A question about finitism

Montgomery Link mlink at bu.edu
Mon Feb 20 13:42:55 EST 2023

Dear Professor Gaifman,

Thank you for your Email, and the "rule that generates the sequence," and
thanks to Professor Pratt for the "gapless geodesic."  May I please ask a

In your first Email below you mentioned "questions that arise within the
framework of the potential infinite."  In your second Email, you mentioned
that one might adopt a suggestion of Professor Tait's and take Skolem's
primitive recursive arithmetic as "the theory of potential, non-actual
infinity."  I agree completely with that suggestion and with your response
to Professor Avron. Let us delay for the moment the issue of the explicit
characterization of the potential infinite.

The main issue might be approached along slightly different lines.  Start
with zero and add one, then repeat the addition.  How many repetitions of
that operation are permitted? One might want to restrict the number of
repetitions to what is humanly feasible.  But there seems to be no clear
demarcation for what is feasible and what is not.  There is no stability
here without a clear limit, nothing with which to work. "Later," as it
were, the infinitary stage of the actual finished totality of all the
natural numbers arises. This is not the first or the biggest jump. The big
jump has already been taken once one moves away from any attempt to limit
the feasible. For what is needed is a unit or stable limit to work with,
and that is not provided by the feasible alone.

You mentioned a finitism in your first Email as sequential extensions based
on proper initial segments of the natural numbers. The Aristotelian
questions about the potential v. the actual infinite and the jump from the
potential to the actual, whatever that entails, already presuppose the main
jump.  The biggest jump is from the strictly finite to the finite, where
the difference would be that there is some n>m that is not allowed to
extend m, or there is no such.  In the second case, there is a rule that
generates sequences from m to any n>m.  A long time ago, when Kurt Goedel
and Hao Wang used to talk about this issue, their angle was to delay for
later talk about the potential and the actual infinite.  This was not
merely a division of labor strategy.  The question of the potential versus
the actual infinite does not even arise at first in the sense that it
cannot yet be formulated.  There are so many elegant statements from a
number of sources, but one version on my desk in front of me Wang gives as
follows (LJ, p. 213):

"It is well known, for instance, that both Hilbert's finitism and Brouwer's
intuitionism take this big jump for granted. Indeed, it is only after the
big jump has been made that familiar issues over potential and actual
infinity, construction and description, predicative and impredicative
definitions, countable and uncountable sets, strong axioms of infinity, and
so on arise in their current form."

Enderton's Elements has different kinds of ordinal operations: monotone,
continuous, and normal (p. 216). The jump from a monotone to a continuous
operation is the jump from successor to limit ordinals, but, and I'll say
it once again before closing, the biggest jump has already been taken with
the formalization of monotone operations.

Sorry to go on so long and thank you for your patience.


Montgomery Link

Quick References

Enderton, H.B. Elements of Set Theory (Boston: Academic, 1977).
Wang, H. A Logical Journey (Cambridge: MIT Press, 1996).

On Sun, Feb 12, 2023 at 11:47 PM Haim Gaifman <hg17 at columbia.edu> wrote:

