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I would like to ask for literature recommendations on foundations of set theory focusing on the treatment of the concepts & the nature of metalanguage & metatheory, so the language&theory used to reason about the object theory. Especially, to what extent can one treat it or parts of it with formal rigor as in the case of formal language & theory? Unfortunately, most books on foundations of set theory sweep these aspects benevolently under the rug, and this raises the question at which "level of formalness" one should actually treat metalanguage & metatheory and if there are some works addressing this issue.

Motivation: In discussion here with Mikhail Katz I learned that usually metalanguage & metatheory are usually considered less formally then the object theory, which we treat in full formal rigorosity, ie we can pin down the the formal system associated with it, ie the formal language in which its sentences are phrased, we have syntax rules, deductive apparatus/rules of inference, etc which actually precisely dictate what we can do and what not as long as working with formal theory. Attempting that same approach to meta things - i.e. just to regard metatheory & metalanguage as another formal theory & language used to make statements about the object theory - seemingly is not so easy to establish.

One of the example indicating that there are seemingly fundamental differences between metatheory and object theory from level of formalness is eg the concept of "metalanguage integers". It appears as an intrinsic object of metatheory which seemingly has no analogon in object theory.
Recall, that object theory consists of formulae in certain formal language. If our formal theory describes set theory then it should have a formula "there exist a unique inductive set". Then, in every model of this theory there exist a set witnessing the truth of this formula, which we would call "internal integers". But note that these integers live in a fixed model; it doesn't make sense to say that the theory itself has integers.
In contrast in metalanguage one can talk about "metalanguage integers" indicating that metalanguage & metatheory happen to have less formal character.

Mikhail Katz
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user267839
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1 Answers1

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There is an article by Nelson that makes some related comments in the context of the distinction between potential and completed infinity:

Nelson, Edward. Hilbert's mistake. 2007

Here Nelson writes:

Let us distinguish between the genetic, in the dictionary sense of pertaining to origins, and the formal. Numerals (terms containing only the unary function symbol S and the constant 0) are genetic; they are formed by human activity. All of mathematical activity is genetic, though the subject matter is formal.

Numerals constitute a potential infinity. Given any numeral, we can construct a new numeral by prefixing it with S.

Now imagine this potential infinity to be completed. Imagine the inexhaustible process of constructing numerals somehow to have been finished, and call the result the set of all numbers, denoted by N.

Thus N is thought to be an actual infinity or a completed infinity. This is curious terminology, since the etymology of “infinite” is “not finished”.

Notice Nelson's reference to "human activity" in describing numerals (which seem to be metalanguage integers).

The most detailed discussion of metalanguage integers that I know of is in Kuhlemann's recent book. Unfortunately this Google Book version does not provide access to pages where metalanguage is discussed.

Mikhail Katz
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  • Thanks that's quite interesting. Have to admit to have some troubles with understanding the "Now imagine this ..." paragraph. Does Nelson make precise what it should mean to "complete" the potential infinity. Similarly, what should it actually mean to "finish" the construction of numerals? Isn't "constructing by human activity" an "finite process" at all? So I'm not sure what Nelson actually refers to in that paragraph. Could you suggest some intuition for this part? – user267839 Sep 17 '24 at 08:46
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    Nelson is taking the position in favor of potential infinity and critical of actual/completed infinity. The latter is so familiar to everybody (almost) from kindergarten that is hard to imagine what the difference is between them. Nelson's take is to relate potential infinity to numerals (which I understand to be metalanguage integers, or else their object-language counterparts), and completed infinity to the infinite totality $\mathbb N$ as formally understood. – Mikhail Katz Sep 17 '24 at 09:59
  • I'm not sure if that hits the nail concerning the difference between potential and actual infinities but can this difference be "intuitively" grasped by means of a thought experiment: Say there is some abstract time dependent process going on and one asks what it actually means to "understand what is going on there"? One could try to answer it in two naive ways which appear to me to be rather similar to this potential vs actual infinity dialectic: One can say one "understands" the process when one "knows" what happens "at the end" - so exhibit a single "limitic" datum - or at least – user267839 Sep 19 '24 at 17:48
  • to believe that it is possible to exhibit such datum.

    But on the other hand one could claim "one understands the process" if one is able to response to any instant query what the process is doing at the moment the query is posed witout beeing capable to answer what it become once in future.

