Re: infinity


Tony Orlow wrote:
> Randy Poe said:
> > The meaning of "Card(*N) = Card(P(*N))" would be "there exists
> > a bijection between *N and P(*N)." Since there does not, you
> > can not draw that conclusion.
> Did you have a specific objection to the mapping through the infinite bit
> strings that I offered? If so, please state it.

You show the first few terms without specifying the mapping
precisely, then you claim (without proof) that the right
hand side and left hand side both have exactly term for
every element in *N.

It isn't true. Provably.

> > > > 2. A bijection between P(N) and R can be shown to exist.
> > > Through the infinite binary strings, I know.
> >
> > Then why did you ask?
> I didn't. I said you derive your results from axioms that basically state them
> with no justification.

The axioms have no justification? What can that possibly mean?

What would constitute "justification" for an axiom?

> > No such vague statements as "other infinity of bits" are
> > needed. Just consider bit strings consisting of one bit in each
> > position n where n is a natural number.
> >
> > It is provable (not an axiom) that every real number in [0,1)
> > corresponds to such a bit string, and every such bit string
> > corresponds to a real number. There is a bijection between those
> > PARTICULAR infinite bit strings and the numbers in [0,1).
> >
> > What do you think is missing there?
> It sounds like a bijection to me. So, what is your objection to my bijection
> between *N and P(*N)?

You are missing the connection between your strings and
your elements. This connection between infinite strings
and real numbers is provable.

Let's attempt to put this on something approaching a rigorous
ground.

Your bit "strings" are of length |*N| I assume. So there is
a function b(x,y), the "y"-th bit of element x, for every
x, y in *N.

Does every such string represent an element of *N?

The answer is no. The elements of *N are those strings which
have one bit for every finite natural number. *N is the
collection of such strings. So in the most natural
mapping between *N and bit strings, b(x,y) is 0 for any
y which is not a finite natural.

Your "bijection" relies on a representation which you have
not proved exists. Remember that I said it is PROVABLE that
every infinite string represents a real number in [0,1)?
You have no such proof that every *N-length string
represents an element of *N.

Can you come up with such a rule? First, what to you are
the elements of *N, and how do I take a *N-length string and
prove it represents a unique such element?

- Randy

.

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