> Dear Arnon,
> You appeal to *understanding—*a  vague, difficult concept.
> Suppose I define a natural number as follows:
> A natural number  is either 0 or any number obtained from 0 by adding  1 a
> finite
> number of times: 0, 0+1, 0+1+1,….
> This  is not  a full definition  since “a finite number of times”
> has not been defined. Nonetheless it provides one with a certain
> understanding:
> Consider the process of adding ones. At each finite stage of this process
> one gets a finite number, *n, *of the form 0+1+1+…+1.
> We might however go further and  imagine what comes after all these
> stages, an *infinitary stage, *at
> which we get the finished totality consisting of all the natural numbers.
> You seem to imply that without taking this last step we do not understand
> what the
> natural numbers are. But this, I claim, is not true. It is merely a matter
> of certain habits. One can understand very well what
> the natural numbers are, without assuming the completed entity  of all the
> natural numbers. It is sufficient
> that we understand the rule that generates the sequence. You can also see
> this  by considering the following theory: Let  ZF be Zermelo Fraenkel set
> theory;
> and let ZF|* be obtained by replacing in it the axiom of infinity by its
> negation (which says that there are no infinite sets).
> This  theory will constitute a rather weak version of finitism. The
> strongest (and much more interesting) version is obtained by adopting
> a suggestion of Bill Tait and using Skolem’s system PRA (Primitive
> Recursive Arithmetic) as the theory of potential, non-actual infinity.
> Best, Haim
> On Feb 4, 2023, at 5:15 PM, Arnon Avron <aa at tauex.tau.ac.il> wrote:
> Dear Haim,
> I was never able to understand the coherence of `finitism'.
> For example, in your first reply to Vaughan you wrote:
> "Only structures based on proper initial segments of the natural numbers:
> {0, 1, 2,…, *m*} are accepted as legitimate, but for every *m, *if *n* >
> *m*, one accepts also the extension based on {0, 1, …., *m*, *m*+1,…., *n*
> }."
> First question: at least to me it seems that if one understands that X is
> an initial segment of the natural numbers, it means that somehow
>    he understands that there is something that X s an initial segment of,
> so he understands that there is the  collection of the natural numbers.
>    So why pretending not to understand that collection?
> Second question: you explicitly wrote that "*for every m*, if n>m ...".
> But if I understood you correctly (almost certainly I did not) a finitist
> is not allowed
>    to make claims about *every m*!
> And the final question: is there any way for a finitist to explain his
> principles (even to himself!)
>    without violating these principles? I doubt it...
> Best regards,
> Arnon
> ------------------------------
> *From:* FOM <fom-bounces at cs.nyu.edu> on behalf of Haim Gaifman <
> hg17 at columbia.edu>
> *Sent:* Sunday, January 29, 2023 5:03 AM
> *To:* Vaughan Pratt <pratt at cs.stanford.edu>
> *Cc:* fom at cs.nyu.edu <fom at cs.nyu.edu>
> *Subject:* Re: Re: Mathematics with the potential infinite
> Dear Vaughan,
> Long time no hear no see, and it is very nice to hear from you.
> The restriction of subscribing only to potential infinities (which can be
> traced back to Aristoteles) is Hilbert’s so called *finitist* position;
> Abraham Robinson agrees with him. Only structures based on proper initial
> segments of the natural numbers: {0, 1, 2,…, *m*} are accepted as
> legitimate, but for every *m, *if *n* > *m*, one accepts also the
> extension based on {0, 1, …., *m*, *m*+1,…., *n*}. The functions and/or
> relations that come with these structures are the usual functions and/or
> relations of PA (Peano Arithmetic). Of course, the functions are partial
> functions  , given the restrictions on the domain.
> PA, which is based on the standard model *N *of natural numbers, is much
> much… stronger than the theories
> that arise within the framework of potential infinity.
> One such interesting theory has been proposed by Skolem
> and is known as PRA for Primitive Recursive Arithmetic.
> Now your question, if I understand you correctly, asks for a way of
> describing an uncountable structure using only potential infinities.This
> would be impossible, unless you allow countable non-standard model for the
> theory linearly ordered groups.
> Best, Haim Gaifman
> On Jan 28, 2023, at 3:13 AM, Vaughan Pratt <pratt at cs.stanford.edu> wrote:
> My apologies for not having previously followed threads on this topic.
> However after seeing Stephen Simpson's message just now (Friday) it
> occurred to me to ask whether an uncountable set could be described using
> only potential infinities, for example the real numbers (R, *, 0, <=) as a
> linearly ordered group under addition, compatibly ordered in the sense that
> each of the group multiplication's two arguments is monotone: if x <= y
> then x*z <= y*z, and likewise for the right argument.  (* = +.)
> Define a *geodesic* to be a nondegenerate linearly ordered group (G, *, 0,
> <=).  (Although G is not assumed abelian, the linear order makes it
> abelian.)  Examples include the integers, the dyadic rationals,  every
> field between the rationals and the reals, and many non-Archimedean
> extensions thereof.
> Call a geodesic G *gapless* when (i) it is dense, and (ii) for every
> nonempty suborder (U, <=) of (G, <=) having no least element, and every
> nonempty suborder (L, <=) of (G, <=) with L < U and having no greatest
> element, such that there is at most one element of G between L and U; then
> there exists an element of G between L and U.
> I claim that every gapless geodesic is isomorphic to R with the above
> structure.
> (Proof outline: Take any element x of G with 0 < x and pair 0 and x with 0
> and 1 in R.  Pair the integers in R with the subgroup of G generated by x,
> cyclic and therefore abelian.  Repeatedly divide the intervals in (n, n+1)
> in G into two equal parts and pair the results with the dyadic rationals in
> (0,1), a dense set.  Pair each dyadic irrational q in R with the unique x
> given by the gaplessness condition for any L and U in G whose counterpart
> in R converges to q from each side.  Lastly, G must be Archimedean or there
> would be an empty gap between the finite and infinite elements of G.)
> 1.  Do these definitions, claims, and constructions meet the criteria for
> only potential infinities?
> 2.  Can R be shown to be uncountable using only potential infinities?
> (Those familiar with Otto Hoelder's 1901 paper showing that every
> Archimedean linearly ordered group is isomorphic to some subgroup of R
> under addition, which may be anywhere between Z and R, may see some
> similarity of ideas in the above.)
> Vaughan Pratt
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