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

Aatu Koskensilta aatu.koskensilta at xortec.fi
Mon Jun 16 01:56:11 EDT 2003

Vladimir Sazonov wrote:

> Aatu Koskensilta wrote:
>>Vladimir Sazonov wrote:
>>>What (as you say) "disinformed" me? Some deeper that in the school
>>>things like Goedel's theorems, especially on incompleteness and
>>>Goedel/Cohen proof on independence of CH, what demonstrated (to me)
>>>that both N and continuum are vague concepts, [ --- ]
>>Like [] I can understand your position with regards to the continuum,
>>but as to N, I'm still baffled. Surely the notion of *non*-standard
>>model of arithmetic is much more illusive, as any such model must
>>necessarily be non-recursive? 
> Here I feel you assume ZFC or the like where what you mentioned 
> makes sense. As I already wrote (actually many times), in this 
> framework I have no problems with understanding the concept of 
> standard or nonstandard models of PA. All of this is defined 
> (or proved) in ZFC quite precisely. I have a serious problem 
> with understanding when "standard" model of PA is mentioned 
> in some ABSOLUTE, metaphysical, quasireligious sense (not 
> RELATIVE to ZFC or the like). 

This is exactly what I fail to comprehend. It seems that since we have 
the metatheorem in systems like ZFC and the like about PA that all its 
recursive models are isomorphic and *standard*, this shows that we 
*can't*, without assuming quite strong platonistic framework even 
produce any non-standard models for display, or to have any point of 
ambiguity. I don't think we need to relativise these notions to any 
specific metatheory, even though the result itself is proved in some 

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).

 > It seems you do not understand
> what I mean. If I would know what is unclear for you when I 
> refer to ABSOLUTE and RELATIVE, I would try to explain. 
> However, I seemingly already explained my views in all 
> essential details. I am just wondering why what I wrote 
> quite explicitly is misunderstood or just ignored. 

I don't think I misunderstand you as thoroughly as you think. In my 
opinion the fact that all standard models of arithmetic are recursive is 
of philosophical importance, especially for someone with a 
finitistic/intuitionistic/constructionistic stance. You might of course 
disagree, but I don't see why. Supposedly recursiveness is an absolute 
epistemological notion. Do you agree with this? Or do you think that 
recursiveness itself is in some sense relative to a formal theory?

What you write is misunderstood or just ignored because it's alien, at 
least to me (insert smiley here). At first I thought you represented 
some sort of Hilbertian formalsim (an eminently sensible view, by the 
way), but your views now seem much more radical.

>>There seems to be no such clear distinction with standard and
>>non-standard models of set theory (let alone the notion of "the"
>>standard model of set theory), and thus I can appreciate the idea that
>>there is something inherently vague to the continuum or the even more
>>substantially infinitistic set theoretic objects. But N? There seems to
>>be a genuine *mathematical* distinction here; the standard model is the
>>recursive model, and the non-standard ones are the non-recursive ones.
> The fact that WITHIN ZFC we have these results has no relation to 
> my question on what is the ABSOLUTE standard N. I strongly believe 
> that this is actually a wrong, fictitious concept having nothing 
> rational behind of it. Usually mentioned abstractions of potential 
> or actual infinity with respect to the ABSOLUTE standard N are 
> themselves very vague. 

But the absolute standard N only becomes vague when you wander outside 
the realm of intuitionistic/constructionistic/finitistic mathematics! 
This is very much unlike the case with ZFC and its standard and 
non-standard models, let alone its "the standard model". You seem to see 
no essential difference here, which baffles me.

> Say, potential infinity of N essentially assumes that we can 
> always add 1 to any number. Moreover, it is assumed that we can 
> ARBITRARILY iterate this our ability. It is this ARBITRARILY what 
> is unclear for me. I understand that this assumes that by iterating 
> the operation x+1 we should have the ability to always fulfill the
> operation x+y. Further iterations lead to multiplication, 
> exponential, superexponential operations, primitive recursive 
> functions, Ackermann's function,..., epsilon-0-recursive functions, 
> 			AND SO ON. 
> Yes, we can continue further and further, but how further? 
> Until we will get tired? What this AND SO ON really means? 
> Can anybody explain? If not, then this is something indefinite, 
> vague. Thus, the "resulting" N is also vague. Let us be honest 
> before ourselves. 

I don't see why this needs to be vague at all. It is true that many 
philosophers in the past have formulated this "going on arbitrarily" in 
a misinformed fashion, even claiming that we are somehow "compulsed" 
mentally to always go on. This compulsion exists, but it is of modal 
nature; we *can* always go on, and this continuation is unambiguous 
*provided* one doesn't actually work in some strong platonistic 
metatheory. The so-on simply means that we *can* always consider the 
natural numbers further ahead, and that they are uniquely determined by 
the production rule (and the fact that whatever recursive function we 
might consider, it acts "correct" for these numbers, i.e. in effect 
Dedekind's recursion theorem).

> Again, what is the "length" of the resulting N? Intuitively, 
> it is much more comfortable for me to think (together with 
> Esenin-Volpin) about many (infinite) Ns of various "length", 
> with various abilities to iterate the ability to iterate the 
> operation x+1. It is intuitively plausible that the simple 
> iteration of x+1 leads us only to feasible numbers where 
> 2^1000 is non-feasible. 

I can't understand how any number we can actually name and work with 
could be non-feasible!

> Of course, we can, in principle, make this AND SO ON explicitly 
> defined WITHIN a formal theory. But this will mean that we 
> relativized this AND SO ON and corresponding version of N to a 
> formal theory. (QED!) In general, the only possibility to do 
> something precise in mathematics is via formalizing. 

Your conclusion only follows if one accepts your thesis that the so-on 
is somehow inherently vague. I don't see why this needs to be so.

>> From your postings I gather this won't satisfy you, but I'd be
>>interested to know whether GЖdel's theorems merely motivated you to
>>question the platonistic picture of mathematics or do you believe they
>>server as arguments against such a position?
> As I wrote in a posting to FOM, they are (may be indirect) witnesses  
> of the vagueness of N. They stimulated me to start doubting and asking 
> the question "what is the standard model of PA?". I have no direct 
> answer, and, I believe, nobody has. Why then to use this "wrong" 
> concept (except explicitly within ZFC) at all? 

There is a kernel of truth in what you say. I believe we don't need a 
concept of "the standard model of PA" unless we're already working in a 
strong theory, simply because there's no way for non standard models to 

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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