[FOM] 455: The Quantifier "most"

[Harvey Friedman]

THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION

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In listening to a talk at Ohio State University by Michael Glanzberg  
(Philosopher at U Cal Davis) on February 18, 2011, I became aware from  
the talk that philosophers have been interested in applying informal  
quantifiers such as "most" to infinite contexts such as "most natural  
numbers". Being fully aware that intuitions with fully satisfactory  
mathematical treatments in finite contexts, may not have fully  
satisfactory mathematical treatments in infinite contexts, I set about  
showing that this is the case for "most".

In the finite context, "most" can, and can only(?), be mathematically  
treated as "more than half".

I came up with the following principles, stated in terms of subsets of  
N, where N is the set of all nonnegative integers.

1. If we partition N into three parts A,B,C, then at least one of the  
following holds:

a. Most numbers are in A union B, and not most numbers are in C.
b. Most numbers are in A union C, and not most numbers are in B.
c. Most numbers are in B union C, and not most numbers are in A.

2. Most numbers are in N.
3. For all A,n, we have: most numbers are in A if and only if most  
numbers are in A union {n}.

Note that 1) (and obviously 2)) holds in the finite context for the  
following reason. Clearly at least one of A,B,C is less than half.  
Take one. The union of the remaining two must be more than half.

Of course, 3) does not hold in any finite context, and is in fact  
unobtainable in any finite context where the whole space is "most".

The following indicates that there is no "contradiction" arising from  
conditions 1-3 above.

THEOREM 1. There is a predicate "most" on all subsets of N for which  
1),2),3) hold. In fact, we can use any finitely additive probability  
measure on all subsets of N, where points have measure zero. The  
nonprincipal ultrafilters correspond to such measures which are 0,1  
valued. If we use a nonprincipal ultrafilter, then we can sharpen  
condition 1) by eliminating clause c). It is well known that ZFC  
proves the existence of nonprincipal ultrafilters on (the subsets of) N.

THEOREM 2. (well known). The existence of such measures in Theorem 1  
is not provable in ZF. Furthermore, there is no set theoretic  
definition which ZFC proves defines a finitely additive probability  
measure on all subsets of N, where point have measure zero. In  
addition, ZF proves that there is no Borel measurable such measure.

Theorem 2 indicates that there is no reasonable mathematical  
construction of a specific finitely additive probability measure on  
all subsets of N, where points have measure zero.

But what about for "most" with 1-3 above?

THEOREM 3. The existence of a predicate "most" on all subsets of N,  
obeying conditions 1-3 above, is not provable in ZF. There is no set  
theoretic definition which ZFC proves defines a predicate "most" on  
all subsets of N, obeying conditions 1-3 above. Furthermore, ZF proves  
that there is no Borel measurable predicate "most" on all subsets of N  
obeying conditions 1-3.

Theorem 3 indicates that there is no reasonable mathematical  
construction of a specific predicate on on all subsets of N, obeying  
conditions 1-3.

The proof of this uses standard technology from forcing. For those  
familiar with this technology, here is the core element of the proof.

LEMMA. There are two Cohen generic partitions (A,B,C), (D,E,A union B)  
of N.

We can obviously weaken condition 1) by using partitions of N into  
more than 3, but finitely many, sets. The same results hold.

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I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 455th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-449 can be found
in the FOM archives at http://www.cs.nyu.edu/pipermail/fom/2010-December/015186.html

450: Maximal Sets and Large Cardinals II  12/6/10  12:48PM
451: Rational Graphs and Large Cardinals I  12/18/10  10:56PM
452: Rational Graphs and Large Cardinals II  1/9/11  1:36AM
453: 453: Rational Graphs and Large Cardinals III  1/20/11  2:33AM
454: Three Milestones in Incompleteness  2/7/11  12:05AM

Harvey Friedman