Homogeneous tree
From Infogalactic: the planetary knowledge core
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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