then an equivalence relation on S 2 \ D : x ⇠ y i ff there is a f 2 F 2 such that f � x = y . (iv) Let X be a selector. So S 2 \ D = U x 2 X F 2 x . (v) But F 2 x = ( G 1 x ] aG 2 x ). ∆ (vi) Then Ω i = U x 2 X G i x for i 2 { 1 , 2 , 3 , 4 } do the job. (vii) Hence, S 2 \ D = Ω 1 ] a Ω 2 = Ω 3 ] b Ω 4 .

  35. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion S 2 s S 2 \ D

  36. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion S 2 s S 2 \ D (i) A higher dimensional Hilbert Hotel trick.

  37. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion S 2 s S 2 \ D (i) A higher dimensional Hilbert Hotel trick. (ii) Fix a line l through the origin not intersecting D . (a) Only countably many angles ✓ such that for some n > 0 ⇢ n l, θ ( D ) \ D 6 = ; .

  38. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion S 2 s S 2 \ D (i) A higher dimensional Hilbert Hotel trick. (ii) Fix a line l through the origin not intersecting D . (a) Only countably many angles ✓ such that for some n > 0 ⇢ n l, θ ( D ) \ D 6 = ; . (b) Choose some other angle. Let R be the corresponding rotation.

  39. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion S 2 s S 2 \ D (i) A higher dimensional Hilbert Hotel trick. (ii) Fix a line l through the origin not intersecting D . (a) Only countably many angles ✓ such that for some n > 0 ⇢ n l, θ ( D ) \ D 6 = ; . (b) Choose some other angle. Let R be the corresponding rotation. (c) Then any two elements of D are in separate R -orbits.

  40. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion S 2 s S 2 \ D (i) A higher dimensional Hilbert Hotel trick. (ii) Fix a line l through the origin not intersecting D . (a) Only countably many angles ✓ such that for some n > 0 ⇢ n l, θ ( D ) \ D 6 = ; . (b) Choose some other angle. Let R be the corresponding rotation. (c) Then any two elements of D are in separate R -orbits. (d) That is, R i D \ R j D = ; whenever i 6 = j .

  41. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion S 2 s S 2 \ D (i) A higher dimensional Hilbert Hotel trick. (ii) Fix a line l through the origin not intersecting D . (a) Only countably many angles ✓ such that for some n > 0 ⇢ n l, θ ( D ) \ D 6 = ; . (b) Choose some other angle. Let R be the corresponding rotation. (c) Then any two elements of D are in separate R -orbits. (d) That is, R i D \ R j D = ; whenever i 6 = j . = S 2 \ Σ 2 . ∆ ∆ = D [ RD [ R 2 D · · · ; and Σ 1 (iii) Then, Σ 2

  42. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion S 2 s S 2 \ D (i) A higher dimensional Hilbert Hotel trick. (ii) Fix a line l through the origin not intersecting D . (a) Only countably many angles ✓ such that for some n > 0 ⇢ n l, θ ( D ) \ D 6 = ; . (b) Choose some other angle. Let R be the corresponding rotation. (c) Then any two elements of D are in separate R -orbits. (d) That is, R i D \ R j D = ; whenever i 6 = j . = S 2 \ Σ 2 . ∆ ∆ = D [ RD [ R 2 D · · · ; and Σ 1 (iii) Then, Σ 2 (iv) It can be verified that S 2 = Σ 1 ] Σ 2 and S 2 \ D = Σ 1 ] R Σ 2 .

  43. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Recap Things we’ve done so far:

  44. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Recap Things we’ve done so far: (i) S 1 s ( S 1 \ { ⇤ } ) (2 pieces).

  45. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Recap Things we’ve done so far: (i) S 1 s ( S 1 \ { ⇤ } ) (2 pieces). (ii) ( S 2 \ D ) s ( S 2 \ D ) ] ( S 2 \ D ) (4 pieces).

  46. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Recap Things we’ve done so far: (i) S 1 s ( S 1 \ { ⇤ } ) (2 pieces). (ii) ( S 2 \ D ) s ( S 2 \ D ) ] ( S 2 \ D ) (4 pieces). (iii) S 2 s ( S 2 \ D ) (2 pieces).

  47. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Recap Things we’ve done so far: (i) S 1 s ( S 1 \ { ⇤ } ) (2 pieces). (ii) ( S 2 \ D ) s ( S 2 \ D ) ] ( S 2 \ D ) (4 pieces). (iii) S 2 s ( S 2 \ D ) (2 pieces). What this gives us:

  48. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Recap Things we’ve done so far: (i) S 1 s ( S 1 \ { ⇤ } ) (2 pieces). (ii) ( S 2 \ D ) s ( S 2 \ D ) ] ( S 2 \ D ) (4 pieces). (iii) S 2 s ( S 2 \ D ) (2 pieces). What this gives us: S 2 s ( S 2 \ D ) s ( S 2 \ D ) ] ( S 2 \ D ) s S 2 ] S 2 .

