Re: Frege: Reason's nearest kin


Owen wrote:
Barb Knox wrote:
In article <1150817678.616165.145920@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"george" <greeneg@xxxxxxxxxx> wrote:

Frege's definition of number is metaphysical tripe. His approach is
inconsistent, and no one today defines the number 3 as the set of all
sets with 3 members.


Barb Knox wrote:
Not so.

It is so too so.

For example, in NFU set theory numbers are defined precisely
that way.

They ARE NOT.

I am not an expert on the NFU literature, but the little that I have
read all represent a natural number N as precisely the set of all the
sets containing exactly N elements. And even if there are other
approaches, my "not so" is still accurate regarding the claim that "NO
ONE today defines the number 3 as the set of all sets with 3 members"
(emphasis added) -- clearly, some NFU folks do just that.

JEEzus.
As a set theory, NFU cannot possibly have any opinion
about what is or isn't a number.

Of course it can (and does). For example, in NFU the usual ZF/NBG/etc.
von Neumann ordinals are definitely NOT numbers, since they are not even
sets in NFU.

ANY old collection of its sets COULD be TAKEN to be numbers.

Not so, since NFU (like other set theories) has rules regarding which
"collections of its sets" can be considered to be sets in the first
place. The notion of arbitrary collections of sets has no place.

THAT is a decision made INdependently of the theory itself.
None of the axioms that define the theory has any opinion about
which sets are vs. aren't numbers. That is superstructure.
This is equally true of ZFC. There, while the natural numbers
are usually defined as the finite ordinals, they also COULD have
been defined
0={},
1={{}},
2={1},
3={2}, etc., etc.

Numbers as we intuit them are ACTUALLY defined by something
closer to Peano Arithmetic (2nd-order PA is in fact COMPLETELY
adequate to the task) than to any set theory.

Sure thing.

Furthermore,
ANY set theory has the property that PA's "successor" operator
could be implemented in it in A GREAT MANY DIFFERENT ways.

Users of NFU may be in the habit of repeating Frege's mistake
of choosing to define natural numbers as proper classes, but that
doesn't make it any less a mistake (and an ugly one at that), and
it certainly isn't part of NFU proper.

In NFU, the numbers are SETS (and indeed NFU has no notion of proper
classes). Whether or not the Fregean representation is an "ugly
mistake" is completely independent of the fact, which was what my reply
addressed, that SOME people today DO define the number 3 as the set of
all sets with 3 members. I personally don't find it ugly, but it
doesn't bother me that you do -- eye of the beholder and all that.

Hello Barbra.
It is clear to me ( it is painful for me to correct what I say, ever
time.

It is surely the case that: the Peano postulates are accurate.

They are not axioms at all. See: Bertand Russell, 'Introducton to
Mathematical Logic

Russell, showed that the Peano axoims, are logical results.


We have had a concen in spite of David Ullrich


Why do you think differently??

.

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