FOM: Feasibility and (in)determinateness of the standard numbers

Torkel Franzen torkel at
Fri Apr 3 18:54:32 EST 1998

 Vladimir Sazonov says:

   >Thus, you insist that an ideal image of a first order model
   >arising in our consciousness when working in a formal system is
   >not an illusion (possibly having some objective roots in the
   >real world, in the formal system, in our psychology, in the
   >objective lows of thinking, etc.), but corresponds to a real
   >non-illusory thing or entity.

  No, I don't insist on anything of the sort - indeed I don't know
what to make of such talk of first order models arising in my
consciousness. Rather, as I explained before, I don't find any
substance in the idea that we are in the grip of an illusion in our
thinking about the natural numbers. This idea, to my mind, is best set
aside in considering the interest and value of the investigation of

  >Yes, *such* a rule cannot generate 2^N.  But does this
  >nondeterministic procedure generate any *determinate* collection
  >of (intuitively more and more complex, say, of arbitrary
  >feasible complexity) binary strings *unlike* deterministic
  >procedure generating *indeterminate* or vague collection of
  >(more and more big) feasible numbers?  What is the difference?

  Feasible complexity has nothing to do with the generation of 2^N
by a rule. You can choose a deterministic rule instead if you like.
The applications of the rule are indeed mostly imaginary, but what
is indeterminate about the totality of objects obtainable (as one
says, in principle) by the rule?

  >You believe that only 2^N is indeterminate.  What is the reason
  >for the difference you make?

  Simply that 2^N isn't generated by a rule.

  >It seems that our beliefs and non-beliefs related with "standard 
  >model" lead us into a vicious circle. A reasonable "neutral" and 
  >fruitful solution in general would consist in representation of 
  >any beliefs and intuitions by a formal system.

  Why do you think so? I don't think formal systems are one bit clearer
or more determinate than the natural numbers. If you insist on
introducing and using "feasible formal systems", that is hardly a
neutral procedure.

  >For example, the limit and 
  >continuity concepts were rather unclear before formalization.  
  >Also, in the framework of formal system ZFC G"odel and Cohen 
  >*actually* have demonstrated in some concrete terms why and how 
  >2^N is indeterminate. 

  We can introduce formal systems and their associated concepts
without relying on limits, continuity, infinitary set theory. We
do rely on our ordinary understanding of recursively generated
structures like the natural numbers.

Torkel Franzen

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