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[Computational Complexity] Voting on Mathematical Truths: The Axiom of Det.

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  • GASARCH
    One of the founders of Conservapedia (a conservative alternative to Wikipedia) said the following on The Colbert Report: There is an absolute truth. People
    Message 1 of 1 , Jan 4, 2010
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      One of the founders of Conservapedia (a conservative alternative to Wikipedia) said the following on The Colbert Report:
      There is an absolute truth. People don't vote on mathematical things like 2+2=4.
      Given the source this quote may be ironic. However, this post is not about Conservapedia or Wikipedia. Its about voting on mathematical truths.

      There is one kind of math where a vote might be appropriate. Some Set Theorists would like to resolve CH. We already know that this cannot be done in ZFC. So they want to add more axioms. What property should an axiom have? It should be obvious. It is unlikely that we will have new axioms of that type. How about that it be reasonable? Some set theorists think it is reasonable to remove FULL AC and add The Axiom of Determinacy (stated below). I want YOU to VOTE on if it is reasonable.

      Definition: Let A be a subset of {0,1}&omega. Let GA be the following game: player I picks b1 &isin {0,1}, then player II picks b2 &isin {0,1}, then player I picks b3 &isin {0,1}, etc. If the final sequence b1 , b2 , b3 ... is in A then I wins. If not then II wins.

      Definition: Let A be a subset of {0,1}&omega. A is determined if either player I or II has a winning strategy for GA.

      The Axiom of Determinacy (AD): For all sets A that are subsets of {0,1}&omega, A is determined.
      1. AD is known to be true for A a Borel Set (Donald Martin proofed that).
      2. AD contradicts the uncountable AC but implies the countable AC.
      3. AD implies that every subset of the plane is measurable.
      4. I don't think AD implies CH or not CH (are both AD + CH and AD + NOT(CH) known to have models?)
      5. AD has a Wikipedia entry. This should not be taken as a sign that it is true of false. It should not even be taken as a sign that its well known. It just means someone put up an entry.
      6. My wife thought AD was false in 1995 but true in 2005. I do not know what changed her mind. She is not a set theorist and had not thought about it in the intervening 10 years.
      7. All finite games are determined so this is taking something true of finite games and assuming it is true for infinite games.
      When you vote note that there is no right or wrong answer.

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      Posted By GASARCH to Computational Complexity at 1/04/2010 11:11:00 AM
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