Differences between potentially infinite and infinite

[aostraff]

A few weeks ago, I was reading "The Philosophy of Set Theory" by Mary Tiles and I just wanted to clarify on the definitions of potential infinite and actual infinite, and their differences. I am guessing there will be different definitions for them, but anything will make them easier to understand. Thanks.

 

[Hurkyl]

I've bumped this over to the philosophy forum, since (to the best of my knowledge) these are philosophical terms, not mathematical ones.

My impression of these terms is that "potentially infinite" comes up when you deal with a sequence of "finite" approximations to some "infinite" idea. For example, rather than thinking of a decimal numeral as an object unto itself, you can instead imagine a process that iteratively writes its decimal digits one-by-one. I think that one would say that the decimal numeral has actually infinitely many digits, whereas the process outputs potentially infinitely many digits.


One thing to be very careful of is not to confuse the individual terms of the approximation with the process itself (or with the thing being approximated). For example, while each of the terms of the approximation

0.9, 0.99, 0.999, 0.9999, ...​

has finite length and is less than 1, the "process" itself represents a number exactly equal to 1, and the "process" itself is not finite in duration. (But it does have a finite description) I've seen it asserted that confusing the process with the individual terms is one of the major sources of confusion regarding the fact that 1=0.999...


However, to my eye, many uses of "potentially infinite" seem to be really dealing with the idea of indeterminate variables, despite never actually being phrased that way.

 

[aostraff]

Intriguing. From your view (Hurkyl), isn't actual infinite more ambiguous that potential infinite? I'm not sure if these two terms can even be compared with much meaning but it seems like actual infinite and potential infinite have a relationship analogous to 'past tense' and 'future tense'. I wonder if I'm even on the right track anymore.

 

[Hurkyl]

As I've seen them used, "potentially infinite" and "actually infinite" are intended to be antonyms: an infinite object is either potential or actual. So if one is ambiguous, then the other is equally ambiguous.



However, in mathematics, "infinite" is just like any other mathematical notion: in whatever context you're discussing, it comes with a completely unambiguous definition. In the usual forms of set theory, that completely unambiguous definition for the term "infinite set" is "there does not exist a bijection between the set and an initial segment of the natural numbers".

If there is a bijection, then your set is finite. If there is not a bijection, then your set is infinite. If you're using some strange logic, and assign some nonclassical truth value to the statement "there is not a bijection", then the statement "your set is infinite" has the exact same truth value.

 

[Werg22]

Wasn't the "potential infinite" view of mathematics championed by Poincaré? To my understanding Poincaré opposed the philosophical stance than mathematical objects exist statically and prior to being constructed. For Poincaré, there is no circle at a particular point with a particular radius until we have constructed one. I think language like "the set of all real numbers" would be absurd to him.

 

[csprof2000]

I tend to agree with Werg22.

I also think that constructive proofs are more satisfying than non-constructive ones, even when the constructive proofs seem more tedious or less elegant.

Potential vs Actual infinity? Well... it's not something mathematics generally concerns itself with. It used to be a big deal, but it's sort of swept under the rug now. You can avoid problems like that in most cases by treating actual infinities as complete and potential infinities as processes. It's only when you invert these two that you start getting into problems...

Sort of like the sum of an infinite series being the limit of its partial sums... the infinite series is an actual infinity, and that's what you're trying to find and make static (say that it's some number). The right hand side is a potential infinity... the sum is never really equal to the actual infinity on the left, but tends to it. I actually write my infinite summations with the limit notation, but most people find this a little arcane... and in general the notation is abused to allow things like

2 = sum of, n from 0 to infinity, (1/2)^n.

I guess it's not that bad, though, since a series is already taken to be the limit of partial sums, but I digress.

I guess potential and actual infinity differ in just the way you'd think: potential infinities could go on forever (but don't... or they're still going on) and actual infinity is already done (if that even makes sense...).

 

[aostraff]

The view of potential infinite as a process of 'getting to actual infinite' that I am getting from your posts raises another question. The discrepancy of these infinites makes it seems quite subtle, but I have a feeling that they have a substantial difference when it comes down to dealing with paradoxical statements involved with motion, sort of like Zeno's Paradoxes. No?

 

[Hurkyl]

Sort of like the sum of an infinite series being the limit of its partial sums... the infinite series is an actual infinity, and that's what you're trying to find and make static (say that it's some number). The right hand side is a potential infinity... the sum is never really equal to the actual infinity on the left, but tends to it.

Ack no! You're confusing the object with a sequence of approximations to an object! The sum of an infinite series is just a number; it's not something that can "tend". It's the sequence of partial sums that "tends" to something.

(At least, if you are using the words like "sum" or "limit" in a fashion resembling what people usually mean by those words. )

 

[Hurkyl]

I think language like "the set of all real numbers" would be absurd to him.

It's interesting, though, because you don't really need set theory at all to talk about the idea. If you have a logical^* proposition P that has the interpretation "P(x) iff x is a real number", then working with P is as good as working with the set of all real numbers.

(Of course, if you don't have such a P, then you can't really talk about the notion of 'real number', can you. )

*: Note that I'm not even requiring classical logic. It's fine to use nonclassical^** intuitionist logic, or even some Turing-machine based.

**: I clarify, because classical logic is an intuitionistic logic.

 

[Werg22]

Hurkyl, I see.

The trouble that I see with views like or similar to Poincaré's is that there is a huge burden to clarify identity. If there is no circle at the point (0,0) of radius 1, then we construct one, can we construct another one? Is the second the same as the first? In modern mathematical language, they are equal in the sense of identity. But to Poincaré, they might only be equal because of an equivalence relation. Similarly, if the values of a function f and a function g are the same at every element in their common range, are they equal in the sense of identity, or are they equal in a lesser sense?

Seems to complicate the whole procedure quite a bit, and I don't imagine there are logical inconsistencies in conducting mathematical discourse as if all constructed things exist a priori.