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Homogeneous tree

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In descriptive set theory, a tree over a product set is said to be homogeneous if there is a system of measures such that the following conditions hold:

  • is a countably-additive measure on .
  • The measures are in some sense compatible under restriction of sequences: if , then .
  • If is in the projection of , the ultrapower by is wellfounded.

An equivalent definition is produced when the final condition is replaced with the following:

  • There are such that if is in the projection of and , then there is such that . This condition can be thought of as a sort of countable completeness condition on the system of measures.

is said to be -homogeneous if each is -complete.

Homogeneous trees are involved in Martin and Steel's proof of projective determinacy.

References

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  • Martin, Donald A. and John R. Steel (Jan 1989). "A Proof of Projective Determinacy". Journal of the American Mathematical Society. 2 (1). Journal of the American Mathematical Society, Vol. 2, No. 1: 71–125. doi:10.2307/1990913. JSTOR 1990913.