  49. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Recap Things we’ve done so far: (i) S 1 s ( S 1 \ { ⇤ } ) (2 pieces). (ii) ( S 2 \ D ) s ( S 2 \ D ) ] ( S 2 \ D ) (4 pieces). (iii) S 2 s ( S 2 \ D ) (2 pieces). What this gives us: S 2 s ( S 2 \ D ) s ( S 2 \ D ) ] ( S 2 \ D ) s S 2 ] S 2 . So S 2 s S 2 ] S 2 .

  50. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Recap Things we’ve done so far: (i) S 1 s ( S 1 \ { ⇤ } ) (2 pieces). (ii) ( S 2 \ D ) s ( S 2 \ D ) ] ( S 2 \ D ) (4 pieces). (iii) S 2 s ( S 2 \ D ) (2 pieces). What this gives us: S 2 s ( S 2 \ D ) s ( S 2 \ D ) ] ( S 2 \ D ) s S 2 ] S 2 . So S 2 s S 2 ] S 2 . And the number of pieces we’ve needed is 8.

  51. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion There or nearabouts Things we’ve done, and things we can do with them.

  52. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion There or nearabouts Things we’ve done, and things we can do with them. We have S 2 s S 2 ] S 2 .

  53. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion There or nearabouts Things we’ve done, and things we can do with them. We have S 2 s S 2 ] S 2 . But B 3 \ { 0 } = S 2 ⇥ (0 , 1]!

  54. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion There or nearabouts Things we’ve done, and things we can do with them. We have S 2 s S 2 ] S 2 . But B 3 \ { 0 } = S 2 ⇥ (0 , 1]! So B 3 \ { 0 } s ( B 3 \ { 0 } ) ] ( B 3 \ { 0 } ).

  55. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion There or nearabouts Things we’ve done, and things we can do with them. We have S 2 s S 2 ] S 2 . But B 3 \ { 0 } = S 2 ⇥ (0 , 1]! So B 3 \ { 0 } s ( B 3 \ { 0 } ) ] ( B 3 \ { 0 } ). So B 3 \ { 0 } s B 3 would do the job.

  56. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion There or nearabouts Things we’ve done, and things we can do with them. We have S 2 s S 2 ] S 2 . But B 3 \ { 0 } = S 2 ⇥ (0 , 1]! So B 3 \ { 0 } s ( B 3 \ { 0 } ) ] ( B 3 \ { 0 } ). So B 3 \ { 0 } s B 3 would do the job. But this easily follows from S 1 \ { ⇤ } s S 1 .

  57. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The Banach-Tarski “Paradox” B 3 is G 3 -paradoxical.

  58. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The Banach-Tarski “Paradox” B 3 is G 3 -paradoxical. This also shows that there is no finitely additive total measure on R n for n � 3.

  59. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The Banach-Tarski “Paradox” B 3 is G 3 -paradoxical. This also shows that there is no finitely additive total measure on R n for n � 3. Stronger version: Let A and B be bounded sets in R n , n � 3 with non-empty interior. Then A s G 3 B .

  60. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The Banach-Tarski “Paradox” B 3 is G 3 -paradoxical. This also shows that there is no finitely additive total measure on R n for n � 3. Stronger version: Let A and B be bounded sets in R n , n � 3 with non-empty interior. Then A s G 3 B . Something weaker than AC su ffi ces, G¨ odel’s Completeness Theorem. Or the Compactness Theorem for First-order Logic. Both of these are equivalent to what is called the Ultrafilter Lemma (Henkin).

  61. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The Banach-Tarski “Paradox” B 3 is G 3 -paradoxical. This also shows that there is no finitely additive total measure on R n for n � 3. Stronger version: Let A and B be bounded sets in R n , n � 3 with non-empty interior. Then A s G 3 B . Something weaker than AC su ffi ces, G¨ odel’s Completeness Theorem. Or the Compactness Theorem for First-order Logic. Both of these are equivalent to what is called the Ultrafilter Lemma (Henkin). Whoops?

