1 Introduction

Although infinity does not appear as a separate topic in university level mathematics classes, it permeates the study of various major concepts. The knowledge of mathematical infinity evolves from describing “anything and everything incomprehensibly large” (Guillen, 1983, p. 42) or small for that matter, to “a rational treatment of the subject” (p. 42) in relation with diverse mathematical concepts, while individuals keep encountering it in different contexts throughout their mathematical journey. This notion does not have a precise definition; it usually appears in definitions as an adjective (for an interesting discussion on how to define “infinite sets”, see Durand-Guerrier, 2023).

As students progress through their study of advanced mathematical ideas, they may experience difficulties in transiting between stages of mental construction and deal with these concepts algorithmically (Arnon et al., 2014). Study of the construction of mathematical infinity from the viewpoint of a cognitive theory can offer explanations as to the nature of this learning process as well as difficulties involved and point to possible instructional strategies.

In this article we hope to shed light on the nature of the mental construction of mathematical infinity, placing our attention on how individuals act on infinite entities. A general question that guides our investigation is the following: What are the characteristics of the mental structures and mechanisms that allow an individual to act on infinite entities? After explaining our theoretical approach, we will reformulate this question in more detail and in line with the terminology of our framework. Here we mention that it involves delving into the very nature of the constructs of our theoretical framework, going beyond traditional explanations and expanding these notions through a novel and stimulating discussion. This way we aim to extend our knowledge about the notion of infinity and APOS theory. We hope that this contribution will inspire reflection in the mathematics education community and that it will lead to further explorations, both theoretical and empirical.

2 Literature review

Some research on understanding mathematical infinity centered around intuitions (Fischbein, 1987) that are generally related to the potential aspect of infinity, that is, forming a process that continues without an end. This kind of intuitions might present an obstacle for learning or accepting the static aspect of infinity typically associated with the cardinality of sets, known as actual infinity (Fischbein, 2001). For example, thinking in terms of every number being followed by its successor as a dynamic process in the context of natural numbers is an indication of potential infinity. According to Fischbein (2001), this interferes with the conception of the set \({\mathbb{N}}\) as a finished entity containing all the natural numbers, which corresponds to actual infinity.

Also related to intuitive reasoning, other studies have shown that students tend to overgeneralize the use of strategies for comparing the cardinalities of finite sets to infinite ones (Fischbein et al., 1979; Tsamir, 1999). For example, when asked whether the set of natural numbers or the set of even numbers contained more elements, a great majority of the students sustained that since the former set contains the latter one, it must have more elements (Fischbein et al., 1979), an argument that works for the comparison of finite sets but not for the comparison of infinite sets.

Students’ difficulties with accepting actual infinity have been the subject of some studies. For example, Sierpinska (1987) reports that students tend to consider the number \(0.\overline{9 }\) as an approximation for the number \(1\). Voskoglou (2013) asserts that the difficulty that students have in accepting the existence of transcendental numbers results from their previous work with irrational numbers that can be represented algebraically; that is, students cannot see the set of irrational numbers as a completed totality.

Investigations about students’ interaction with mathematical paradoxes also touch on the potential and actual aspects of infinity. Mamolo and Zazkis (2008) identified students’ tendency to think in potential terms while engaging in questions about limiting situations. They mention that in order to deal with the cognitive conflict resulting from work on paradoxes, some students tried to reconcile their intuitive ideas with the formal theory—which caused more conflict—while others chose to dissociate their beliefs or intuitions from what is established by the mathematical theory so as to accept the formal solution. One of the paradoxes they used was the ping-pong ball conundrum. In this context, fitting infinitely many time intervals in a limited time presents a challenge resulting from the interaction of potential and actual infinities, apart from coordinating three processes that correspond to: (1) placing the balls, (2) taking the balls out, and (3) the time passing (Mamolo & Zazkis, 2008). These authors observed that even after providing the students with the solution of this paradox, “students demonstrated an overwhelming intuitive resistance to the possibility of an empty barrel” (p. 178).

In the context of the same conundrum, Ely (2011) refers to different properties of finite states involved in an infinite iterative process, warning that not all of the properties of an intermediate state can be projected to the final state.

Understanding of mathematical infinity continues to be an intriguing topic for math education researchers, perhaps partially because infinity continues to fascinate mathematicians, as well.

3 Theoretical framework

From the viewpoint of APOS theory that frames our study, construction of mathematical knowledge passes through the stages known as Action, Process, Object and Schema, by means of mental mechanisms such as interiorization and encapsulation. Actions, Processes, Objects and Schemas are sometimes denoted as structures, and other times referred to as conceptions. Oktaç et al. (2022) offer the following sentence in order to distinguish the nuances between their usage: “An individual with a Process conception of function has integrated both the Action and Process structures related to the function concept into his/her understanding and is progressing towards an Object stage” (pp. 1172–1173).

