🚀 go-pugleaf

On Mar 27, 3:36 pm, MoeBlee <jazzm...@hotmail.com> wrote:
> On Mar 27, 2:23 pm, Transfer Principle <lwal...@lausd.net> wrote:
> > On Mar 25, 7:50 am, MoeBlee <jazzm...@hotmail.com> wrote:
> > > On Mar 24, 7:05 pm, Albrecht <albst...@gmx.de> wrote:
> > > > I have the axiom of infinity in mind. And the combination of the
> > > > power set axiom with the axiom of infinity. Without these axioms the
> > > > interpretation of the diagonal argument of Cantor is completely
> > > > different from the common interpretation.
> > > Statements like that aren't constructive mathematics or finitist
> > > mathematics or ultra-finitist mathematics. Just ignoramus mathematics.
> > So merely stating that all known proofs in ZF of the existence
> > of uncountable sets (i.e., what Cantor's diagonal argument is
> > designed to prove) require both Infinity and Powerset is now
> > considered "ignoramus mathematics"?
> No. I have no idea why you ask that question. I did not say nor imply
> that "stating that all known proofs in ZF of the existence of
> uncountable sets require both Infinity and Powerset [makes one an
> ignoramus]."

Why did you call Albrecht an "ignoramus"? Let's look again at
_exactly_
what you (MoeBlee) wrote above:

> > > Statements like that aren't constructive mathematics or finitist
> > > mathematics or ultra-finitist mathematics. Just ignoramus mathematics.

So you clearly describe some statement made by Albrecht to be
"ignoramus mathematics." Let's look again at that statement which you
so describe:

> > > > I have the axiom of infinity in mind.

OK, so he's referring to the Axiom of Infinity of ZFC. That doesn't
make
Albrecht an "ignoramus."

> > > > And the combination of the
> > > > power set axiom with the axiom of infinity.

Now he's referring to both the Axioms of Infinity and Powerset. Those
don't make Albrecht an "ignoramus."

> > > > Without these axioms the
> > > > interpretation of the diagonal argument of Cantor is completely
> > > > different from the common interpretation.

What is the common interpretation of Cantor's diagonal argument? Its
common interpretation is that there exists an uncountable set --
namely,
the set R of real numbers.

So Albrecht is saying that without _both_ (i.e., the "combination of")
the
Powerset Axiom and the Axiom of Infinity, the proof that there exists
an
uncountable set R no longer holds.

In theories other than ZFC, there really can be a different
interpretation
of Cantor's diagonal from the common interpretation. In another
thread,
for example, there's discussion of a theory ND (zuhair) in which the
correct interpretation of Cantor is that there is no _set_ that's a
bijection
from N to R, but there could be a proper class that's such a
bijection.

Returning to Albrecht, what do ZF-Infinity or ZF-Powerset prove about
the existence of uncountable sets? Not very much, I'd think. And I
feel
that merely pointing this out doesn't make one an "ignoramus."