Re: the binary tree

Are there values in the nodes? If so, what are they? If not, where do all the real numbers lie?

If you have an infinite binary tree, then every node can be labeled
with a finite binary sequence or a dyadic rational in Z[1/2] /\ [0,1).

The n-th level can be labeled with finite binary sequences from {0,1}^n.

Every path can be labeled with an infinite binary sequence
from {0,1}^N, which with denumerably many duplicates, maps
onto the reals.

If you extend this tree to the omega_0-th level, you've reached
a level with uncountable many nodes which can be labeled with the
infinite binary sequences of {0,1}^N.


.

Relevant Pages

  • Re: Sequence of {0,1} onto [0,1] in R
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  • Binary Tree and Pairs of Nodes
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  • Re: The Computable Reals (alpha version)
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  • Re: Answer to Dik T. Winter
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  • "Gaps" again (was: how to list all of the real numbers)
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