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The Katowice Problem

Kp Hart
Kp Hart

This document discusses the Katowice Problem, which asks whether two sets X and Y must have the same cardinality if the power sets modulo finite sets P(X)/fin and P(Y)/fin are isomorphic as rings. It presents two results showing that P(1)/fin and P(2)/fin are not isomorphic, leaving open the question of whether P(0)/fin and P(1)/fin could ever be isomorphic. It then shows that if P(0)/fin and P(1)/fin were isomorphic, it would imply the existence of a non-trivial automorphism of P(0)/fin.

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Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism The Katowice Problem T卒a sc卒eil卒脹n agam K. P. Hart Faculty EEMCS TU Delft Amsterdam, 5 October, 2016: 16:00 16:45 K. P. Hart The Katowice Problem 1 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Easy exercise one Exercise Let X and Y be two sets and f : X Y a bijection. Make a bijection between P(X) and P(Y ). Solution: A f [A] does the trick. K. P. Hart The Katowice Problem 3 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism The hard exercise Exercise Let X and Y be two sets and F : P(X) P(Y ) a bijection. Make a bijection between X and Y . Solution: cant be done. Really!? K. P. Hart The Katowice Problem 4 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism How can that be? But, if we have sets with the same number of subsets then they have the same number of points. For if 2m = 2n then m = n. True, for natural numbers m and n. But that was not (really) the question. The proof for m and n does not produce a bijection. It does not use bijections at all. K. P. Hart The Katowice Problem 5 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism On to in鍖nity We have a scale to measure sets by: 0, 1, 2, 3, . . . 0 refers to countable. 1 refers to the next in鍖nity and so on . . . I teach this stu鍖 every Friday afternoon in SP 904 (C1.112) K. P. Hart The Katowice Problem 6 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism On to in鍖nity Remember Cantors Continuum Hypothesis? It says: 20 = 1: the number of subsets of N is the smallest possible uncountable in鍖nity. When Cohen showed that the Continuum Hypothesis is unprovable, his method actually showed that 20 = 21 = 2 does not lead to contradictions. This is a situation with a bijection between P(X) and P(Y ) but no bijection between X and Y . K. P. Hart The Katowice Problem 7 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Easy exercise two Exercise Let X and Y be two sets and F : P(X) P(Y ) a bijection that is also an isomorphism for the relation . Make a bijection between X and Y . Solution: if x X then {x} is an atom (nothing between it and ), hence so is F {x} . But then F {x} = {y} for some (unique) y Y . Theres your bijection. K. P. Hart The Katowice Problem 8 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Some algebra We can consider P(X) as a group, or a ring. Addition: symmetric di鍖erence Multiplication: intersection A -isomorphism is also a ring-isomorphism. K. P. Hart The Katowice Problem 10 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism There is a nice ideal in the ring P(X): the ideal, 鍖n, of 鍖nite sets. You can see where this is going . . . K. P. Hart The Katowice Problem 11 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism The problem The Katowice Problem Let X and Y be sets and assume P(X)/鍖n and P(Y )/鍖n are ring-isomorphic. Is there a bijection between X and Y ? Equivalently . . . If the Banach algebras (X)/c0 and (Y )/c0 are isomorphic must there be a bijection between X and Y ? Equivalently . . . K. P. Hart The Katowice Problem 12 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism The problem . . . the original version The Katowice Problem If X and Y are homeomorphic must X and Y have the same cardinality. Our sets carry the discrete topology and X = 硫X X, where 硫X is the Cech-Stone compacti鍖cation. Actually: X is also the structure space of (X)/c0 and the maximal-ideal space of P(X)/鍖n. So it all hangs together. K. P. Hart The Katowice Problem 13 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Two results Theorem (Frankiewicz 1977) The minimum cardinal 虜 (if any) such that P(虜)/鍖n is isomorphic to P(了)/鍖n for some 了 > 虜 must be 0. Theorem (Balcar and Frankiewicz 1978) P(1)/鍖n and P(2)/鍖n are not isomorphic. K. P. Hart The Katowice Problem 14 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Consequences Corollary If 1 虜 < 了 then P(虜)/鍖n and P(了)/鍖n are not isomorphic, and if 2 了 then P(0)/鍖n and P(了)/鍖n are not isomorphic. So we are left with Question Are P(0)/鍖n and P(1)/鍖n ever isomorphic? K. P. Hart The Katowice Problem 15 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Why ever? The Continuum Hypothesis implies that P(0)/鍖n and P(1)/鍖n are not isomorphic? So, we can not prove that they are isomorphic. But, can we prove they they are not isomorphic? The are they ever translates to: is there a model of Set Theory where P(0)/鍖n and P(1)/鍖n are isomorphic? K. P. Hart The Katowice Problem 16 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Consequences We want P(0)/鍖n and P(1)/鍖n are isomorphic to be false. We have many consequences. But not yet 0 = 1. Heres a nice one . . . K. P. Hart The Katowice Problem 18 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism An automorphism of P(0)/鍖n Work with the set D = Z 1 so we assume 粒 : P(D)/鍖n P(0)/鍖n is an isomorphism. De鍖ne 裡 : D D by 裡(n, 留) = n + 1, 留 . Then = 粒 裡 粒1 is an automorphism of P(0)/鍖n. In fact, is non-trivial, i.e., there is no bijection : a b between co鍖nite sets such that (x) = [x a] for all subsets x of K. P. Hart The Katowice Problem 19 / 20
Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Light reading Website: fa.its.tudelft.nl/~hart K. P. Hart, De ContinuumHypothese, Nieuw Archief voor Wiskunde, 10, nummer 1, (2009), 3339 D. Chodounsky, A. Dow, K. P. Hart and H. de Vries The Katowice problem and autohomeomorphisms of , (arXiv e-print 1307.3930) K. P. Hart The Katowice Problem 20 / 20

