🚀 go-pugleaf

On Mar 13, 5:43 pm, Jim Burns <burns...@osu.edu> wrote:
> On 3/13/2011 5:03 PM, Transfer Principle wrote:
> It is not very difficult to come up with a theory
> where "there are as many reals as naturals", if
> you're willing to misrepresent what people mean
> to that extent. Just call the naturals "the reals",
> and "there will be as many reals as there are
> naturals". Happy now?

We need something less trivial then simply redefining
"reals" to mean "naturals" to satisfy Herc's desideratum,
since Herc refers to reals that aren't naturals in some
of his posts.

> > ... or an unimaginably "bizarre" _theory_,
> > like NFU, or zuhair's ND, and so on ...
> Are you saying that Cantor's diagonal theorem
> can be disproven in NFU or zuhair's ND? I
> would like to see that. Obviously, given what
> I have written above, I would not expect that.

For NFU, here's a link to a page about NFU:

http://math.boisestate.edu/~holmes/holmes/nf.html

We scroll down to the discussion on Cantor:

"Cantor's paradox of the largest cardinal: Cardinal
numbers are defined in NF as equivalence classes of
sets of the same size. The form of Cantor's theorem
which can be proven in Russell's type theory asserts
that the cardinality of the set of one-element subsets
of A is less than the cardinality of the power set of
A. Note that the usual form (|A| < |P(A)|) doesn't
even make sense in type theory. It makes sense in NF,
but it isn't true in all cases: for example, it
wouldn't do to have |V| < |P(V)|, and indeed this is
not the case, though the set 1 of all one-element
subsets of V is smaller than V (the obvious bijection
x |-> {x} has an unstratified definition!)."

So in NFU, V is a non-Cantorian set -- i.e., it's a
counterexample to Cantor's Theorem. Here V refers to
the _set_ (not proper class, but _set_) of all sets,
which exists in NFU.

But note that in NFU+Infinity, I believe that the set
of natural numbers exists and is Cantorian, so we
don't yet have as many naturals as reals. For that,
we need to go to zuhair's ND.

Now zuhair wrote his posts about ND during the first
week in March. I choose to give a link to the zuhair
thread using Mathforum (though I could have used
Google instead). But this thread "Skolem's paradox
internalized" should be visible no matter what
newsreader you have (unless you killfiled it):

http://mathforum.org/kb/message.jspa?messageID=7397980&tstart=0

"Now, the idea is that the bijection between for example Omega (the
set
of all natural numbers) and its power, would be a proper class,
and it is not definable by a formula with set parameters,
so the diagonal will be a proper class also,
since otherwise we'll arrive at a true paradox."

> Well, yes, you SAY that, but SHOWING me some
> mathematics (a disproof of Cantor's diagonal
> theorem, for example) would so much more interesting
> than more CLAIMS.

Now zuhair would be much better than I to prove that R is
countable in his theory, so you might want to ask zuhair
for the full proof in the revelant thread.