Actions are externally guided transformations where the individual follows explicit steps. In the context of the infinite repeating decimal \(0.\overline{9 }\), with an Action conception “[t]he student recites, verbally or in writing, an initial sequence of 9 s” (Dubinsky et al., 2013, p. 234). The mechanism through which Processes are formed is known as interiorization. This happens when the individual repeats and reflects on Actions, and gradually gains control over them; this way it becomes possible to reason in a general manner. In the context mentioned before, with a Process conception “[a]n infinite repeating decimal emerges as the student acknowledges that an initial sequence of 9 s continues forever” (p. 234). The need to apply Actions on Processes leads to the construction of mental Objects through the mechanism of encapsulation. Encapsulation can be compared to packing the contents of a Process in order to be able to handle it as a whole and transform it when necessary. Continuing with the same example, with an Object conception “[t]he student sees an infinite sequence of 9 s as an entity and can imagine or actually apply mental actions or processes to that sequence” (p. 234). The application of Actions on these Objects gives rise to new Processes, and so the construction of knowledge continues in a spiral manner (Arnon et al., 2014). When relationships are established between a concept and other structures associated with diverse mathematical notions, a Schema is formed. Figure 1 shows the main elements of APOS theory.

Fig. 1
figure 1

Main elements of APOS theory

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A model that explains the construction of knowledge in terms of mental structures and mechanisms is called a genetic decomposition (Arnon et al., 2014). It forms an essential part of APOS theory since it is intimately related to the design of research instruments and pedagogical strategies.

We now turn to the use of APOS theory in the construction of knowledge related to mathematical infinity.

3.1 APOS and infinity

Previous studies from the lens of APOS theory explored the mental structures and mechanisms developed by students working with mathematical situations in infinity-related contexts such as paradoxes, comparison of sets and the equality \(0.\overline{9 } = 1\) (Dubinsky et al., 2005a, b, 2013). According to Brown et al. (2010), the Objects that are obtained from encapsulating infinite processes do not necessarily inherit the characteristics of the process; they transcend the associated Process and are called Transcendent Objects.

An example of the kind of object described in the previous paragraph is the fractal known as the Sierpiński triangle. An iterative transformation of the initial triangle produces, at each finite step, a geometric figure with finite perimeter and non-zero area. However, the Sierpiński triangle that is obtained as a finished state of this process has infinite perimeter and zero area. Furthermore, it cannot be represented geometrically. It is a transcendent object, that is, it transcends the associated process; it does not maintain the characteristics of the shapes obtained at each iteration.

The nature of the Object resulting from an infinite Process is not the only feature that makes the learning of infinity different, from an APOS viewpoint. Another theoretical development is a possible new mental structure called Totality, observed in students working with infinity-related tasks (Dubinsky et al., 2013). Since the beginnings of the theory, it has been assumed that when an individual conceives a Process as a complete whole, they can also transform it by applying Actions on it (see for example Asiala et al., 1996). However, recent investigations offer empirical evidence in some infinity-related contexts that this is not necessarily the case (Dubinsky et al., 2013; Villabona et al., 2022).

Dubinsky et al. (2013) report that although some students clearly accept the equality of the numbers \(0.\overline{9 }\) and 1, hence seeing the process associated with the number \(0.\overline{9 }\) as completed, they cannot solve the equation \(0.\overline{9 }+x=1\). Finding the values of \(x\) that satisfy the equation requires the application of a specific Action. In terms of APOS, these students have constructed a Process and furthermore they are able to see it as a static entity, but they have not constructed an Object conception, since they cannot apply Actions on that entity. For Dubinsky et al. (2013) this implied the existence of a possible new structure called Totality, between Process and Object.

This in turn led to the consideration of a new mechanism called completion that indicates transition from Process to Totality in infinity-related contexts (Villabona et al., 2022). It also suggested the possibility of employing the mechanism of encapsulation on a Totality instead of on a Process in certain situations. Weller et al. (2009) were already referring to this idea by noting that “[b]ecause an infinite process has no final step, and hence no obvious indication of completion, the ability to think of an infinite process as mentally complete is a crucial step in moving beyond a purely potential view” (p. 10). Hence Totality can be defined as a static structure constructed by completing an infinite Process, on which Actions cannot be applied yet. When an individual can apply Actions, it means that they have encapsulated the Totality into an Object. We should mention that in order for Totality to be accepted as a new structure, there is need for research evidencing its plausibility in contexts other than infinity. If this happens, then APOS theory might be organized as shown in Fig. 2.

Fig. 2
figure 2

(Adapted from Villabona et al., 2022)

APOS theory including the possible structure Totality and its mechanism completion

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Mamolo (2014) considers that beyond acting on an object, the way the object is acted upon is important, and affects the construction of a conception about infinity. She mentions two ways through which one can think of actual infinity: “as the encapsulated object of a completed infinite set (to which bijections can be applied) and as the encapsulated object of a transfinite number representing the cardinality of an infinite set (to which arithmetic operations can be applied)” (p. 2). In the context of transfinite cardinal numbers, Mamolo’s questions about “How does a learner act on infinity?” and “What can the ‘how’ tell us about an individual’s understanding of infinity?” (p. 9) are important both for deepening our knowledge about student learning and for the advancement of APOS theory. She also refers to the importance of the “appropriateness of the transformations being constructed” (p. 6), an issue which concerns us in this study from a different angle, as we explain further.

From the perspective of APOS theory it is generally considered that an Object conception is determined by the ability to apply Actions. However this explanation is quite general and does not provide information about a situation where an individual can apply certain Actions and not others, as well as the complexity of Actions. Our study provides an original contribution to this discussion.

