Re: Well Ordering the Reals

In article <MPG.1de3cf0f70754a0698a6fb@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> Virgil said:
> > In article <MPG.1dde9c15bccebfd398a6d7@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> >
> > > Virgil said:

> > > > What is your index set for *N?

> > > *N
> > > >
> > > > Every endless string of binary digits of form abc... or ...cba
> > > > can be represented as a function, f:N -> {0,1}, whose domain is
> > > > the index set of its digits.
> > > >
> > > > What is the index set for the members of *N?

> > > *N. There is no end to the digits. They go to infinite positions,
> > > like the values of the set.
> >
> > Is TO claiming that the position of each digit in each member of *N
> > is indexed by *N itself? In that case, To is claiming that not only
> > is his *N bijectable with its power set, but that it IS its own
> > power set.


> No, the power set is a set of sets of naturals, while *N is a set of
> naturals.
But if *N is representable by srings of binary digits, as TO claims,
then that set of srings must have some index set to indicate the
positions of digits in those strings.
TO has said that the index set for *N is *N itself, which is an
impossibility, and has given no other answer to the questin of what the
index set is.

So one must conclude that his *N does not exist at all.



> >
> > Now previously TO has allowed as how the binary digits in members
> > of *N were indexed by members of N, so that there were infinitely
> > many digits in each member of *N. Such a construction is quite
> > possible, even reasonable.

> No, I just didn't bother responding to that, since it was irrelevant,
> and you know it won't fly with me. In order for *N to have infinite
> natural values, the bit strings in *N must have bits with infinite
> positions in the string, since the sum of a finite number of finite
> numbers is always finite, and any bit in a finite position has a
> finite number of preceding bits, each representing a finite value.

> Since no element of N is infinite, N is not sufficient to index *N.
> *N is the only set of indexes which will suffice, and the bit strings
> in *N are ultimately indistinguishable from those in P(*N).
> >
> > But a set of "binary strings" that IS its own index set?

If *N is bijectable with P(*N), as TO implies, then every string of *N
is representable by a string in P(*N) with a single 1 bit, but then, by
bijection, it must be representable back in *N by a string with a single
1 bit.

So every string in *N needs only a single 1 bit to represent it.
.

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