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Alternative Theories and Higher Infinite

  • Chapter
Towards a General Theory of Classifications

Part of the book series: Studies in Universal Logic ((SUL))

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Abstract

We have presented in the previous chapter, a theory able to generate the infinite set of classifications as the continuum, each classification being in a one-to-one correspondence with a real number. But, as we know, there are, in the mathematics of the infinite, since the works of Cantor, Suslin and others, a lot of possible views of the continuum. Moreover, since the undecidability results of Cohen, there exist also a lot of possible set theories. So, it may be useful to ask some questions about what happens concerning the existence of classifications in those alternative theories, in particular, when they admit higher forms of the infinite. Though the risk is obvious, there, to end up at some undecidability results, several arguments speak for such an extension.

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Notes

  1. 1.

    In the whole chapter, definitions and theorems come, the most of the time, from Jech (see [264]). We also borrowed some results from Kanamori (see [268]) and, exceptionally, from Shelah (see [448]). Applications of these definitions and theorems in the case of infinite classifications are generally due to Parrochia (if not to Neuville).

  2. 2.

    Naturman and Rose (see [356]) have extended this result to full relation algebras and function monoids: For two cardinals λ,κ>0, the full algebras, partition lattices and function monoids on λ,κ are, in fact, elementary equivalent iff λ and κ are second-order equivalent.

  3. 3.

    The proof may be found in [332]. See also [445].

  4. 4.

    The notion of “order type” is defined as follows: let W 1 and W 2 be two well-ordered sets (i.e. a well-ordered relation is defined on each of them). If W 1 and W 2 are isomorphic, we say that they have the same order type.

  5. 5.

    Recall also that a linear order P is a “well order” if every non-empty subset of P has a least element.

  6. 6.

    Let T be a tree. Recall that an antichain in T is a set A, contained in T, so that any two distinct elements x and y of A are incomparable, i.e. so that we never have x<y nor y<x.

  7. 7.

    The reader will find the proof in Jech (see [264], 115).

  8. 8.

    Recall that if α is an ordinal, α=β+1 is a successor ordinal. If α is not a successor ordinal, then α=sup{β:β<α}=α. α is said to be a limit ordinal.

  9. 9.

    Theorems 9.9.3 and 9.9.4 are proposed as exercises by Jech (see [264], 263).

  10. 10.

    The proof may be found in Jech (see [264], 551).

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Authors and Affiliations

  1. Department of Philosophy, Université Jean Moulin – Lyon III, Lyon, France

    Daniel Parrochia

  2. Ecole Nationale Supérieure des Sciences de l’Information et de la Bibliothèque, Villeurbanne, France

    Pierre Neuville

Authors
  1. Daniel Parrochia
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  2. Pierre Neuville
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Parrochia, D., Neuville, P. (2013). Alternative Theories and Higher Infinite. In: Towards a General Theory of Classifications. Studies in Universal Logic. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0609-1_9

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