We illustrate this situation with three different Actions that can be performed on the Sierpiński triangle. The first two involve finding its perimeter and area. In both cases, iterative processes are considered in their limiting states. Calculating the perimeter involves the identification of the number of sides and their length at each step of the iteration, and finding the value at the limit when the process is terminated. Calculating the area on the other hand requires determining the areas of the triangles generated at each iteration, and then considering what transpires when the process is finished. A third Action involves determining the fractal dimension; for that, the learner needs to have knowledge about specific techniques or rules that come from fractal geometry, measure theory and topology. Equally important in this case is the necessity for one to accept the existence of an object whose dimension is not an integer. Obviously, this latter Action turns out to be much more complex than the previous two.

In all the cases mentioned in the previous paragraph, the Totality that invites the acting does not change. However, these Actions explore different aspects of the concept, where, in some cases, a deeper comprehension or a connection with more advanced domains of mathematics is needed. This led us to question the type of conception constructed by an individual who can apply certain Actions but cannot apply others. Can we say that they have constructed an Object conception? Or is it more appropriate to refer to the construction of only certain aspects of that object that allows acting on it in a somewhat limited manner? Breidenbach et al. (1992, pp. 257–258) refer to tasks that motivate “the construction of a particular mathematical concept, or a new aspect of it”. In the following paragraph we explore further the notion of ‘aspects’ related to an object, that we call facets, and illustrate it with an example.

The Objects that emerge from different types of infinite Processes have different characteristics that in turn determine the kinds of Actions that can be performed on those Objects. When a certain Action is applied by an individual, a certain facet of the cognitive Object is constructed. For example, an individual who determines the area of the Sierpiński triangle constructs a certain aspect of this Object related to the geometric pattern involved in its formation. On the other hand, in this case it is not necessary to be aware of its topological properties, an aspect which is constructed if one calculates its dimension.

The cognitive structures and mechanisms through which an individual confronts the complexity of an infinity-related task are different in different types of contexts, be it dynamic or static (Roa-Fuentes & Oktaç, 2014). Even in dynamic contexts, the convergence or divergence of the process affects the kind of constructions developed. Therefore a genetic decomposition of infinity, independent of the context in which it appears, can only touch overall characteristics. Roa-Fuentes and Oktaç (2014) proposed this kind of a model where they synthesized the generic cognitive elements that can be used when faced with mathematical infinity. They called it a Generic Genetic Decomposition of Infinity, which can be adapted with the purpose of presenting Particular Genetic Decompositions for different contexts.

Our aim in this paper is to unfold the characteristics of the Object that is related to an infinite process, as well as the associated mental mechanisms through which an individual constructs this conception. By focusing on the passage from a Process to Totality and from Totality to an Object conception of infinity as the individual applies increasingly complex Actions on the Totality, we aim to offer a novel theoretical perspective on what is meant by an Object conception within APOS theory. We posit that this conception is not constructed in a one-time application of a certain Action, but rather in a progressive manner by means of facets that develop when each time a new type of Action is applied. Two questions that guide our research are the following: What is the nature of the Actions that can be applied on a Totality? What are the characteristics of the encapsulation mechanism and the related Object in infinity related contexts?

In order to reach this goal, we propose a Particular Genetic Decomposition for the context of “the power set union problem” (explained in detail in the Method section). We also consider the application of three types of Actions on the Totality of the Process related to this context, with the purpose of exploring the facets of the resulting Object.

4 Method

The methodology associated with APOS theory incorporates components of theoretical analysis; design and implementation of instruction; and collection and analysis of data, which together form the research cycle (Arnon et al., 2014). In line with our research aim, we adapted the research cycle to involve the theoretical analysis and data components (Fig. 3). The theoretical analysis consisted in generating a genetic decomposition, which in turn informed the design of the interview for data collection. Even though we did not develop an instructional component, we conducted didactical interviews during which reflection about the topics in question is motivated, and furthermore construction of new mental structures can be observed (Oktaç, 2019).

Fig. 3
figure 3

Research cycle

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Our initial theoretical analysis consisted of a preliminary genetic decomposition for the particular context of the infinite union of power sets. It takes into account the theoretical analysis reported in Brown et al. (2010) and offers a new perspective by including Totality as well as possible Actions that can be applied on it, together with the corresponding facets of the resulting Object. The interview was designed for the purpose of verifying through data whether this genetic decomposition is valid. The information related to data collection is provided in Sect. 6.

In the process of the validation of a genetic decomposition, data obtained from all research participants are considered. Interviewees’ constructions were examined against the ones predicted by the theoretical analysis. We note that all the participants that we interviewed provided evidence related to Action, Process, Totality or Object structures. Empirical evidence for different parts of the genetic decomposition can be generated by different individuals; for example, one student might show an Action conception, while another shows evidence of using Processes. The aim of this analysis is not to classify the participants, rather, it is to look for evidence of the structures that make up the genetic decomposition. Hence, in order to focus the attention on the validation of the genetic decomposition in line with the methodology of APOS theory, we report on those participants that best illustrate the elements of our genetic decomposition to demonstrate its validity.

