Metalanguage Integers

Question

In context on set theory & model theory of set theory what does exactly mean "metalanguage integer(s)"? Recall a metalanguage ( where one reasons about object theory phrased in object language) is as well a formal language which can can ( eg in ZFC) but not need ( object lan ZFC, metalan homotopy type language) coinside with object language.

When one says "$n$ is a metalanguage integer" or "$\omega$ are metalanguage integers" what is plainly asked concretely meant by this? Can it be formalized in precise terms?

A naive guess: per definitionem a language has an underlying alphabet + syntactic grammar which decides with terms can be syntactically legally constructed from atomic pieces, ie letters of alphabet, predicate symbols, etc. Does it just mean that $n$ and $\omega$ are parts of underlying alphabet- or terms made io atomic symbols - of the metalanguage and that's it?

#EDIT (added later motivated by Mauro ALLEGRANZA's point): Here a consequent question: I heard never the term "object language integers". If my naive guess above is correct and a "metalanguage integer" $n$ ( eg $n=3$ is indeed abbrev of $s(s(s(0)))$) is indeed just a term of atomic objects of metalanguage: then what is the problem with doing the same in object language, ie to construct in analogous manner "object language intergers"? Or, are symbols (regarded as "atomic" pieces) $s, 0$ always only available in the alphabet of metalanguage, but not of object language?

Answer 1

There are three distinct meta's as I have seen being used.

1. Formal metatheory. This situation arises if you are working within some formal object theory, say ZFC, and wish to study the object theory itself (in some sense) from an external perspective. For instance, when working up to forcing you will encounter various theorem schemas which could be encoded as theorems in a formal metatheory, often PA, fragments thereof, or some mixture of a weak set theory and number theory. Note that there is no contradiction in terms even if the formal metatheory is the same as the formal object theory. So both theories could be ZFC.

   If the formal metatheory is PA in the signature $\langle \{0\}, \{', {+}, {\cdot} \}, \{ = \} \rangle$, then the non-negative metaintegers in this sense are (modulo equality) the syntactic expressions $0$, $0'$, $0''$, and so forth. One may define auxilliary expressions for convenience, such as $ 1 :\overset{\cdot}{=} 0'$, etc, with the assumption that such expressions are expanded into their definitions if need be or the signature expanded if useful.

2. Platonistic metatheory. This is a sort of vaguely assumed to "exist" kind of collection of reflexive and intuitive knowledge about some mathematical object. In the case of non-negative integers, this is the theory of $\mathbb{N}_0$. We have the Chinese remainder theorem, infinitude of primes, and various other widely accepted results.

   The nonnegative metaintegers here are the Platonistic non-negative integers. We denote them as $0$, $1$, and so on, but the Platonistic integers are more conceptual objects rather than the syntactical objects.

3. Minimal metatheory. A strictly formal theory still requires setting up. There is no path to a formal theory if we pretended not to understand what syntactic expressions are, how to parse them, what a well-defined formula is, or what constitutes a syntactic proof. We need to be literate. This is where mathematics must touch the physical world, and is a mandatory minimal background in setting up a formal theory.

   In setting up a standard formal theory, one will already have some collection of variables, sentence variables or individual variables (or even functional and predicate variales in higher order logic). In first order theories in particular, one typically assumes a (potentially) countably infinite metacollection of individual variables $(x_0), x_1, x_2, x_3, \dotsc$. Hence one will ask: what are these $0, 1, 2, 3$ in the subindices?

   They are again syntactic objects, and how exactly you set them up is up to you. I like the base one approach, where the integers are sticks. So $1 :\overset{\cdot}{=} |$, $2 :\overset{\cdot}{=} ||$, $3 :\overset{\cdot}{=} |||$.

   I refuse to pretend not to understand when two stacks of sticks are equal (same number of sticks when counted), how to add sticks (write them side-by-side), comparing is allowed (which has more sticks, taking minima-maxima), and so forth. I really treat them as minimal metatheory non-negative integers. Meaning that I do not consider these sticks necessarily to possess the full properties of the Platonistic metaintegers, rather I assume only as much as I need, the bare necessities. If it turns out on page 394 that I need a tad more, I happily pretend to know some extra (obvious) fact about how these sticks behave, and pretend that I had already assumed this additional fact from page 1 if need be.