[FOM] Slater's latest claim...

Randall Holmes holmes at diamond.boisestate.edu
Tue Oct 7 13:50:23 EDT 2003

Slater  says:

>In any case I can make my main point - against von Neumann sets being 
>numbers - quite independently of this, using Hazen's, or Holmes' 
>offered logic in which the numeral in '(nx)Fx' can be quantified 
>over.  For if at least this is so, then one can form 'iota-n(nx)(x 
>isin {{}, {{}}})' in the Russellian manner, and how is this to be 
>read?  It is to be read as 'the number n such that there are n 
>members in {{}, {{}}}' i.e. 'the number of members of {{}, {{}}}'. 
>But it is clear that, on either Hazen's or Holmes' construal, this 
>number (which is the same as the number 2) is distinct from the set 
>{{}, {{}}}, indeed from any set.

I can't imagine what Slater is thinking.  Note that Slater is talking
about _my_ suggested logical interpretations of (nx)(Px).

In one of these, (nx)(Px) is interpreted in NFU as the assertion
{x | Px} \in n, where n is a Frege natural number (that is, a _set_).

Slater might object that he wants my higher-order logic interpretation
in which (nx)(Px) is to be interpreted as asserting that a certain
second-order property n is to be ascribed to the _property_ [x:Px].
That's still fine from my standpoint: higher order logic (of order
omega, which is my stated milieu for this interpretation) admits a
further interpretation in NFU in which all properties, of whatever
order, are _sets_ (and type distinctions and distinctions between
properties with the same extension are collapsed).  For details see my
"Foundations of mathematics in polymorphic type theory", in a recent
special issue of Topoi.  So it is not impossible to interpret the
number as a set (and that is all I need to support my position -- I do
not need to show that it _has_ to be interpreted as a set).

I'll grant that 2 is distinct from {{}, {{}}} in my interpretations :-)

In the other interpretation (in omega-logic), an interpretation
of (nx)(Px) is offered in which it is indeed possible to quantify over
the numeral, but, on careful analysis, the numeral doesn't refer to
anything at all (even when it is quantified over!!!!!)  The reason
for this is that the "quantification" over numerical quantifiers
is interpreted as infinitary conjunction or disjunction, so no variable
binding is actually involved.

Re earlier remarks which I was planning to ignore, but which I might
as well deal with now:

With Hazen I observe that the use of numbered variables involves no
reference to numbers.  One can just as well use x, x', x'', x''',
and so forth.  Slater will probably then object that Hazen's definition
can't be stated using this notation.  But it can (with a little work).

Re this remark:

"What Holmes has to understand is that the foundations of mathematics 
have been completely misrepresented for the last 100 years."

The foundations of mathematics are in good hands.  Leave them there.

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at math.boisestate.edu
not glimpse the wonders therein. | http://math.boisestate.edu/~holmes

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