  62. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The Banach-Tarski “Paradox” B 3 is G 3 -paradoxical. This also shows that there is no finitely additive total measure on R n for n � 3. Stronger version: Let A and B be bounded sets in R n , n � 3 with non-empty interior. Then A s G 3 B . Something weaker than AC su ffi ces, G¨ odel’s Completeness Theorem. Or the Compactness Theorem for First-order Logic. Both of these are equivalent to what is called the Ultrafilter Lemma (Henkin). Whoops? Aside: The Downward L¨ owenheim-Skolem theorem is equivalent to AC (Tarski).

  63. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Notions of size

  64. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Notions of size (i) f : A ! B , injective, then say | A |  | B | .

  65. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Notions of size (i) f : A ! B , injective, then say | A |  | B | . (ii) Clearly, the notion is transitive and reflexive.

  66. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Notions of size (i) f : A ! B , injective, then say | A |  | B | . (ii) Clearly, the notion is transitive and reflexive. (iii) AC is equivalent to: A and B are sets, then either | A | < | B | or | B | > | A | or | A | = | B | .

  67. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Notions of size (i) f : A ! B , injective, then say | A |  | B | . (ii) Clearly, the notion is transitive and reflexive. (iii) AC is equivalent to: A and B are sets, then either | A | < | B | or | B | > | A | or | A | = | B | . (iv) That is, this notion of size is a total order.

  68. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion Notions of size (i) f : A ! B , injective, then say | A |  | B | . (ii) Clearly, the notion is transitive and reflexive. (iii) AC is equivalent to: A and B are sets, then either | A | < | B | or | B | > | A | or | A | = | B | . (iv) That is, this notion of size is a total order. (v) Aside: AC says that R has a definite size in this sense, but this size can be almost anything by Cohen.

  69. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The horrors in earnest

  70. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The horrors in earnest There can be infinite sets which have no subset of size @ 0 (Cohen).

  71. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The horrors in earnest There can be infinite sets which have no subset of size @ 0 (Cohen). @ 0 is no longer the smallest infinite cardinality, since there may be infinite sets which are incomparable with @ 0 .

  72. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The horrors in earnest There can be infinite sets which have no subset of size @ 0 (Cohen). @ 0 is no longer the smallest infinite cardinality, since there may be infinite sets which are incomparable with @ 0 . In fact, all partial orders can be embedded as cardinalities.

  73. Introduction Vitali Construction Banach-Tarski Horrors without AC Conclusion The horrors in earnest There can be infinite sets which have no subset of size @ 0 (Cohen). @ 0 is no longer the smallest infinite cardinality, since there may be infinite sets which are incomparable with @ 0 . In fact, all partial orders can be embedded as cardinalities. Let I be a set and for every i 2 I , X i a non-empty set. Then without AC , Q i 2 I X i may be empty.

Recommend

BSS RAMs with \ nu-Oracle and the Axiom of Choice Christine Ganer Hamburg 2016 BSS RAMs

BSS RAMs with \ nu-Oracle and the Axiom of Choice Christine Ganer Hamburg 2016 BSS RAMs

BSS RAMs with \ nu-Oracle and the Axiom of Choice Christine Ganer Hamburg 2016 BSS RAMs with \ nu-Oracle and the Axiom of Choice (History and Outline) Stephen C. Kleene Recursion Theory based on recursion and -operator Yiannis

478 views • 34 slides
Computational social choice Lirong Xia Todays schedule Computational social choice: the

Computational social choice Lirong Xia Todays schedule Computational social choice: the

Computational social choice Lirong Xia Todays schedule Computational social choice: the easy-to- compute axiom voting rules that can be computed in P satisfies the axiom Kemeny: a full proof of NP-hardness IP formulation

569 views • 26 slides
Martins Axiom and Choice Principles Eleftherios Tachtsis Department of Mathematics University

Martins Axiom and Choice Principles Eleftherios Tachtsis Department of Mathematics University

Martins Axiom and Choice Principles Eleftherios Tachtsis Department of Mathematics University of the Aegean Karlovassi, Samos, GREECE SWIP Set Theory Workshop in Pisa June 13, 2017 Department of Mathematics E. Tachtsis Martins Axiom

1.02k views • 74 slides
Ultrafilters, Closure operators and the Axiom of Choice Gonc alo Gutierres

Ultrafilters, Closure operators and the Axiom of Choice Gonc alo Gutierres

Ultrafilters, Closure operators and the Axiom of Choice Gonc alo Gutierres CMUC/Universidade de Coimbra It is well known that, in a topological space, the open sets can be characterized using filter convergence. In ZF , we cannot replace

