[FOM] Sharp mathematical distinction between potential and actualinfinity?

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Mon Sep 29 16:06:41 EDT 2003

"Timothy Y. Chow" wrote:
> Years ago, Stephen Simpson proposed a list of "the most basic mathematical
> concepts," which included number, set, function, etc.  I wonder if
> "infinity" belongs on this list?
> It seems to me that at minimum, "potential infinity" of some sort is basic
> to mathematics.  Consider the following caricature of a debate that has
> been rehearsed innumerable times on the FOM list and elsewhere:

Although this is a "caricature", it is somewhat misleading. 

> P: The concept of N, the set of all natural numbers, is crystal-clear.
> F: No it isn't.  For example, first-order PA has nonstandard models.

The point was that there is no such an ABSOLUTE concept as 
"the standard model" of PA at all. 

> P: Huh?  By appealing to mathematical logic, you betray your belief
>    that the concept of a *rule* is clear, and N is just as clear as
>    the concept of a rule.
> F: Nonsense.  A rule is finite, or at most potentially infinite, but
>    N is a completed infinity.

The point was that a rule or a formalism or a deduction is not 
only finite, but feasible (physically presentable). (It does not 
matter here whether N is thought of as completed or only potential 
infinity.) Formalisms and the intuitive, imaginary worlds like N 
which they "describe" have completely different status if we are 
not considering formalisms metamathematically by including them 
(by Goedel numbering) into, say PA. We should be quite explicit 
about this if we do not want to go into a vicious circle. 
Additional essential point is that WE DO NOT NEED any theory of 
formal systems to use their rules: we only need to be well trained 
for that. (Likewise, we do not need any theory of bicycles to ride.) 
These notes allows us to avoid any serious infinite regress in 
understanding the nature of mathematical rigorous reasoning. 

What is interesting, that the above caricature so strongly 
misrepresent the views of F only. Caricature should rather 
strengthen some features, may be up to absurd. That would 
be normal. 

> I confess that my sympathies lie with P here; I tend to think that
> if we take skepticism towards N seriously, then we also need to take
> "Kripkensteinian" skepticism towards rules seriously, and this leads
> quickly to a wildly, and in my opinion unacceptably, ultraskeptical
> view of virtually all mathematics and logic.

I do not know what is "Kripkensteinian" skepticism. The views 
outlined above have nothing ultraskeptical. We just should be 
careful enough to not mix everything in one big heap where 
we would lost ourselves forever. Is this skepticism or a request 
to be sufficiently precise? 

Skepticism is rather an antipode of a belief. But should science 
be based on beliefs at all? Say, in physics we have some experiments 
confirming a law. We can say that this gives us the right to believe 
in this law. But more strictly speaking, this means that this is a 
good reason to include this law in corresponding physical theory. 
Thus, it is not a belief at all. (Will we pray to this law?) 
It is rather a hope that this law is reliable. If some couterexample 
will be found, there will be no tragedy concerning losted beliefs, 
only a problem how to correct the theory. "Belief" is rather 
a figure of speach. We can use it, but recall what is really 
staying behind this word. 

> However, I might change my mind if someone could demonstrate a sharp
> distinction between potential and actual infinity.  The distinction
> seems to have pretty much evaporated in modern mathematics, and it
> seems that only the philosophers still talk about it.  Or am I just
> underinformed?  Is there, for example, a way of drawing a clear
> mathematical distinction between potential and actual infinity that
> blocks the move from skepticism-towards-N to skepticism-towards-rules?

The last line is unclear to me. However, see the above notes. 

I do not hope on any possibility of MATHEMATICAL distinction 
between potential and actual infinity. Mathematical means to 
give a formal definition in a fixed formal system. These ideas 
seem too general for that. I hope only on some clarification. 

>From my point of view, potential infinity is much less clear 
that the actual one. It consists in our (intuitive, imaginary) 
ability to always add 1 to any natural number. Also it consists 
in our (imaginary) ability to iterate the previous ability. 
Say, the first iteration of the successor operation gives rise 
to the addition operation x + y. Iterating addition gives 
multiplication. Then we can come to the ability to multiply, 
to take exponential, any primitive recursion, then Ackermann 
function appears, and so on. The main problem is with this 
"and so on". We usually slurring over this question. But this 
is really the most problematic question on potential infinity 
(called also potential feasibility). We could fix this "and so on" 
by fixing a (quantifier-free) formalism like PRA (where already 
Ackermann function is not presented). But I cannot imagine how to 
grasp this "and so on" in its "complete" strength. I even do not 
understand what does it mean. 

By the way, these considerations already lead us to the idea that 
(potential infinite) N is something indefinite. We cannot tell 
how "long" it is. Thus, PRA does not guarantee its closure under 
the Ackermann function whereas PA guarantees. Everything depends 
on the formalism considered. Moreover, all the above informal 
considerations on potential infinity are possible ONLY when we 
already have some formalisms, each on "an" (not "the") N. 
Otherwise we would have no way even to discuss on N in any 
non-trivial way (mentioning exponential, etc.). 

Taking into account that natural numbers in mathematics are 
ALWAYS relativized to a (semi)formal theory, we should conclude 
we have no sufficient reason to take the general colloquial term 
(natural) "numbers" as mathematically meaningful. It rather denotes 
a vague concept in our discussions around mathematics. 

On the other hand, actual infinity arises already in Peano Arithmetic 
where quantification over "all" natural numbers is allowed together 
with the classical logic. Intuitionists and constructivists also 
allow quantification over N understood as potentially infinite. 
But this is rather subtle point. I am not sure that they do not 
come to actual infinity here. If our intuition allows us to 
confirm that they "really" use only potential infinity, then 
the well-known reduction of clasical PA to intuitionistic or 
constructive theories of N would "eliminate" in a sense the 
actual infinity from PA. 

Actual infinity arises even more explicitly when we consider N as 
a "first class citizen" of set theory. This seems clear. I see 
potential infinity much more problematic. It arises also in set 
theory when we consider, say, the class of "all" ordinals or 
cardinals. But I, personally, would rather concentrate on potential 
infinity of N. 

Vladimir Sazonov

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