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> The boundary does not pass between some huge finite number and the next, infinitely large one. Rather, it separates two kinds of mathematical statements: “fin

> The boundary does not pass between some huge finite number and the next, infinitely large one. Rather, it separates two kinds of mathematical statements: “finitistic” ones, which can be proved without invoking the concept of infinity, and “infinitistic” ones, which rest on the assumption — not evident in nature — that infinite objects exist. Let Y = some ridiculously huge finite number. The notion that ∞ = Y + 1 has to be one of the most insidious defects in reasoning. Even to acknowledge the defect by dismissing it is almost like saying things could have turned out that way when they _never_ could. Finite means limited or bounded or quantifiable. Infinite means unlimited or unbounded or unquantifiable. They are two wholly separate classes of things. Another way to say this is discrete versus continuous (in nature). Imagine I said that I could bridge the coloured/non-coloured divide. What could I possibly mean by that. It would mean that I have found a way to talk about both coloured things and non-coloured things that applies to both class of things. Put another way. Imagine I separate coloured things into green ones and not green ones. Both subclasses belong to the class of coloured things. What class then do the subclasses of finite things and non-finite things belong to? Negation constructs the distinction, negation _is_ the bridge in a way and all the could be said of both subclasses of things is that they are both things. (Which, to be clear, is not saying very much at all, is it?) If those things are numbers then all we are saying is that both things are numbers, if both things are procedures then all we are saying is that both things are procedures. And so on. If I'm thinking about this properly then the paper is trying to more precisely define the term "countably". That's as most as it can do. To say otherwise is not just vastly overstating what the paper is about but outright misleading.

Smaug123 10y ago
You're not thinking about this properly. There are some facts about finite objects which simply cannot be proved without reasoning about infinite objects - for example, the well-definedness of the TREE function. These facts are true if you have access to infinite objects, but not necessarily if you don't. This paper shows that a certain class of statements about finite objects, which we knew were true by virtue of reasoning about a certain infinite object, in fact remain true if you're restricted only to reasoning about finite objects.
- igravious 10y ago
  I tried to give my reasons by using only appeals to objects of our common understanding. You have countered my claim by appealing to "the well-definedness of the TREE function". Do you expect me (and everyone else) to know what that is? In order for us to be able to follow your argument you would need to spell that out in a bit more detail. If I'm not thinking about this properly, which of the assertions I made was incorrect? Where is the error in my reasoning? Is there an error in how I am conceiving things? > This paper shows that a certain class of statements about finite objects From what I can gather, it's not that simple. For a start, the initial set is the set of natural numbers. This is not finite. The procedure for generating/enumerating them is. The paper deals with pairs of inequalities based on the natural numbers and sub-sequences to be found therein. This set of pairs is also non-finite (but I'm happy with the assertion that in some sense it is a different order of infinity from the natural numbers). We are now trying to reason about the nature of the sequencing of these sub-sequences. Are you saying that sometimes the sub-sequences are finite? If so, their complement would be infinite. And it is the partitioning that makes this so.
  - Smaug123 10y ago
    Your understanding of the point of the theorem is very different to mine, and I'm moderately sure my understanding is pretty close to correct. It is a fact of mathematics that there are some statements which are solely about finite objects, but to prove them requires reasoning about an infinite object. For a more accessible example than TREE, I think the Ackermann function falls into this category. The Ackermann function A(n+1, m+1) = A(n, A(n+1, m)) is well-defined for all n and m (we prove this by induction over NxN), but the proof relies on considering the lexicographic order on NxN which is inherently infinite. (I'm not totally certain that all proofs of Ackermann's well-definedness rely on an infinite object, but the only proof known to me does.) Ackermann's function itself is in some sense a "finite" object, but the proof of its well-definedness is in some sense "infinite". Whatever the status of my conjecture that "you can't prove that Ackermann's function is well-defined without considering an infinite object", it is certainly a fact that Ackermann is not primitive-recursive, and "primitive-recursive functions" corresponds to the lowest level of the five "mysterious levels" the article talks about. So the analogy is as follows. Imagine that we knew of this "infinitary" proof that Ackermann is well-defined, but we hadn't proved that no "finitary" proof exists. (So finitists are not happy to use Ackermann, because it might not actually be well-defined according to them: any known proof requires dealing with an infinite object.) Now, this paper comes along and proves that actually a finitary proof exists. Suddenly the finitists are happy to use the Ackermann function. The actual definition of TREE is a bit too long for me to explain here, but it is an example of a function like Ackermann, which is well-defined, but in fact if you're not allowed to consider infinite objects during the proof then it is provably impossible to prove that TREE is well-defined. So the statement "TREE is well-defined" is, in some sense, "less constructive" or "more infinitary" than R_2^2.