In order to verify our genetic decomposition, semi-structured didactical interviews were performed, videotaped and transcribed. In line with the methodology associated with APOS theory, the transcriptions were examined by each researcher separately before a triangulation was performed and an agreement was reached.

As a result of their interviews with graduate students and mathematicians, Wilkerson-Jerde and Wilensky (2011) conclude that an educational implication of their study is the identification of the need for “explicit attention to the different fragments that make up a mathematical object of study” (p. 40) and for helping undergraduate students to gain related experiences. Along with these lines and also informed by our genetic decomposition, we decided to interview individuals who had completed all coursework in an undergraduate mathematics program, since they needed to have varying levels of maturity in advanced set theory and experience with infinite iterative processes. We interviewed 8 graduate students in Mathematics Education, 3 graduate students in Mathematics and 3 university instructors.

The data presented here correspond to four of the interviewees that best illustrate the elements of our genetic decomposition.

5 Theoretical analysis

In this section, we first describe the infinity-related context used in our study, followed by the genetic decomposition that we propose for it.

5.1 The context of the union of power sets

The mathematical context that concerns us is the union of power sets given by: \({\cup }_{k=1}^{\infty }P(\left\{1, 2, 3, \dots , k\right\})\). Within this context, an adaptation of the question reported in Brown et al. (2010) was used as the initial interview situation (the complete interview is presented in Appendix A):

Describe the elements generated by \({\cup }_{k=1}^{\infty }P(\left\{1, 2, 3, \dots , k\right\})\). \(P\) denotes the “power set”, which is the set of all subsets of a given set.

Answering this question requires the structures of Action, Process and Totality, constructed around the notion of countable infinity. On the other hand, the ability to apply Actions on the Totality of the iterative process gives rise to the Object structure. Since we are interested in the characteristics of an Object conception, we decided to work with different types of Actions that might be applied in this context. We established three kinds of Actions that can be applied to the infinite family of finite sets as we explain further.

The first type of Action is related to considering the infinite union set as an element of other sets, as in the case \(A=\left\{-3, 1, {\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)\right\}\). It is named Totality as an element. The second and third types of Action emerge from two possible paths that can be used to determine if the following equality is true: \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)=P\left({\mathbb{N}}\right)\). In order to reject this equality, one can find elements of the set on the right-hand side that do not belong to the set on the left-hand side; this corresponds to the second type of Action, called verifying the double inclusion. The third type consists in determining that the cardinals of the two sets are not equal, denoted as verifying cardinality.

As can be appreciated, these three types of Action involve different levels of complexity and explore different facets of the same Object resulting from the same infinite process. We posit that performing more complex Actions may require a more robust Object conception that includes more facets.

Performing Actions in this context implies previous constructions about fundamental concepts of set theory such as an Object conception of power set, a Process conception of an infinite union of sets, an Object conception of subset, among others. The problem statement asks for a description of the elements generated by the iterative union. This provides an incentive for the construction of an infinite iterative process which in turn leads to determining the characteristics of the elements produced in each iteration. Now we present our cognitive model for this particular context, followed by the data that validate it in Sect. 6.

5.2 Genetic decomposition

The construction of the infinite iterative process starts by employing Actions to determine the elements of the sets that correspond to the first few values of \(k\). When \(k=1\), we obtain the set \(P(\{1\})=\{\varnothing ,\{1\}\}\). For \(k=2\), we get \(P(\{\text{1,2}\})=\{\varnothing ,\{1\},\{2\},\{\text{1,2}\}\}\), and so on. When the individual repeats these Actions and reflects on them, they can realize that in each iteration, the last set contains the ones obtained in previous iterations:

$$P(\{1\})\subset P(\{\text{1,2}\})\subset P(\{\text{1,2},3\})\subset \cdots$$

Being aware of this property is closely related to the mechanism of interiorization. It facilitates the formulation of an expression for the set generated by the union of power sets for any value of \(k\). The general case, for \(k=n\), that is, the set generated in the \({n}^{th}\) iteration is given by:

$${\cup }_{k=1}^{n}P\left(\left\{1, 2, 3, \dots , k\right\}\right)=P(\left\{1, 2, 3, \dots , n\right\})$$

Awareness of this property indicates a more general comprehension that reaches beyond the Actions involved in each step of the iterative process. That is, it implies the construction of a Process conception. For evidence that one has used the mechanism of completion—enabling the construction of Totality—there has to be a change in the way one reasons. This reasoning would allow the individual to imagine the Process as finished, as opposed to seeing it in a dynamic manner. This way the person might argue that although infinitely many sets are being united, this Process can yield an entity that contains infinitely many finite sets.

Construction of an Object conception requires application of Actions. One type of Action that can be performed is to think about the Totality as a finished set that can form part of other sets, called Totality as an element. To look for evidence of this type, in the interviews we asked the participants to determine the cardinality of some sets, for example:

$$C=\left\{{\mathbb{N}},{\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)\right\}$$

This task inspires the individual to think about the Process involved in \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)\) or in \({\mathbb{N}}\) as elements of a set, which implies an Object structure related to infinity.

On the other hand, determining if the following equality is satisfied involves Actions that require a deeper understanding of the Object:

$${\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right) =P({\mathbb{N}})$$

Here, the individual needs to reflect about certain properties of the set given by \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)\).

