Re: New countable infiniity logic

From: tinyurl.com/uh3t (rem642b_at_Yahoo.Com)
Date: 11/21/04 

Date: Sun, 21 Nov 2004 12:36:15 -0800

> From: "|-|erc" <spam@fodder.abc>
> Say you have an *infinite* list of computable real numbers.

Do you mean a mapping from the positive (or non-negative) integers such
that the result of each mapping is a computer program which when run
would generate more and more digits of some particular real number,
such that if the program were let run forever it would eventually
produce any digit whatsoever of that particular real number?

If that's what you mean, it's impossible for your mapping to include
*all* computable real numbers, and simultaneously your mapping itself
to be computable.

> if you can't iterate through a tree data structure you never studied
> programming

If the tree structure you're using is binary, where each branch
determines the next bit in the binary representation of some real
number (or you can have 10-way branching to directly generate decimal
digits, etc., any finite base of number representation binary ternary
etc. works the same), then this tree traversal has nothing to do with
your original question.

Yes you can surely traverse such an infinite tree in the sense that you
can do a breadth-first search which gets the root node then the two
1-deep nodes then the four 2-deep nodes etc., eventually reaching any
desired node you want at any finite depth.

But real numbers aren't these nodes. Real numbers are the ends of the
infinitely long zigzaggy branches, which exist only in the sense of an
epsilon-delta limit definition, they are not actually part of your
tree, just a bunch of additional points derived from your tree by
feeding an infinitely-long zigzaggy branch into the epsilon-delta
definition of limit. Your tree-traversal will reach every finite prefix
of the binary (or other base) representation of every real number, but
will never reach even *one* of the actual real numbers, much less all
of them.

Relevant Pages

  • Re: New countable infiniity logic
    ... Do you mean a mapping from the positive integers such ... If the tree structure you're using is binary, ... Yes you can surely traverse such an infinite tree in the sense that you ... infinitely long zigzaggy branches, which exist only in the sense of an ...
    (sci.math)
  • Re: Galileos Paradox
    ... set theory. ... There are sets that are both infinite ... but without a bijection between them. ... the mapping: ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... mapping does or does not exist. ... It is not excluded "a priori" that a model of ZFC might not map '0' to ... Geometry is USED for physics. ... than your tree. ...
    (sci.math)
  • Re: Binary Tree and Pairs of Nodes
    ... Every node in your infinite binary tree occurs at the ... So you say you have a mapping of every path to a node. ... You do not map My path to 1, because whichEVER path YOU mapped to 1, ...
    (sci.logic)
  • Re: Epistemology 201: The Science of Science
    ... >> as rationals WITHOUT mangling the number system. ... The mapping has nothhing whatsoever to do ... > Bob Kolker ... the entire infinite set as a whole to another set. ...
    (sci.cognitive)
Loading