512 views • 36 slides
g -first countable spaces and the Axiom of Choice Gon calo Gutierres CMUC/Universidade de

g -first countable spaces and the Axiom of Choice Gon calo Gutierres CMUC/Universidade de

g -first countable spaces and the Axiom of Choice Gon calo Gutierres CMUC/Universidade de Coimbra In 1966, A. Arhangelskii introduced the notion of weak (local) base of a topological space, and consequently defined what are a g -first

699 views • 50 slides
The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle/ZF Lawrence C. Paulson

The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle/ZF Lawrence C. Paulson

The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle/ZF Lawrence C. Paulson Computer Laboratory Why Do Proofs By Machine? Too many been done already! Gdels incompleteness theorem ( Shankar ) thousands of

327 views • 19 slides
Partition Relations for Linear Orders without the Axiom of Choice 03E02, 03E60, 05C63 Thilo

Partition Relations for Linear Orders without the Axiom of Choice 03E02, 03E60, 05C63 Thilo

Choicelessly Partitioning Quadruples Partition Relations for Linear Orders without the Axiom of Choice 03E02, 03E60, 05C63 Thilo Weinert Department of Mathematics, Ben-Gurion-University of the Negev, Isra el Joint work with Philipp L

444 views • 21 slides
Axiom of Choice, Maximal Independent Sets, Argumentation and Dialogue Games Christof Spanring

Axiom of Choice, Maximal Independent Sets, Argumentation and Dialogue Games Christof Spanring

Axiom of Choice, Maximal Independent Sets, Argumentation and Dialogue Games Christof Spanring Department of Computer Science, University of Liverpool, UK Institute of Information Systems, Vienna University of Technology, Austria PhDs Tea Talk

696 views • 36 slides
Axiom of Choice, Maximal Independent Sets, Argumentation and Dialogue Games Christof Spanring

Axiom of Choice, Maximal Independent Sets, Argumentation and Dialogue Games Christof Spanring

Axiom of Choice, Maximal Independent Sets, Argumentation and Dialogue Games Christof Spanring Department of Computer Science, University of Liverpool, UK Institute of Information Systems, Vienna University of Technology, Austria London,

505 views • 14 slides
Mechanizing Set Theory: Cardinal Arithmetic and the Axiom of Choice Krzysztof Gra bczewski ,

Mechanizing Set Theory: Cardinal Arithmetic and the Axiom of Choice Krzysztof Gra bczewski ,

Gra bczewski &amp; Paulson Mechanizing Set Theory 1 Mechanizing Set Theory: Cardinal Arithmetic and the Axiom of Choice Krzysztof Gra bczewski , Copernicus University, Torun, Poland Lawrence C Paulson , Computer Laboratory, Cambridge

223 views • 10 slides
Mathematics for Computing Equational Logic as Calculus of Boolean Expressions COMP SCI 1FC3

Mathematics for Computing Equational Logic as Calculus of Boolean Expressions COMP SCI 1FC3

Axioms Fill in the Blanks! Anything Wrong? false (3.1) Axiom, Associativity of : = (3.15) p p false (3.2) Axiom, Symmetry of : p p (3.3) Axiom, Identity of : = (3.11) p q p q

487 views • 4 slides
Axiom Holdings Inc. Business Presentation 2016 Private and Confidential About Axiom Holdings

Axiom Holdings Inc. Business Presentation 2016 Private and Confidential About Axiom Holdings

Axiom Holdings Inc. Business Presentation 2016 Private and Confidential About Axiom Holdings Ltd. COMPANY OVERVIEW KEY FIGURES Independent power producer and real estate Share Price* US$ 2.34 developer that develops, builds, owns and

269 views • 23 slides
The Time Slice Axiom in Perturbative Quantum Field Theory Klaus Fredenhagen 1 II. Institut f

The Time Slice Axiom in Perturbative Quantum Field Theory Klaus Fredenhagen 1 II. Institut f

Introduction Time slice axiom for a free scalar field The algebra of Wick polynomials Time slice axiom for Wick polynomials Time slice axiom for interacting field theories Conclusions and Outlook The Time Slice Axiom in Perturbative Quantum

228 views • 18 slides
Choice Theory A Synopsis 14.123 Microeconomic Theory III Muhamet Yildiz Road map 1. Basic

Choice Theory A Synopsis 14.123 Microeconomic Theory III Muhamet Yildiz Road map 1. Basic

2/2/2015 Choice Theory A Synopsis 14.123 Microeconomic Theory III Muhamet Yildiz Road map 1. Basic Concepts: 1. Choice 2. Preference 3. Utility 2.Weak Axiom of Revealed Preferences 3. Preference as a representation of choice

339 views • 6 slides
Computational social choice Combinatorial voting Lirong Xia Feb 23, 2016 Last class: the

