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Vladimir_Nesov comments on 5 Axioms of Decision Making - Less Wrong
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Let X() be a consistent probability assignment (function from statement to probability number).
Let Y() be a probability assignment including: Y(2+2=5) = X(Y is consistent), and otherwise Y(z)=X(z)
What's X(Y is consistent)?
If X(Y is consistent)=1, then Y(2+2=5)=1, and Y is blatantly inconsistent, and so is X is inconsistent according to basic laws of mathematics.
If X(Y is consistent)=0, then Y(2+2=5)=0=X(2+2=5), and by definition X=Y, so X is inconsistent according to itself.
What does it mean for this function to be "consistent"? What kinds of statements do you allow?
If "probability assignment" is a mapping from statements (or Goedel numbers) to the real interval [0,1], it's not a given that Y, being a "probability assignment", is definable, so that you can refer to it in the statement "Y is consistent" above.