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The Katowice Problem

  • 1. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism The Katowice Problem T卒a sc卒eil卒脹n agam K. P. Hart Faculty EEMCS TU Delft Amsterdam, 5 October, 2016: 16:00 16:45 K. P. Hart The Katowice Problem 1 / 20
  • 2. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Easy exercise one Exercise Let X and Y be two sets and f : X Y a bijection. Make a bijection between P(X) and P(Y ). Solution: A f [A] does the trick. K. P. Hart The Katowice Problem 3 / 20
  • 3. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism The hard exercise Exercise Let X and Y be two sets and F : P(X) P(Y ) a bijection. Make a bijection between X and Y . Solution: cant be done. Really!? K. P. Hart The Katowice Problem 4 / 20
  • 4. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism How can that be? But, if we have sets with the same number of subsets then they have the same number of points. For if 2m = 2n then m = n. True, for natural numbers m and n. But that was not (really) the question. The proof for m and n does not produce a bijection. It does not use bijections at all. K. P. Hart The Katowice Problem 5 / 20
  • 5. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism On to in鍖nity We have a scale to measure sets by: 0, 1, 2, 3, . . . 0 refers to countable. 1 refers to the next in鍖nity and so on . . . I teach this stu鍖 every Friday afternoon in SP 904 (C1.112) K. P. Hart The Katowice Problem 6 / 20
  • 6. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism On to in鍖nity Remember Cantors Continuum Hypothesis? It says: 20 = 1: the number of subsets of N is the smallest possible uncountable in鍖nity. When Cohen showed that the Continuum Hypothesis is unprovable, his method actually showed that 20 = 21 = 2 does not lead to contradictions. This is a situation with a bijection between P(X) and P(Y ) but no bijection between X and Y . K. P. Hart The Katowice Problem 7 / 20
  • 7. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Easy exercise two Exercise Let X and Y be two sets and F : P(X) P(Y ) a bijection that is also an isomorphism for the relation . Make a bijection between X and Y . Solution: if x X then {x} is an atom (nothing between it and ), hence so is F {x} . But then F {x} = {y} for some (unique) y Y . Theres your bijection. K. P. Hart The Katowice Problem 8 / 20
  • 8. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Some algebra We can consider P(X) as a group, or a ring. Addition: symmetric di鍖erence Multiplication: intersection A -isomorphism is also a ring-isomorphism. K. P. Hart The Katowice Problem 10 / 20
  • 9. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism There is a nice ideal in the ring P(X): the ideal, 鍖n, of 鍖nite sets. You can see where this is going . . . K. P. Hart The Katowice Problem 11 / 20
  • 10. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism The problem The Katowice Problem Let X and Y be sets and assume P(X)/鍖n and P(Y )/鍖n are ring-isomorphic. Is there a bijection between X and Y ? Equivalently . . . If the Banach algebras (X)/c0 and (Y )/c0 are isomorphic must there be a bijection between X and Y ? Equivalently . . . K. P. Hart The Katowice Problem 12 / 20
  • 11. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism The problem . . . the original version The Katowice Problem If X and Y are homeomorphic must X and Y have the same cardinality. Our sets carry the discrete topology and X = 硫X X, where 硫X is the Cech-Stone compacti鍖cation. Actually: X is also the structure space of (X)/c0 and the maximal-ideal space of P(X)/鍖n. So it all hangs together. K. P. Hart The Katowice Problem 13 / 20
  • 12. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Two results Theorem (Frankiewicz 1977) The minimum cardinal 虜 (if any) such that P(虜)/鍖n is isomorphic to P(了)/鍖n for some 了 > 虜 must be 0. Theorem (Balcar and Frankiewicz 1978) P(1)/鍖n and P(2)/鍖n are not isomorphic. K. P. Hart The Katowice Problem 14 / 20
  • 13. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Consequences Corollary If 1 虜 < 了 then P(虜)/鍖n and P(了)/鍖n are not isomorphic, and if 2 了 then P(0)/鍖n and P(了)/鍖n are not isomorphic. So we are left with Question Are P(0)/鍖n and P(1)/鍖n ever isomorphic? K. P. Hart The Katowice Problem 15 / 20
  • 14. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Why ever? The Continuum Hypothesis implies that P(0)/鍖n and P(1)/鍖n are not isomorphic? So, we can not prove that they are isomorphic. But, can we prove they they are not isomorphic? The are they ever translates to: is there a model of Set Theory where P(0)/鍖n and P(1)/鍖n are isomorphic? K. P. Hart The Katowice Problem 16 / 20
  • 15. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Consequences We want P(0)/鍖n and P(1)/鍖n are isomorphic to be false. We have many consequences. But not yet 0 = 1. Heres a nice one . . . K. P. Hart The Katowice Problem 18 / 20
  • 16. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism An automorphism of P(0)/鍖n Work with the set D = Z 1 so we assume 粒 : P(D)/鍖n P(0)/鍖n is an isomorphism. De鍖ne 裡 : D D by 裡(n, 留) = n + 1, 留 . Then = 粒 裡 粒1 is an automorphism of P(0)/鍖n. In fact, is non-trivial, i.e., there is no bijection : a b between co鍖nite sets such that (x) = [x a] for all subsets x of K. P. Hart The Katowice Problem 19 / 20
  • 17. Two easy exercises and a hard one The Katowice Problem A non-trivial automorphism Light reading Website: fa.its.tudelft.nl/~hart K. P. Hart, De ContinuumHypothese, Nieuw Archief voor Wiskunde, 10, nummer 1, (2009), 3339 D. Chodounsky, A. Dow, K. P. Hart and H. de Vries The Katowice problem and autohomeomorphisms of , (arXiv e-print 1307.3930) K. P. Hart The Katowice Problem 20 / 20