[FOM] Ultrafinitism - some comments

Vladimir Sazonov vladimir.sazonov at yahoo.com
Wed Jan 23 20:58:18 EST 2013

Vaughan Pratt wrote:

>mentioning several ideas including Rohit Parikh's generous upper bound of 2^{1000} on ultrafinite numbers, which as far as I know has not been improved in the meantime (but check with Rohit who is good at answering email).
I think, using the word "improving" is somewhat ambiguous here - in which sense?. Also, in which sense the "bound"? Particularly, the most valuable technical part of Parikh's work was about establishing of a different, much higher upper bound. 
I realise that the above quotation was probably not intended to be very precise, but let me try to clarify. 

In his JSL'1971 paper, Parikh used the term "feasible numbers". These numbers were postulated to be closed under successor, addition and multiplication and, under his approach, the upper bound for them is 2_{2^1000}, i.e. much higher than  2^{1000}: 

(1)     all feasible numbers < 2_{2^1000}.

Note that this bound is an *exponential tower* of 2's of the exponential height 2^1000. Even the tower 2_{1000} (of the height 1000) would be too low, so that the axiom 

(2)     all feasible numbers < 2_{1000}

is actually contradictory in his approach. (It can be formally derived that 2_{1000} is feasible.) Anyway, Parikh gave a first mathematically rigorous demonstration that feasibility can be formalised at least in the form (1) and the bound in (1) was actually the best.  
On the other hand, even the axiom 

(3)     all feasible numbers < 2^{1000}

(with a more plausible and much lower, "improved" upper bound 2^{1000} or even 2^{100}) is indeed intuitively "true". As I remember from a personal discussion with Parikh in 1995, he considers this as a "truth", but only (1) - with much higher upper bound was possible to consistently postulate. 

Now, for the version (3) of the axiom, feasible numbers can only be postulated to be closed under successor and addition. Multiplication will (evidently) lead to a contradiction. (2^{1000 = 2*2*...*2, and so would be feasible.)

But the last version requires even more restrictive approach (skipped here) to make it formally consistent. See 
"On Feasible Numbers" in http://cgi.csc.liv.ac.uk/~sazonov/papers.html .
Finally, a more general comment on the topic as a whole for a newcomer (such as a student):
Unfortunately, many considerations on feasibility (or ultrafinitism) have rather speculative character. It was good and indeed stimulating at initial stages (e.g. for me personally such was reading of earlier very informal Essenin-Volpin's works). But if we really want to make something mathematical from this, it is crucial and inevitable to think in terms of their formalisation - as it was firstly demonstrated by Parikh. Otherwise, I believe, too informal (and even semi-formal, insufficiently rigorous) considerations can easily become fruitless and actually discrediting the whole (I believe, nice) idea. 
Of course, clarifications presented above are not a complete formalisation, should not be considered as such, and so they are insufficient without considering the necessary missing details.  

Vladimir Sazonov

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