🚀 go-pugleaf

Transfer Principle wrote:
> 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.

And also because it is very unlikely (unlikely to
point that we should actually ASK HERC!) that he
thinks the real numbers and the natural numbers are
the same set. Isn't that right, Transfer?

>>>... 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.

Okay, this stuff looks interesting. It makes a
nice change of pace from discussions of how many
letters it is proper to use in refering to posters.

However, as I see you mention below and as
Aatu Koskensilta also tells us, we can prove the
reals are uncountable in NFU.

And, anyway, we don't think Herc is using either
NFU or zuhair's ND, do we?

The important difference I see between NFU and ZFC
is that NFU excludes many sets that ZFC includes,
in particular the bijection (in ZFC) x |-> {x}.

It seems to me that NFU has not so much eliminated
the perplexing relationships called paradoxes as
moved the perplexity to other relationships.
For example, the NFU set-of-allsets V is, let us
say, non-Cantorian, which eliminates the paradox
of P(V) being both a proper subset of and larger
than V. However, another consequence of this is
that |V| > |P_1(V)|, where P_1(V) is the subset
of P(V) consisting of all one-element sets. I don't
feel any happier with this. Or, more to your
point, I suppose, I don't imagine Herc being any
happier by trading one paradox for another.

I wonder if there is some law of meta-logic that
conserves perplexity.

I see in the Wikipedia entry that a set A for which
  |A| = |P_1(A)| is called /Cantorian/. I suspect that
all of the sets that have counterparts in ZFC
are Cantorian and that all the non-Cantorian NFU
sets would be proper classes in, for example, NBG.

> 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

I haven't been following that thread, but thank you
for pointing it out.

You seemed to have missed that zuhair is explaining
his just-added Axiom of Countability,
| For every set x ( x is countable -> Px is countable ).

Every countable set is indeed a counter-example to
Cantor's general theorem. However, this is done by
brute force, so to speak. I am at least a little
skeptical that the resulting set theory is still
coherent enough to contain a complete ordered field.

> "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.

Perhaps you could do that. You were the one suggesting
that that R is countable in zuhair's system, I believe.

A question for myself (and, PLEASE, anyone who might
know the answer) is whether there is a proof of
"If there exists a complete ordered field (that is, the
real numbers) then there is no complete list of the
reals." a conjecture I made up thread.