     – user267839 Sep 19 '24 at 17:48
  • Can this analogy be used to reflect the difference between actual and potential infinities or does it missing the point? – user267839 Sep 19 '24 at 17:53
  • A nitpick concerning the concept of numerals in general you are refering to: So far I understand these are just the terms of form $S(S(S(...(0))))$. How do you deal with "circularity issues" for numerals? Or, would you argue that you treat numerals as a meta concept, so circularity problems can be "sweeped at meta level under the rug"? – user267839 Sep 20 '24 at 18:22
  • I think we have been through this one before. Using the ellipsis $\ldots$ raises all sorts of red flags because such notation is ambiguous. And yes, I do believe that Nelson uses numeral in the sense of metalanguage integer. – Mikhail Katz Sep 22 '24 at 10:02
  • Yes, I remember, the ellipsic expressions should be as well avoided even at meta language level due to the unavoidable ambuguities as you emphasised there. And, as you also wrote there the metalanguage integers can be thought as collection possessing sorites-like properties. When you say "collection" to distinguish it from "set" you intend to emphasise by this that we cannot find a "distinguishing predicate" determining all members,right? We can only say that eg a picked term of form S(S(S(S(0))) is a metalanguage integer/ numeral, and thats it, but not characterize all? – user267839 Sep 22 '24 at 18:00
  • More concretely what I not completely got from the mentioned discussion is do you allow circularity phenomena on metalanguage level? Namely, as you wrote there, on level of object theory & formal integers to define latter via "there exist a n with $S^{\circ n}$" is clearly circular and so cannot be used to define formal integers. But, do you allow terms on metalevel like $S^{\circ n}$ for n metalanguage integer\numeral? Of course, this would again lead to circularity if we would try to declare numerals in that way. So the question is, if in realm of meta language you allow circularities? – user267839 Sep 22 '24 at 18:16
  • By the way a different point: does the presented 'thought experiment' from comment #3 captures more or less acurately on informal level correctly the "gist" of difference between potential vs actual infinities in Aristotele's resp Nelson's sense? Or is this picture flawed? – user267839 Sep 22 '24 at 18:17
  • @user267839 This seems to be related to recursive stuff and CS, but I am not too familiar with that. By the way, you may find reading Formalism 64 enlightening. – Mikhail Katz Sep 23 '24 at 09:56
  • You are refering to the suggested "thought expleriment"? Yes, was just an ad hoc idea, let just ignore it. What I would still like to understand is what I adressed in "Yes, I remember..." comment. You wrote in that conversation that reasonings in meta language should not tend to keep too formal - in contrast to the work inside object theory. Therefore I'm wondering - do you allow circular constructions inside metalanguage? The "running example" is this flawed "definition" of integers via term $S^{\circ n}$ which to formal theories is clearly invalid is would fall into circularity – user267839 Sep 24 '24 at 09:08
  • By "circular instructions" do you mean recursive definitions? This is addressed to a certain extent in Robinson's text Formalism 64. It's a haunting text and I can't say I understand it fully. By the way notice this update with a Robinson quote: https://math.stackexchange.com/questions/2801663/minimal-requirements-for-platonist-views-of-the-standard-model-of-set-theory-lea – Mikhail Katz Sep 24 '24 at 09:11
  • [cont] as if we want to make expression $S^{\circ n}$ formal, we would need to know where arguments $n$ live, but in turn we would use $S^{\circ n}$ to define this, and that's the circularity. So for formal theories that's invalid. But my question is if such circular constructions/ definitions are "tolerated" for meta language? Eg when dealing with numerals/ metalanguage integers, can we "allow" there the term $S^{\circ n}$ where $n$ numeral even this also by same reasoning would be obviously circular. – user267839 Sep 24 '24 at 09:14
  • Well defining the $S^{\circ n}$ would necessarily be a recursive definition, so in that sense what you seem to be asking is whether recursive definitions are allowed at the meta level. Seems risky to me. – Mikhail Katz Sep 24 '24 at 09:17
  • Ad "By "circular instructions" do you mean recursive definitions?" I was inspired at that point by the concept like of "lazy evaluation". Say we want hypothetically define something like numerals naively in meta language. We have a $0$ and seccessor fct. Then we just write down the - in strictly formal framework nonsensic - object $Nu:={S^{\circ n}(0) \ \vert \ n \in Nu }$. Clearly, in formal sense its highly illegal due to circularity reasons. But my concern was if in meta framework - due to its less strict formalness - this still – user267839 Sep 24 '24 at 09:53
  • can be "tolerated" if we refer to this lazy evaluation principle (which is clearly spongly, but that's why I appeal to less formalness in meta framework). The lazy evaluation would say that although $Nu:={S^{\circ n}(0) \ \vert \ n \in Nu }$ ("wanna be numerals") is circular as $S$ would be a function on an object for which definition it is actually used to - but at every "call" of $S$ to define next member - say $S(n)=S^{\circ n}(0)$ we don't actually need to know how $S$ is defined on $Nu$ - which we haven't still completely constructed - but only on – user267839 Sep 24 '24 at 09:54
  • predecessors of $n$, so we actually at this stage make "lazy" usage only of $S \vert _{\le n}$ which we at this stage already know. – user267839 Sep 24 '24 at 09:55
  • Hm, as you said that looks risky, that's why my retrat to meta framework as I don't think that formal theories would allow such lazy evaluation constructions. But was curious to which amount this "makes sense" at meta level. – user267839 Sep 24 '24 at 09:55
  • I don't really know much about programming languages so I can't comment meaningfully. – Mikhail Katz Sep 24 '24 at 09:57