There are many well-known equivalences in reverse mathematics between statements of the form "Every set is contained a countable coded $\omega$-model of $T$" and $S$, where $S, T$ are subsystems of second order arithmetic and $T < S$ in the usual hierarchies. For example:


* $\mathsf{ACA}\_0^+$ is equivalent over $\mathsf{RCA}\_0$ to "Every set is contained a countable coded $\omega$-model of $\mathsf{ACA}$". This is theorem 1.7(i) of Rathjen [2012]; the proof states it follows from lemma 3.4 of Afshari and Rathjen [2009].
* $\mathsf{ATR}\_0$ is equivalent over $\mathsf{RCA}\_0$ to "Every set is contained a countable coded $\omega$-model of $\Delta^1\_1\text{-}\mathsf{CA}$" (or $\Sigma^1\_1\text{-}\mathsf{DC}$). This is theorem 1.7(ii) of Rathjen [2012] and is given as following from lemma VIII.4.19 of Simpson's book.
* $\Pi^1\_1\text{-}\mathsf{CA}\_0$ is equivalent over $\mathsf{ACA}\_0$ to "Every set is contained in a countable coded $\beta$-model". This is in Simpson's book as theorem VII.2.10 and doesn't have a prior attribution associated.


(I have left out results that concern $\beta\_n$-models and stronger systems. If there are other equivalences below the strength of $\Pi^1\_1\text{-}\mathsf{CA}\_0$ I'd like to hear of them.)


There is an obvious omission from this list, namely that $\mathsf{ACA}\_0$ is equivalent over $\mathsf{RCA}\_0$ to the statement "Every set is contained in a countable coded $\omega$-model of $\mathsf{RCA}\_0$".


In some current work I need a similar lemma, and would like to properly attribute the result. However, I haven't been able to track one down, or indeed find anywhere that this theorem is written down. My knowledge of the literature is not as extensive as it could be, so I might just have missed it, in which case a citation would be very much appreciated. Alternatively, perhaps it is simply too obvious a fact for anyone to have bothered. In that case it would be helpful to know, so I could simply note it as folklore.


Relatedly, is this case like that of $\Pi^1\_1\text{-}\mathsf{CA}\_0$ in the list above, in that we can drop the requirement that the $\omega$-model satisfies any particular theory? (Obviously there are some delicacies here regarding the precise statement of the equivalence.)




---


This question was inspired by a talk of Michael Rathjen on well-ordering principles and $\omega$-models, as well as François Dorais's [blog post](http://dorais.org/archives/1080) which mentions theorems of this sort.


[Afshari and Rathjen 2009] B. Afshari and M. Rathjen: Reverse Mathematics and Well-ordering Principles: A pilot study, Annals of Pure and Applied Logic 160 (2009) 231-237.


[Rathjen 2012] M. Rathjen. $\omega$-models and well-ordering principles. In N. Tennant, editor, Foundational Adventures: Essays in Honor of Harvey M. Friedman. College Publications, 2012.



Attribution of an equivalence of the existence of omega-models of RCA0
There are a lot of subtleties here. For technical reasons, I'll use $\mathsf{WKL}\_0$ instead of $\mathsf{RCA}\_0$ to explain them.


The following two theorems are found in Simpson's book:


Theorem VIII.2.6. *The following is provable in $\mathsf{WKL}\_0$. For all $X \subseteq \mathbb{N}$, there exists a countable coded strict $\beta$-model $M$ such that $X \in M$.*


Theorem VIII.2.11. *The following is provable in $\mathsf{ACA}\_0$. For all $X \subseteq \mathbb{N}$, there exists a countable $\omega$-model $M$ of $\mathsf{WKL}\_0$ such that $X \in M$.*


By Theorem VIII.2.2 countable coded strict $\beta$-model is a countable coded $\omega$-model which satisfies $\mathsf{WKL}\_0$. In Theorem VIII.2.6, it is true that the $\omega$-model satisfies $\mathsf{WKL}\_0$ but this is not provable in $\mathsf{WKL}\_0$. Indeed, $\mathsf{WKL}\_0$ is too weak to properly make sense of "satisfies" in this context.


Because of the subtleties I outlined in my blog post (see also this [MO answer by Carl Mummert](https://mathoverflow.net/a/159845)), the statement "every set is contained in a countable coded $\omega$-model of $\mathsf{WKL}\_0$" has two possible meanings. This leads to a conundrum:


* If you interpret "satisfies" using the sort of translation as I outline at the end of my blog post, "every set is contained in a countable coded $\omega$-model of $\mathsf{WKL}\_0$" is actually equivalent to $\mathsf{WKL}\_0$.
* If you interpret "satisfies" using valuations as in Simpson's book, then the statement "every set is contained in a countable coded $\omega$-model of $\mathsf{WKL}\_0$" is equivalent to $\mathsf{ACA}\_0$.


The reason for this solely depends on the meaning of "satisfies" and not on the existence of the models in question. So please be careful when saying that $\mathsf{ACA}\_0$ is equivalent to "every set is contained in an $\omega$-model of $\mathsf{RCA}\_0$" (and variants).


That said, to address your reference request, Simpson writes in the notes to VIII.2:



> 
> Theorem VIII.2.11 and lemma VIII.2.15 are well known, but their origins seem difficult to trace. See the references in Shoenfield [220], e.g., Kleene [142, §72].
> 
> 
> 


These references are:


* J. R. Shoenfield, [Degrees of models](http://www.ams.org/mathscinet-getitem?mr=139525), J. Symbolic Logic 25 (1960), 233–237.
* S. C. Kleene, [Introduction to Metamathematics](http://www.worldcat.org/oclc/523942), Van Nostrand, 1952.


It is also unclear to me how to correctly attribute the result you have in mind.