In order to argue that the equality is not satisfied, one can determine that the infinite union on the left-hand side cannot generate infinite sets. This implies that there exist sets in \(P({\mathbb{N}})\) that are not in \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)\). This Action is called verifying the double inclusion. In this case it is not sufficient to think about the Process of the infinite union as a static entity; it is also necessary to consider certain properties related to this Process. Another implication is being aware that some of the properties of the Process may not hold for the resulting Object, which has to do with its transcendence as explained in Sect. 3.1.

Verifying cardinality to determine that \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)\) and \(P({\mathbb{N}})\) are not equal, is carried out by establishing the cardinality of each set. While \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)\) has the cardinality of countably infinite sets \(({\aleph }_{0})\), \(P({\mathbb{N}})\) has the cardinality of uncountable sets \(({\aleph }_{1})\). Comparing the set \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)\) with another set is an Action on the cognitive Object that it represents. On the other hand, determining the cardinalities of these infinite sets implies specific knowledge about Cantor’s theory.

The level of complexity involved in each Action is different. We can consider the Action Totality as an element as being the simplest among the ones mentioned. Verifying the double inclusion can be thought of as having an intermediate complexity, and verifying cardinality is the most complex.

As mentioned earlier, it is possible for an individual to be able to perform certain types of Actions and not others. This might have to do with different levels of evolution of their operative capacity on the same entity. For example, someone may determine that \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)\) and \(P({\mathbb{N}})\) are not equal, by verifying the double inclusion. However, this does not mean that they can determine the cardinality of these sets. The factors involved in the encapsulation of the Totality when performing the three types of Actions are different. We explain this phenomenon by referring to the relationship between each type of Action and the exploration of a distinct facet of the cognitive Object. An individual who faces a particular situation may construct some facet of the Object, and hence, can apply a certain type of Action. But there might be other facets, not yet developed, and for this reason, the individual may not be able to apply the corresponding Actions.

Referring to the types of Actions, if an individual applies a type 1 Action, we will say that they have constructed the facet 1 of the Object, etc. The more facets constructed, the more robust the Object conception will be. We should clarify that we are not restricting the number of facets that an Object might entail, and in most cases, it will be impossible to determine all of them. As an individual gets more involved in the study of mathematics, the relations that they establish between different elements will continue to evolve, hence adding more facets to those already constructed. In fact, mathematics as a discipline evolves, as new discoveries introduce novel ways of acting on previously constructed Objects; this in turn leads to the development of yet new facets of these Objects.

Figure 4 shows in a diagrammatic manner the genetic decomposition related to the context of the infinite union of power sets, where \({O}_{t}\) denotes the transcendent Object.

Fig. 4
figure 4

Genetic decomposition of the infinite union of sets

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This genetic decomposition has been validated through empirical data obtained by means of interviews, as detailed in the next section.

6 Data analysis and results: validation of the genetic decomposition

The interview revolved around the context of the infinite union of sets given by \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)\), and consisted of questions that allowed the study of the three types of Actions explained in the previous section.

The data presented in the subsections that follow validate our genetic decomposition. Performing interviews with people showing different levels of experience in set theory was crucial. We describe interviews with Marian, Miguel, Ivón and Kevin. Marian is a master's student in mathematics; Miguel holds a master's degree in mathematics education and is an instructor of basic university mathematics courses; Ivón is a master's student in mathematics education; Kevin holds a doctoral degree in mathematics and is an experienced instructor of advanced mathematics courses. The interviews enabled us the opportunity to analyze the evolution of the participants’ mental constructions in terms of the proposed genetic decomposition.

Now we present the evidence for Process, Totality and Object structures.

6.1 Evidence of Process

There are certain properties that an individual notices when they construct a Process conception. Marian (all the names are fictitious and I denotes the interviewer) recognizes that the sets satisfy the inclusion property given by \(P\left(\left\{1\right\}\right)\subset P\left(\left\{\text{1,2}\right\}\right)\subset \dots\). She also describes the set generated by the \({k}^{th}\) iteration:

Marian: Well, so if we continue, then naturally it would be \(P\left(\left\{\text{1,2},3,\dots ,k\right\}\right)\). The cardinal of that set would be \({2}^{k}\). And its elements, what are they? Well, it would be the empty set, all the subsets formed by one element, all the subsets formed by two elements, three, until \(k\).

Subsequently, Marian tries to determine what the elements generated by the infinite union are. However, she doesn’t realize that the property \({\cup }_{k=1}^{n}P\left(\left\{1, 2, 3, \dots , k\right\}\right)=P(\left\{1, 2, 3, \dots , n\right\})\) that she detected for finite values of \(k\) does not hold when the process is completed. She considers that at some moment the values of \(k\) will not only describe finite subsets of \({\mathbb{N}}\) but also the infinite subsets, for example, the set of even numbers, as seen in Fig. 5.

Marian: It is an infinite set of finite and countably infinite subsets. Well, I was thinking, why finite? Well, we know why, because of how I am generating them. And why also infinite? Well, because we know that within this [signals \(P\left(\left\{1,\dots ,k\right\}\right)\)], there will be infinite ones also, for example, the set of even numbers, and those are infinite.

