Re: Is continuum completely filled up?

Virgil wrote:
In article <45a3ab84@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

Virgil wrote:
In article <45a260c3@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

Virgil wrote:
In article <45a184ae@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

0 e R
x e R -> 2^x e R ^ -(2^x) e R
Which allows omission of most of R.
No every xeR is a successor in this tree, even if infinitely far down the tree.
For what x in your successor generated R is 2^x = 3?
log2(3)

And how does that trace backwards to 0?
If you cannot trace it back to 0 then you cannot have it in your system, as according to your own rules, everything must trace back to 0.

For what x in your successor generated R is 2^x = p, for any odd prime p?
log2(p)

And how does that trace backwards to 0?
If you cannot trace it back to 0 then you cannot have it in your system, as according to your own rules, everything must trace back to 0.
Or for any positive integer power of any odd prime p?
etc
Or for any odd integers except -1 or 1 ?
etc

Not in any finite sequence. It suffices to say that every unique real produces a unique pair of reals, and that there are no cycles in the process. Granted, I have to get back to proving that this covers the reals, but intuitively, it does. :)
.

Relevant Pages

  • Re: Is continuum completely filled up?
    ... Tony Orlow wrote: ... And how does that trace backwards to 0? ... If you cannot trace it back to 0 then you cannot have it in your system, ... it does not do so at all, and your "R" is not the standard set of reals, ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... Tony Orlow wrote: ... No every xeR is a successor in this tree, ... If you cannot trace it back to 0 then you cannot have it in your system, ... Or for any positive integer power of any odd prime p? ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... > Tony Orlow wrote: ... > reals, so it can't possibly denumerate all of the uncountable reals. ... infinite bitstring to represent in that system, ... will require bit strings of infinite length. ...
    (sci.math)
  • Re: An uncountable countable set
    ... Tony Orlow wrote: ... And what is the smallest finite distance? ... That does not match anyone else's set of reals. ... LUB has a LUB which is not a member of the sequence. ...
    (sci.math)
  • Re: An uncountable countable set
    ... Tony Orlow wrote: ... David R Tribble wrote: ... then we'll talk about "covering the reals". ... formal proof of the equivalence between the H-riffics and the reals. ...
    (sci.math)
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