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Hereditarily Finite Lists

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Abstract

This chapter uses the everyday notion of list to introduce the universe of hereditarily finite sets.

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Notes

  1. 1.

    For references and further discussion, see Pollard [11], Chap. 3.

  2. 2.

    You already encountered the type/token distinction in Chap. 1. The relation between a single list-type and its many paper or pixel tokens is like the relation between a single numeral-type and the many numeral-tokens of that type.

  3. 3.

    This is a strategy mathematicians sometimes employ. For example, see Edwards [4], p. 139, for a treatment of multi-sets as lists. A multi-set is a set whose members can occur more than once. For a theory of ordered lists whose entries can occur more than once, see Deiser [3]. Such lists are not unusual. For instance, here is how the list of Wimbledon Gentlemen’s Singles champions begins.

    1. 1.

      Spencer Gore

    2. 2.

      Frank Hadow

    3. 3.

      John Hartley

    4. 4.

      John Hartley

    This is not a ranking. (A gentleman cannot outrank himself.) In mathematical parlance, it is a tuple (in particular, a 4-tuple). The lists we will be discussing in this chapter are not just unranked; they are not even ordered: they are not tuples. Among ranked lists there are some with ties and some without. Our lists are not of either sort.

  4. 4.

    Kanamori [7], Landini [9], Rang and Thomas [12] give some of the history.

  5. 5.

    In the folk song “Bad Man’s Blunder,” the bad man “got surrounded by a sheriff down in Mexico.” I understand this to be a slightly odd way of saying that the sheriff arranged for the bad man to be surrounded, not that the sheriff did the surrounding all by himself.

  6. 6.

    Readers who like a challenge, might tackle Frege’s own exposition in Frege [5]; English translation in van Heijenoort [13], pp. 5–82.

  7. 7.

    See Bennett [2] (available via http://projecteuclid.org).

  8. 8.

    See Ackermann [1]. For a more recent discussion, see Kaye and Wong [8] (available via http://projecteuclid.org).

  9. 9.

    Recall that 6 is 110 in binary notation because \(6=1\cdot 2^2+1\cdot 2^1+0\cdot 2^0\).

  10. 10.

    In set theoretic notation: \(Sn=n\cup \{n\}\).

  11. 11.

    See Hallett [6], pp. 185–194, and Mirimanoff [10].

  12. 12.

    I should warn you, though, that the term ‘super-number’ is not standard. Do not expect anyone else to refer to these objects in this way.

  13. 13.

    See Zermelo [14]; English translation in van Heijenoort [13], pp. 199–215.

  14. 14.

    OK, I am being sloppy here. Stated more precisely, the Listers have made tokens of the two types \(\emptyset \) and \(\{\emptyset \}\). We do not need to imagine that they make the types. At each stage, they list types that have tokens produced at earlier stages.

  15. 15.

    Among the details I am skipping over is the question of whether Axioms 3.1–3.3 allow us to introduce the notion of finite sequence. You might find it interesting to show that they do.

  16. 16.

    To see how unlikely that is, take a look at the “Mathematics Subject Classification” at http://www.ams.org/msc/pdfs/classifications2010.pdf. Note that this document is 47 pages long.

References

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Authors and Affiliations

  1. Department of Philosophy and Religion, Truman State University, Kirksville, MO, USA

    Stephen Pollard

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  1. Stephen Pollard
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Correspondence to Stephen Pollard .

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Pollard, S. (2014). Hereditarily Finite Lists. In: A Mathematical Prelude to the Philosophy of Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-05816-0_3

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