Computational social choice Combinatorial voting Lirong Xia Feb 23, 2016 Last class: the

Computational social choice Combinatorial voting Lirong Xia Feb 23, 2016 Last class: the easy-to- compute axiom We hope that the outcome of a social choice mechanism can be computed in p-time P: positional scoring rules, maximin,

550 views • 29 slides
Games in Descriptive Set Theory, or: its all fun and games until someone loses the axiom of

Games in Descriptive Set Theory, or: its all fun and games until someone loses the axiom of

Games in Descriptive Set Theory, or: its all fun and games until someone loses the axiom of choice Hugo Nobrega Cool Logic 22 May 2015 Descriptive set theory and the Baire space Presentation outline [0] 1 Descriptive set theory and the

1.01k views • 52 slides
Little Shop of Performance Horrors Brendan Gregg Staff Engineer Sun Microsystems, Fishworks

Little Shop of Performance Horrors Brendan Gregg Staff Engineer Sun Microsystems, Fishworks

USE IMPR MPROVE EV EVANGEL ELIZE Little Shop of Performance Horrors Brendan Gregg Staff Engineer Sun Microsystems, Fishworks FROSUG 2009 USE IMPR MPROVE EV EVANGEL ELIZE Performance Horrors I usually give talks on: how to

561 views • 34 slides
Toward dependent choice: a classical sequent calculus with dependent types Hugo Herbelin 1 ,

Toward dependent choice: a classical sequent calculus with dependent types Hugo Herbelin 1 ,

Curry-Howard Axiom of Dependent Choice Sequent calculus State of the art Toward dependent choice: a classical sequent calculus with dependent types Hugo Herbelin 1 , Etienne Miquey 1,2 1 Team r 2 (INRIA), PPS, Universit e

404 views • 24 slides
Site Audits and Support Horrors Overcoming the Bad, the Ugly, and the Atrocious when Taking over

Site Audits and Support Horrors Overcoming the Bad, the Ugly, and the Atrocious when Taking over

Site Audits and Support Horrors Overcoming the Bad, the Ugly, and the Atrocious when Taking over Sites Who am I? Doug Dobrzynski Drupal Developer at Promet Source Managed Web Help Desk Made websites since fifth grade

456 views • 19 slides
Formal Models of Narrative Benedikt L owe Logic, Language and Computation . 19 September 2011

Formal Models of Narrative Benedikt L owe Logic, Language and Computation . 19 September 2011

Formal Models of Narrative Benedikt L owe Logic, Language and Computation . 19 September 2011 Set Theory. Infinite games. Set theory without the axiom of choice (in particular with the axiom of determinacy). Set theory of the reals. PhD

990 views • 81 slides
From AXIOM down to IMP Dominique Duval with the help of Jean-Guillaume Dumas, Burak Ekici,

From AXIOM down to IMP Dominique Duval with the help of Jean-Guillaume Dumas, Burak Ekici,

From AXIOM down to IMP Dominique Duval with the help of Jean-Guillaume Dumas, Burak Ekici, Alexis Laouar, Damien Pous, Jean-Claude Reynaud Algebraic Algorithms and Applications Pisa 31 March 2017 Happy birthday Patrizia! AXIOM

763 views • 35 slides
The constructible universe of ZFA Matteo Viale KGRC University of Vienna 1 WHY THE FOUNDATION

The constructible universe of ZFA Matteo Viale KGRC University of Vienna 1 WHY THE FOUNDATION

The constructible universe of ZFA Matteo Viale KGRC University of Vienna 1 WHY THE FOUNDATION AXIOM? Axiom 1. is a well founded relation. Kunens textbook justifies the axiom of foun- dation on the ground that it is an essential

526 views • 19 slides
The Axiom of Infinity and The Natural Numbers Bernd Schr oder logo1 Bernd Schr oder

The Axiom of Infinity and The Natural Numbers Bernd Schr oder logo1 Bernd Schr oder

Axiom of Infinity Natural Numbers Axiomatic Systems The Axiom of Infinity and The Natural Numbers Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science The Axiom of Infinity and The

968 views • 93 slides
On Voevodskys Univalence Axiom 6th July, 2011, Edinburgh Third European Set Theory Conference

On Voevodskys Univalence Axiom 6th July, 2011, Edinburgh Third European Set Theory Conference

On Voevodskys Univalence Axiom 6th July, 2011, Edinburgh Third European Set Theory Conference . Peter Aczel petera@cs.man.ac.uk Manchester University On Voevodskys Univalence Axiom p.1/25 Plan of Lecture I(1): Introduction II(3):

550 views • 25 slides

More recommend