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This study considers the question of whether individuals build mental structures for the set P(N) that give meaning to the phrase, "all subsets of N." The contributions of our research concerning this question are two–fold. First, we identified constructivist perspectives that have been, or could be used to describe thinking about infinite sets, specifically, the set of natural numbers N. Second, to determine whether individuals' thinking about the set P(N) can be interpreted in terms of one or more of the perspectives we considered, we analyzed the thinking of eight mathematicians. Beyond negative conceptions, that is, what P(N) is not, the results of our analysis cast doubt on whether individual understanding of the set P(N) extends beyond the formal definition. We discuss the possible implications of our findings, and indicate further research arising from this study.

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