FOM: Arbitrary Objects

[Andrzej Derdzinski]

My understanding of "arbitrary objects" is most likely naive, but, having 
followed the discussion, I cannot help the impression that many people read 
more meaning into this concept than there actually is.

First, in mathematicians' and logicians' normal usage "arbitrary" does not 
serve as a predicate, as illustrated by the fact that nobody cares about 
theorems of the form "If ..., then  x  is arbitrary". Trying to fashion 
"arbitrary" as a predicate we end up with something of little interest: a 
predicate the extension of which is the whole universe, so that one might 
decree, for instance, that "x  is arbitrary" really stands for  "x = x".

Thus, "arbitrary" is a concept really appearing only in proofs: suppose that a 
mathematician/logician is proving a statement of the form  Ax P, where  P  is 
a fixed formula (in cases of interest, containing  x  as a free variable). In 
the course of the proof, the mathematician/logician tells himself/herself "Let 
x  be an arbitrary object" and proceeds to use that assumption to derive  P. 
What `arbitrary' means here is entirely subjective, and the intuition behind 
it may very from one person to another. The whole process becomes anchored in 
objective reality only if, at the end, the mathematician/logician is able to 
formalize the proof he/she just found. That means obtaining the formula  Ax P 
as the last entry in a string built on some agreed-upon collection of logical 
axioms and rules of inference, plus some set of premises (such as the "axioms" 
of group theory if, for instance, the statement is about groups). The last 
step in this formal proof is universal quantification, leading from  P  to  
Ax P, which is allowed as long as  x  does not appear as a free variable in 
any of the premises.

Note that this last property of  x  may be viewed as an "objective meaning" of 
an arbitrary object: something about which we have made no assumptions.

Andrzej Derdzinski
Department of Mathematics
Ohio State University