Re: What's the best language to start with? [was: Re: Should I learn C or Pascal?]

[jsa@alexandria (Jon S Anthony)]

In article <52eha1$o7h@krusty.irvine.com> adam@irvine.com (Adam Beneschan) writes:

>  >Actually, when people use the side-ways "8" notation, the typical
>  >intent is simply that of indicating "arbitrarily large" or (somewhat
>  >less so) denumerably infinite (size of naturals - aka Aleph0).  Your
>  >"Aleph2" is a denotation for the size of sets the size of the power
>  >set of R (the reals) where Aleph1 denotes the size of sets
>  >equinumerous with R.  In general you have this whole "backbone" of
>  >transfinite numbers constructed (typically) via the power set
>  >operation.
> 
> As long as we're getting pedantic: Is it really true that Aleph1
> denotes the sets equinumerous with R?

Well, really, that is going too far as it would presume the continuum
hypothesis.  So, you have hoisted me on my own petard.  The Aleph
sequence denotes the ascending ordered sequence of the ordinal _class_
of infinite cardinal numbers.  So, Aleph0 is the smallest, then Aleph1
is least such greater than Aleph0, and so on.  The "and so on" can be
made rigorous.  Now, Aleph0 is not only the least cardinal it is also
the cardinality of N (the naturals) or any other countably infinite
set.

It is also easy to show (after the machinery is set up) that for any
set S

   card(P(S)) = 2^card(S)

So, the card(P(N)) = 2^card(N) = 2^Aleph0.  It also falls right out of
the Shroder-Berstein Thm, that card(R) = 2^Aleph0.  So, the question
is, are there any infinite cardinals between Aleph0 and 2^Aleph0???
If not, then 2^Aleph0 = card(R) = Aleph1.  But this is the Continuum
Hypothesis (CH).  Some have claimed it is true and some have claimed
it is false.  What we do know is that K.Goedel (1939) showed that it
is _consistent_ with standard set theory (Zermelo-Frankel axioms) and
that Paul Cohen (1963) showed that it was _independent_ (both via
formal model theory).  So, it is neither provable nor refutable from
the standard axioms.  So, should it be accepted or not?

Well, that's just not obvious and further you could even argue that it
may not make sense to claim its truth or falsity wrt _standard_ set
theory.

Of course, the situation is even "worse than this", since it is not
any more obvious that for any infinite cardinal number C, there is not
some infinite cardinal between C and 2^C.


> cardinality of the smallest set that is larger than N); but from what
> I've read, it's been shown that the truth of the statement
> 
>     C = Aleph1
> 
> is independent of the other axioms of set theory.  I asked sci.math

Yes, given your defs for C and Aleph1, this is correct.


> theory don't really give you a definition of what a 'set' is, but if
> you come up with a suitable definition that satisfies all the axioms,
> you may be able to determine from your definition whether C=Aleph1 is
> a true statement or not.

The definitions do give a notion of set which is formal.  The axioms
then are specified to produce the/a desired theory of sets.
Interpretation of the axioms provides the semantics (here we are
hinting at formal model theory).  Basically, KG provided a model where
all the standard axioms are true as well as CH, hence CH is consistent
with ZF.  Cohen (with true genius level ingenuity) provided one
showing the standard ones true but CH not, hence CH is _independent_.
This situation is not unlike the independence and consistency of the
so called "fith postulate" of Euclid (two parallel lines never
intersect) and the resolution of that "nut" also involved model
theory.


> However, no one told me whether it was known whether C=Aleph1 or not
> in the "usual case" of the real numbers and integers.

That's probably because it is not known :-), and may be unknowable in
any meaningful sense.


> I believe the question is equivalent to the following: Is there an
> infinite subset of R that is not equivalent (equinumerous) to either R
> or N?

Yes, or simply: is every uncountable set of real numbers equinumerous
to R.


> My impression is that the answer may depend on how you define
> the concepts "real number" and "natural number" and how you define the
> sets R and N.  I don't know whether there's a definite answer . . .

See above...



/Jon

-- 
Jon Anthony
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