[FOM] Slater and numbers

Randall Holmes holmes at diamond.boisestate.edu
Fri Oct 3 13:34:34 EDT 2003

Dear FOM colleagues,

Slater said this:

>Allen Hazen has tried to be 'tangential' to the Holmes/Slater debate, 
>but despite himself has hit the bullseye!  He says (FOM Digest Vol 10 
>Issue 3):
>>         (A) Numerically definite quantifiers.  For any fixed n, FOL with
>>Identity can express "There are n things such that ...".  (Aside: the most
>>efficient way of doing this is apparently due to David Lewis.  To say
>>"There are at least n x such that Fx," start with (n-1) UNIVERSAL
>>         AyAz...Aw
>>then put in one EXISTENTIAL
>>                  Ex
>>then a conjunction of (n-1) negated identities saying that x is not one of
>>                    (~x=y&...~x=w
>>and finally say that x is an F thing:
>>                                  &Fx).
>>The length-increase of these formulas is linear as n increases, which is a
>>lot more user-friendly than other methods.)
>Here, although Hazen tries to think he is dealing with 'any fixed n', 
>he has produced a formula in which 'n' is a variable, and which 
>therefore indicates a place which can be quantified over.  Yes, 
>indeed, *for any n* start with ...n-1 universal quantifiers then 
>...then a conjunction of n-1...'  If he numbered his individual 
>variables, so that y,z,...w were x1, x2,...x[n-1], the point would be 
>even plainer.
>In a representation of 'x has n elements' as I said before (FOM 
>Digest Vol 10 Issue 2), one cannot get rid of the 'n'.  Even if, for 
>any fixed 'n', the numeral 'n' does not occur in the formula (because 
>numbered variables are not used), still the number it denotes must be 
>previously known, to fix the number of various elements, and so 'the 
>set of those sets with n elements' cannot define the number n, since 
>one must know what that number is beforehand.

So (a) Holmes' 'Frege numerals' no more define numbers than von 
Neumann ordinals like {{}, {{}}}, and (b) quantification over the 
numeral place in expressions like (nx)Fx is entirely possible.  Q.E.D.

I reply: Slater should read Hazen more carefully.  In the same post
Hazen says that the numerical quantifier is "inert" and _cannot_ be
quantified over (which is hardly a neutral assertion in the argument
between Slater and myself).  Slater should consider why Hazen is able
to say this.

In fact, there is no single definition of (nx)(Px) in the formal
theory under construction.  There are definitions of (1x)(Px),
(2x)(Px), (3x)(Px), and so forth, which have nothing to do with each
other on the formal level.  Hazen has provided a nice meta-level
discussion of how these definitions are to be constructed.  This
meta-level discussion does contain a mention of the natural number n
-- but the meta-level "definition" (pace Slater) is not a "formula" of
the formal system under discussion at all and so doesn't contain any
damning quantifiable occurrence of n.  (3x)(Px) contains no reference
to the number 3 at all -- you can write out the formula following
Hazen's recipe, and look for the reference in vain.  Similarly,
(117x)(Px) contains no reference to 117.  This is exactly what I meant
when I said that to attempt to quantify over the numerical variables
is to confuse theory and meta-theory (but I thought better at that
time of going into it in detail).

The "definition" of (nx)(Px) that Hazen gives is a _definition
scheme_.  Another point which Slater should consider: if a model of a
theory of arithmetic contains nonstandard natural numbers N (using
whatever coding one likes), we will not be able to instantiate the "n"
in (nx)(Px) with N: only a concretely given natural number can be fed
into Hazen's procedure.  The fact that we cannot quantify over the n
in (nx)(Px) is related to the fact that in the axiom _schemes_ of
replacement and separation in first-order Zermelo set theory we are
not quantifying over predicates (as we do in the single axioms of
separation and replacement in second-order Zermelo set theory).

I agree with Slater that we need to be acquainted with the natural
numbers already in some sense in order to read Hazen's definition.
Everyone is acquainted with the natural numbers (or at least with
certain kinds of talk about the natural numbers) before they learn any
formal definition of (for example) natural numbers as sets; they are
certainly acquainted with the natural numbers before they learn a
formal definition of the numerical quantifiers, as well.  It is only
this level of acquaintance (the ability to count) that is required to
understand Hazen's definition scheme.  This is not evidence against
the natural numbers being sets (or against any other theory of what
they are), any more than the fact that we are acquainted with our dog
Fido before we discover that he is composed of elementary particles is
evidence that Fido is not composed of elementary particles.

My comment on being "standard" is this.  I have never maintained that
NFU (or New Foundations) is standard set theory (in fact, I have
carefully pointed out that it is not).  Slater _has_ maintained or
appeared to maintain that logic with numerical quantifiers is standard
logic.  Both NFU and logic with numerical quantifiers are systems
which admit proofs of consistency relative to standard systems.  Both
are systems worthy of study.  Nonetheless, especially when trying to
support an unusual position in a public forum, one should know what is
standard and likely to be accepted by one's audience without special
justification.  I have been very clear about what is nonstandard about
positions I have defended in this discussion; Slater has not (it is not
clear that he knows in what respect his positions are nonstandard).

An obvious comment which I should have made already re numerical
quantifiers: all formulas with numerical quantifiers make sense if one
allows countably infinite conjunctions and disjunctions in one's logic
(that is, if one is practicing omega-logic).  In this context, one can
quantify over numerical quantifiers with complete impunity (even if
the same bound numerical variable is used to count base type objects
and numbers).  A set theorist should have no particular problem with
omega-logic; a philosopher might (actually should) find it alarming.
It is also worth noting that the formalization of (forall n)(nx)(Px)
in omega-logic (with all definitions fully expanded out) makes no
reference to natural numbers whatsoever; Slater should contemplate
this fact, which is intimately connected with the reasons why there is
no theory-level reference to the natural numbers in Hazen's definition

Sincerely, Randall Holmes

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