Smaug123 10y ago

You're not thinking about this properly. There are some facts about finite objects which simply cannot be proved without reasoning about infinite objects - for example, the well-definedness of the TREE function. These facts are true if you have access to infinite objects, but not necessarily if you don't. This paper shows that a certain class of statements about finite objects, which we knew were true by virtue of reasoning about a certain infinite object, in fact remain true if you're restricted only to reasoning about finite objects.

igravious 10y ago
I tried to give my reasons by using only appeals to objects of our common understanding. You have countered my claim by appealing to "the well-definedness of the TREE function". Do you expect me (and everyone else) to know what that is? In order for us to be able to follow your argument you would need to spell that out in a bit more detail. If I'm not thinking about this properly, which of the assertions I made was incorrect? Where is the error in my reasoning? Is there an error in how I am conceiving things? > This paper shows that a certain class of statements about finite objects From what I can gather, it's not that simple. For a start, the initial set is the set of natural numbers. This is not finite. The procedure for generating/enumerating them is. The paper deals with pairs of inequalities based on the natural numbers and sub-sequences to be found therein. This set of pairs is also non-finite (but I'm happy with the assertion that in some sense it is a different order of infinity from the natural numbers). We are now trying to reason about the nature of the sequencing of these sub-sequences. Are you saying that sometimes the sub-sequences are finite? If so, their complement would be infinite. And it is the partitioning that makes this so.
- Smaug123 10y ago
  Your understanding of the point of the theorem is very different to mine, and I'm moderately sure my understanding is pretty close to correct. It is a fact of mathematics that there are some statements which are solely about finite objects, but to prove them requires reasoning about an infinite object. For a more accessible example than TREE, I think the Ackermann function falls into this category. The Ackermann function A(n+1, m+1) = A(n, A(n+1, m)) is well-defined for all n and m (we prove this by induction over NxN), but the proof relies on considering the lexicographic order on NxN which is inherently infinite. (I'm not totally certain that all proofs of Ackermann's well-definedness rely on an infinite object, but the only proof known to me does.) Ackermann's function itself is in some sense a "finite" object, but the proof of its well-definedness is in some sense "infinite". Whatever the status of my conjecture that "you can't prove that Ackermann's function is well-defined without considering an infinite object", it is certainly a fact that Ackermann is not primitive-recursive, and "primitive-recursive functions" corresponds to the lowest level of the five "mysterious levels" the article talks about. So the analogy is as follows. Imagine that we knew of this "infinitary" proof that Ackermann is well-defined, but we hadn't proved that no "finitary" proof exists. (So finitists are not happy to use Ackermann, because it might not actually be well-defined according to them: any known proof requires dealing with an infinite object.) Now, this paper comes along and proves that actually a finitary proof exists. Suddenly the finitists are happy to use the Ackermann function. The actual definition of TREE is a bit too long for me to explain here, but it is an example of a function like Ackermann, which is well-defined, but in fact if you're not allowed to consider infinite objects during the proof then it is provably impossible to prove that TREE is well-defined. So the statement "TREE is well-defined" is, in some sense, "less constructive" or "more infinitary" than R_2^2.