[FOM] To Vladimir Sazonov and others doubting the unambiguity of N

Aatu Koskensilta aatu.koskensilta at xortec.fi
Tue Jun 17 01:46:36 EDT 2003

Andrew Boucher wrote:

> Aatu Koskensilta wrote:
>> You think that N is in some sense vague. How does this vagueness 
>> manifest itself? Are there some theorems of arithmetic that don't have 
>> a specific truth value? Are there some identities that aren't 
>> "decided"? I can't see how N can be vague *unless* one assumes quite 
>> strong a theory, in which it's supposed to be so (say ZFC).
> I'm afraid I don't understand this point of view - perhaps it's just a 
> reply to Sazanov's comments on models and standard models, which I'm 
> afraid I also don't understand.

This is the case. I don't see how one can even begin to speak about 
standard and non-standard models without already venturing beyond 
finitistic considerations, let alone ultrafinitistic ones.

> But - again although I'm not a card-carrying ultrafinitist -, it would 
> seem the obvious candidate for a vague theorem of arithmetic would be:
>    "For every natural number there exists a successor."

Ok. This is then an entirely different sort of vagueness than the one 
introduced by non-categoricity of PA.

> This is equivalent to many arithmetic theorems over a well-chosen base 
> theory, e.g.
>    "There are an infinitude of primes"
> or the Chinese Remainder Theorem.  So these theorems would also be vague.
> Why might they be considered vague?  Well, it could be argued to have 
> the same characteristics of a Sorites heap.  We can't say exactly when a 
> non-heap becomes a heap.  And we might not be able to say exactly when 
> the "adding 1" becomes difficult/impossible.

I believe that feasibility has much more to do with, say, the Kolmogorov 
complexity of a number than it has to do with the absolute magnitude of 
the number. Assuming that there is no "sharp" boundary between feasible 
and non-feasible numbers, then it seems obvious that the statement

  for all feasible numbers x ( x+1 is feasible)

is true, since just adding one to a number increases the complexity of 
the number very slightly. We could perhaps use other sorts of 
complexity, taking into account not only the "structural" Kolmogorov 
complexity, but also the time and space complexities of the producing 
algorithms. This would square far better with my intuitions about 
feasibility of numbers than the simple definition

  x is non-feasible <--> x >> some boundary value

as there obviously are (almost?) arbitrarily large feasibly numbers.

This is better because when dealing with large numbers we very seldom 
deal with actual base n-representation of those numbers; instead we rely 
on indirect schemes of naming numbers.

>> I can't understand how any number we can actually name and work with 
>> could be non-feasible!
> Well, I don't adhere to the feasible/non-feasible distinction, but 
> here's one way to name and work with numbers which don't exist, in the 
> framework of second-order logic.  Suppose there is a maximum number M 
> and numbers are first-order entities.  Choose any n between 2 and M. 
> Then one can define "virtual numbers" to be second-order entities, which 
> are sequences from numbers {0 ... M} to {0 ... (n-1)}.  That is, this is 
> the base n expansion.  There are (well, looking from the outside anyway, 
> since one cannot count big-letters in the system) (n+1) ^ (M+1), 
> obviously a much bigger number than M.  It's an exercise to show that 
> one can define a sort of addition, multiplication, ordering, etc. on 
> these second-order "virtual" numbers.  So one reason about them (and 
> name them) in a definite way. 
> One can use similar techniques to name even larger numbers (which don't 
> exist!).

And an ultrafinitist is satisfied with this sort of procedure? Is this 
device supposed to model our actual practice?

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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