I: How would you use the union to generate the set of even numbers?

Marian: Grouping the even numbers, then all the ones with the form: [writes the set in Fig. 5]

When I make the subsets, for example, the subsets of \(k,\) there will be that subset of \(\left\{2, 4,\dots \right\}\), right? Because they are all of the combinations.

I: But are you always uniting finite ones?

Marian: No, because when I make here the infinity [signals the infinity sign in the union], within that the infinite ones will come out, the even ones, the odd ones, the primes, I don’t know…

Fig. 5
figure 5

Subset of even numbers generated by the union, according to Marian

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Marian seems to grant a status of natural number to infinity, by mentioning that when she performs the iteration at infinity, she will generate the infinite subsets of natural numbers. When it comes to seeing the process as a whole, she considers that the “step” at infinity forms part of the Process that she constructed. Something similar occurs with Miguel who affirms that the set \(P({\mathbb{N}})\) is generated when all the iterations of the infinite union are carried out.

Miguel: If I see everything at the end, if \(k\) has run through all the naturals, then not all of them have a finite number of elements. But if I say that, you will ask me which one of those does not have a finite number of elements? And that is the problem, that I cannot say it, because there is no final set there. […] If \(k\) runs through all the naturals, then I cannot say that all the sets would have a finite number of elements. […] Even to generate the odd numbers I cannot think of a natural number \(k\), because all the odd numbers are infinite and if I think of a natural number \(k\), then I have finite elements. So then, I will obtain \(P({\mathbb{N}})\) when \(k\) has run through all the naturals.

Miguel, like Marian, considers that \(k\) not only takes natural numbers as values, but when the Process finalizes, it should also generate infinite sets. This implies that the Totality of the Process should be generated by the same Process, although one cannot establish a particular \(k\) for that to happen. Although the interviews with Marian and Miguel illustrate the construction of a Process conception, there is no evidence of the mechanism of completion for the construction of Totality. This is because they cannot describe the elements generated by the infinite union.

After Ivón establishes that the sets obtained at each iteration contain the previous ones, the interviewer asks her to determine the iteration at which the set \(\left\{1, 3, 10\right\}\) is generated. Ivón’s work is shown in Fig. 6.

Ivón: I took it in 10 because it is the largest, here I should say up to where \(k\) should go, right? […] Yes, this element is here in this union [writing \({\cup }_{k=1}^{10}P\left(\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\right\}\right)\)]. If I do it starting with \(k=1, \dots , 10\), this element will be generated by the parts of this set. Yes, because here all of them are possible. This [signals the 1] with this [signals the \(3\)] with this [signals the 10].

Fig. 6
figure 6

Finite iterative union to generate the set \(\left\{\text{1,3},10\right\}\), according to Ivón

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By solving this task, Ivón shows that she comprehends the Process associated with the infinite union. She also shows evidence of a Process conception, by establishing the property given by \({\cup }_{k=1}^{n}P\left(\left\{1, 2, 3, \dots , k\right\}\right)=P(1, 2, 3, \dots , n)\), although she extends this property to the Totality, as shown in Fig. 7.

Fig. 7
figure 7

Property of the infinite union, according to Ivón

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6.2 Evidence of Totality

With the intention to help Ivón construct a Totality conception, the interviewer asks her to describe the elements generated by the infinite union and tell how many there are.

Ivón: Infinitely many, there are infinitely many. And it has finite and infinite sets.

I: At what moment are you going to have an infinite set?

Ivón: Wait, let me see [long silence]. Finite! [long silence] No, yes. The elements of that set are finite. I mean, the sets that are in that set are finite, but the set is infinite.

By considering the infinite union as an infinite set of finite sets, Ivón shows that she can see the iterative process as an entity in itself, which evidences a Totality conception. On the other hand, Kevin demonstrates that he can think about the union as a static entity, as shown in Fig. 8.

Kevin: There are an infinite number of sets with that characteristic. Well, I would think about it this way [writes the sets in Fig. 8].

Fig. 8
figure 8

Characterization of the union, according to Kevin

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I also added the empty set. I mean, better said, all the finite subsets of \({\mathbb{N}}\) are here. Well, let’s say, here are the parts of this set [signalling the power set within the union]. What I think is that, if we take any finite subset, for example, take [writes] (Fig. 9):

So I say that one can find a \(k\) such that this [signalling the union] contains this element here [signalling the finite set in Fig. 6]. This means, that set will be an element here [signalling the union]. So all the finite subsets of \({\mathbb{N}}\) are there.

Fig. 9
figure 9

Finite set generated for some \(k\), according to Kevin

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Kevin makes a complete description of the type of sets that will be included at the end of the process related to the infinite union of power sets. When asked if the union can generate infinite sets, he responds:

Kevin: Well, I would think no, because given any \(A\) […] The simple fact of belonging there [signalling the power set within the union], implies that \(A\) is a subset of a finite set, so it cannot be infinite. They will be finite sets of any cardinality.

I: In some way, when the union contemplates here [signalling the infinity sign of the union] that goes up to infinity, doesn’t it imply that it can generate an infinite set?

Kevin: No, no. That infinity there refers to the fact that \(k\) can take an arbitrary value among the naturals. It doesn’t mean that it takes an infinite set.

