Re: ​Finitism / potential infinity requires the paraconsistent logic NAFL

[Radhakrishnan Srinivasan]

Vaughan Pratt said:
*****************************************
The limitation I see to FOL is that it can't express transitive closure of
a binary relation.  If it could, the following reasoning would permit
defining the notion of a potentially infinite set entirely within FOL.

Extensions of FOL can define transitive closure.  To the best of my
knowledge, the earliest such extension is first order dynamic logic, which
I developed in 1974 for a course at MIT and later published in FOCS'76 as
"Semantical Considerations on Floyd-Hoare Logic". [...]
******************************************

Thanks for all the information and references. I can see that an awesome amount of work has been done on FOL and I should not make statements lightly about it. I will certainly look up first order dynamic logic and try to understand it in detail.

I said in my previous post:

***************************************
But I do think that you have a similar freedom in FOL to deny the existence of nonstandard natural numbers ...
***************************************

There is a bad typo here. I meant "But I do NOT think that ...".

Matthias said:

***************************************
For me, too, finite means "standard finite". My approach does not
enlarge the syntax of FOL, as e.g. First Order Dynamic Logic mentioned
by Vaughan Pratt, but it is purely model theoretic.
 [...] When I said that the nonstandard numbers are indefinitely large finite
numbers, I mean the following: [...] In short, you have non-standard models also from a potentialist point of
view, but each non-standard number occurs at a specific stage and not at
the limit step (e.g. the union of all finite sets). And since you do not
have a completed, infinite set of formulas (exists m. m = c) and c > n,
where n = 0,1,..., you do not need non-standard numbers in this case.
***************************************

I have read your post several times, but I still have issues.

First of all, as a potentialist you have omitted the limit step that determines the existence of nonstandard numbers in the classical sense. But it seems to me that it is mandatory to demonstrate such existence in a nonstandard model of PA, which is required for the consistency of PA. Note that consistency is a syntactic notion and you have used FOL syntax.

Secondly, you say that c is an indefinitely large finite number, and you insist that "finite" means "standard finite". Yet you seem to assume that c cannot be reached by a counting process. That is, when you refer to "specific stage" in the above quote, why can't that stage be one where your potentially infinite set has grown to size c? The point is that if c is standard finite, then even for an arbitrary c, the stages c, c+1, ... of your construction have to exist.

Thirdly, you do recognize the existence of c as an indefinitely large standard finite number. Even if somehow c is not accessible by a counting process, you still have to answer whether the axiom number (c+1) exists in the infinite list of axioms. If it does exist, it would be the contradiction "Exists m (m = c) and c > c", The proof of this contradiction, which shows that c cannot be a standard finite number, should be acceptable to the potentialist because it uses only standard finitely many sentences out of the infinite set, by your own assumption that c is standard finite.

It seems to me that the correct conclusion from the point of view of FOL is that c is nonstandard finite, and there can be at most c axioms in the above list. Therefore, you have to bite the bullet and accept that there exist nonstandard models for your potential infinity in which numbers like c are treated as finite, and there exist stages c, c+1, etc. in your model construction. This is the only way you can avoid falling foul of FOL theorems like Goedel's incompleteness, compactness and completeness theorems.

I however, insist that there are infinitely many axioms in the above set, which makes c infinitely large. Further, for a finitist, nothing can exceed infinity and hence the existence of c, mandated by the compactness theorem, contradicts the axioms of FOL theories like PA and PRA. Therefore, I have to abandon FOL and move to the paraconsistent logic NAFL.

Regards,
R. Srinivasan