```Date: Dec 15, 2012 1:34 AM
Author: ross.finlayson@gmail.com
Subject: Re: On the infinite binary Tree

On Dec 14, 10:17 am, forbisga...@gmail.com wrote:> On Friday, December 14, 2012 6:30:11 AM UTC-8, Charlie-Boo wrote:> > On Dec 12, 5:26 am, Zuhair <zaljo...@gmail.com> wrote:>> > > Lets take the third degree binary tree>> > >      0>> > >    /    \>> > >   0     1>> > >  / \    / \>> > > 0  1 0  1>> > > Now this has 7 nodes BUT 8 paths, those are>> > > 0-0>> > > 0-1>> > > 1-0>> > > 1-1>> > > 0-0-0>> > > 0-0-1>> > > 0-1-0>> > > 0-1-1>> > Only 2 start with 1 and 6 start with 0.  Shouldn't it be 50-50?  If>> > you start with 0 then all start with 0.  You seem to be inconsistent>> > in your manipulations of the tree.>> four of his paths start at second level nodes.>> > You need to diagonalize the binary tree rather than the list of real>> > numbers - and account for multiple trees representing the same>> > number.  Or just ignore the real number interpretation and deal with>> > binary strings without regard to numbers.>> > My many examples of diagonalization in different contexts is a model>> > to do just that.  It's always good to generalize.>> My concern is how one shows there are countably many paths through> the infinite nodes.  I can't move to the second path though infinity> until I've completed the first, that is unless an algorith is proposed> that lets me idenify which infinite paths I've taken without completely> taking them.  I might identify a path by its representation as a rational> number.  Unfortunately that leaves out the irrational numbers.In the binary tree, of nodes of finite distance from the root, ittakes all the nodes to represent the rationals, of the unitinterval.Now, consider diagramming the tree this way, in a breadth-firsttraversal, lay out each row in the y-axis at the corresponding x-coordinate of the row, with its y value being the rational value ofthe expansion.  Then, the maximum value of this point set at each x ismax(x) = 1- 1/2^x.  In the limit that's one.  Now, the paths areordered lexicographically, in the breadth first traversal, for eachthey are.  And, at each row, the difference and distance between pathsis constant.  Then, the infinite binary tree, would have that, in thelimit or asymptotically, there are are infinitely many paths,lexicographically ordered, in the unit interval, in their naturalordering as representations of reals, or rationals and irrationals, inthe unit interval.  These paths are dense in the unit interval.  And,for any node, there is the simple branch to the left or right thatrepresents a rational.Now this is all without some completed infinity or "at leastinfinitely many" number of rows or here columns.  Yet, as a function,modeled by each row, the function so standardly modeled has range[0,1], here without simply declaring that at the point at infinity therange is the continuous line segment.  Via Cauchy, these expansion areall the real numbers of the unit interval, and then some, with dualrepresentation of real in their binary expansions.Then, one way to look at the rows of the binary tree is as a family offunctions BT_p = b^p for b as the domain from zero to 2^p. Then,connecting the dots of each BT_p(n) to BT_p+1(2(n+1)) and BT_p(2(n+1)-1), or along those lines, the resulting diagram, has that theinfinite trees are rooted in those.So, there doesn't exist a breadth-first traversal of the infinitebinary tree.  Yet, that is a countable enumeration, of the nodes inbreadth first order, and the paths in their natural order, of theirvalues.There are infinitely many natural integers.Regards,Ross Finlayson
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