Kevin is aware of the role that \(k\) plays in the union, which allows him to establish that the infinite union does not contain infinite subsets of \({\mathbb{N}}\).

6.3 Evidence of transcendent Object

Ivón evidences that she can see the infinite union of power sets of the natural numbers as a totality, when she refers to it as an infinite set of finite sets. However, she continues to accept the equality \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)=\) \(P({\mathbb{N}})\) as true. This leads to various moments of conflict, which even make her consider, initially, that the subsets of \({\mathbb{N}}\) are all finite.

Ivón: Well, it generates an infinite set, but that infinite set has finite elements. So for me, it generates finite sets, because moreover they are the elements that are in \(P({\mathbb{N}}\)) [signalling \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)=P({\mathbb{N}})\)].

I: In the set \(P({\mathbb{N}})\) are there only finite sets?

Ivón: Yes, I think so.

I: Remember that the set \(P({\mathbb{N}})\) is the set of all subsets of \({\mathbb{N}}\).

Ivón: Yes.

I: Are all subsets of \({\mathbb{N}}\) finite or are there also infinite ones?

Ivón: [Silence] No, they are all finite.

When asked which subsets of \({\mathbb{N}}\) she knows, Ivón immediately remembers that some subsets are infinite.

Ivón: Subsets of naturals… Ah, of course! Even numbers, odd numbers, like that. Of course, that’s it.

Since in \(P({\mathbb{N}})\) there are infinite sets, and Ivón established that in the iterative union there are only finite sets, this means that the two sets are not equal. As a result, Ivón starts questioning herself about whether the union really generates only finite sets. The interviewer asks her to explain how the set of even numbers can be generated from the iterative union of power sets. After various attempts in which Ivón tries to determine an iteration which would generate the set of even numbers, the interviewer asks a more explicit question.

I: You know how this process of union works, you told me, for example, that a set is always contained in the next one. How would you generate the even numbers? Can it generate the even numbers?

Ivón: [Long silence] It is not possible, it cannot generate them, neither the even numbers, nor the odd numbers.

Since Ivón cannot find an iteration that would generate the set of all even numbers, she starts thinking about whether the union can generate infinite sets, at all. For a moment, she thinks that it is possible to obtain infinite sets since \(k\) takes values from \(1\) to infinity, but her new approach does not seem to convince her completely. After thinking about the way in which the iterative union can generate infinite sets such as the sets of even or odd numbers, Ivón decides that the infinite union cannot generate infinite sets. This leads her to question the truth of the equality \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)=\) \(P({\mathbb{N}})\).

Ivón: Oh, God! Is this going to be equal? [signalling \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)=P({\mathbb{N}})\)] […] I think that’s it… [long silence]

I: How do you know if two sets are equal?

Ivón: The double inclusion. And I would say that the set of even numbers is not in the union, right? Because here I will always have finite elements [signalling the union]. They will always be finite elements, always.

Ivón finally sees that her contradictory reasoning arose by assuming that the equality \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)=\) \(P({\mathbb{N}})\) is true. However, the strength of the Totality structure that she constructed permitted her to reject the truth of the equality that she had affirmed at the beginning. Ivón performs the type 2 Action, verifying the double inclusion, and this way constructs the facet 2 of the transcendent Object.

Now we present evidence that even though Ivón was able to perform a type 2 Action, she could not do the same with a type 3 Action, that is verifying cardinality, and hence she could not construct the facet 3 of the same Object. The interviewer asks Ivón to determine the cardinality of the set \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)\).

Ivón: [Long silence] Since that process continues, so within it I will have infinity. Therefore that is an infinite set, cardinality is infinite, I don’t know if it can be said like that.

Since Ivón does not seem to remember some fundamental elements of Cantor’s set theory, the interviewer tries to lead her to realize that the iteration process through which the set \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)\) is constructed permits the numeration of its elements. But Ivón stays with the idea that this set is not countable. As a result, she cannot carry out the type 3 Action of verifying cardinality so cannot yet construct facet 3 of the transcendent Object.

Ivón has no problem with type 1 Action. This Action requires seeing the Totality of the iterative process as an entity in itself. This way, Ivón can identify the Totality as belonging to another entity that can be enumerated.

Type 2 Action, verifying the double inclusion that leads to facet 2, requires the comparison of two entities, namely the sets \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)\) and \(P\left({\mathbb{N}}\right)\). For that, it is necessary to return to the Process associated with \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)\) and to determine whether some particular iteration permits generating an infinite set. Although the mathematical object associated to the facets 1 and 2 of the transcendent Object is the same (an infinite family that contains all finite subsets of \({\mathbb{N}}\)), the construction of facet 2 requires the individual to be aware of the properties of the underlying Process and of the types of entities generated during its application.

The type 3 Action of verifying cardinality requires knowledge that incorporates specific elements from Cantor’s theory, such as determining the cardinality of an infinite set by establishing bijections, and accepting that two infinite sets do not necessarily have the same size. This refers to a facet of the transcendent Object where the individual can work with properties related to advanced knowledge.

Kevin shows that he can perform all three types of Actions, thus providing evidence of constructing all three facets of the transcendent Object. When asked to determine the cardinality of the sets \({\cup }_{k=1}^{\infty }P\left(\left\{1, 2, 3, \dots , k\right\}\right)\) and \(P\left({\mathbb{N}}\right)\), he responds:

Kevin: Well, the cardinal of the union is \({\aleph }_{0}\). [ …] It has the cardinal of the naturals, it is countable. The cardinal of \(P({\mathbb{N}})\) is \({\aleph }_{1}\).

Kevin is familiar with concepts of Cantor’s theory that allow him to accept that two infinite sets can have different cardinalities. The construction of facet 3 implies this kind of knowledge that is evidenced when Kevin can apply specific Actions related to advanced mathematics on an infinite family of finite sets.

7 Discussion

Our observation that students can perform certain Actions but cannot carry out others on the same Object led us to become interested in studying different kinds of Actions and their effect on the resulting conception. An Object conception is constructed when an individual can apply Actions. Our results in the context of infinity demonstrate that construction of a robust Object conception may require more than a unique encapsulation. That is, although an individual may encapsulate a Totality by applying a specific Action, it is very probable that there are other Actions that can be applied to the same Totality, perhaps more complex ones, that the individual may not be able to perform. In this case, the individual has not developed facets of that Object that are related to the Actions not yet performed. These considerations add a new perspective to the Object conception of infinity in terms of APOS theory.

As our results show in a context related to infinity, application of an Action on the Totality of a Process allows the surfacing of an aspect of the Object, related to that Action. These aspects or facets are illustrated in Fig. 10 by fragments that compose the sphere that represents the Object. The aspects, being the product of the application of an Action, cannot be considered independent of the corresponding Action. That is, the aspects do not only characterize the Object, they also characterize how it can be acted on it.

Fig. 10
figure 10

Construction of an Object starting with aspects that emerge from the application of different types of Actions

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In the case of a transcendent Object, different facets emerge as the individual continues establishing relations between that Object and other concepts. These relations allow new ways of transforming the Object, favoring the construction of new knowledge. This might be related to the levels of evolution of Schemas. Study of the construction of an Object through its different aspects may be enriched by considering the evolution of the associated Schemas in future research. This would imply that an Object, similar to the Schema that contains it, evolves.

8 Conclusions

As individuals progress in their mathematical studies, they develop a more advanced comprehension of the implicated concepts. As Breidenbach et al. (1992, p. 253) put it, “[a] particular conception can also exist in different forms, some of which are of greater mathematical sophistication and some less so”. Our research contributes to our understanding of the nature of this sophistication in the context of infinity. It achieves this by pointing to Actions with differing complexities that lead to the construction of multiple facets of the cognitive Objects involved.

Dubinsky et al. (1988) indicate that if a person applies a previously constructed Process to a previously constructed Object in the context of a novel situation, that is “[i]f this has not been done before, then, for the subject, both the process and the object are transformed into slightly more powerful versions and this is also a kind of construction” (p. 45). Our findings suggest a new approach to using APOS theory in infinity-related contexts to analyze and explain how different versions of an Object conception are constructed, something that has not been done until now. The encapsulation mechanism that allows an individual to act upon an infinite entity has different characteristics depending on the type of Action applied. Our study provides evidence that a single encapsulation—that is the application of a single Action—does not necessarily mean that the resulting Object conception is robust. Rather, by applying a specific Action, a facet of that Object is constructed. The more Actions applied to an Object, the more potential facets are built.

Another contribution of our study is a validated genetic decomposition that incorporates Totality in an infinity-related context, different from those reported in Dubinsky et al. (2013) and Villabona et al. (2022). It can serve as a basis for an instructional treatment on set theory.

Of course, as any study, ours also has some limitations. On the one hand, it has been conducted within the context of infinity, where a possible new structure Totality has been taken into consideration, which has not yet been observed in relation with other topics. In this sense, our results cannot be generalized, although it is possible that an Object conception related to other concepts also develops by means of constructing different aspects; research is needed in this direction. On the other hand, our study was performed considering only the analytic/algebraic representation; how the facets are related to different representations can be an interesting line of investigation. We should also mention that the participants in our study were individuals with experience in advanced mathematical ideas; other backgrounds might have an influence on the facets constructed. In particular, interviews with undergraduate students can reveal specific difficulties related to the mental constructions involved.

Further research can investigate other structures such as Processes and Actions together with their evolution while the individual keeps adding more facets to their Object conception. Another line of research is investigating the existence of a possible Totality structure in contexts involving mathematical notions other than infinity.

We consider that the construction of the various facets of an object can be effectively accomplished through a spiral curriculum and learning pathways. This focus implies going back and forth between concepts, constructing different aspects at different moments, depending on the progress of the individuals, deepening their comprehension and establishing connections between notions. We posit that the identification of possible facets related to infinity in different contexts, and the kind of Actions that motivate their construction, can help learners to confront their intuitive ideas and establish solid connections with other notions.

The development of future research in different contexts can further our understanding of the complexity of constructing the various facets of an advanced mathematical object. This would continue providing valuable insights for researchers and instructors about the types of mathematical activities that promote comprehension at different educational levels.

We reiterate our hope that this study will inspire reflection in the mathematics education community and that it will lead to further explorations, both theoretical and empirical.