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    NFU's type-level distinctions between sets and their orde... — Carmelics
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    Challenges→The order type Omega of the natural order on ordinal numbers is itself one of the ordinal numbers

    NFU's type-level distinctions between sets and their order types mean Omega is an ordinal only in a deflated, type-shifted sense, not the same ordinal concept it purports to extend.

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    Key Terms

    Deflated sense(describing the nature of Omega in this system)
    A weakened or reduced version of something; here meaning Omega works as an ordinal in a less straightforward or less complete way than expected.
    NFU(in set theory)
    A modified version of Quine's NF system that Jensen created; it adds extra objects called 'urelements' to make the system work more flexibly.
    Omega(NFU set theory, where the collection of ordinals forms a set)
    The order type of the natural order on the ordinal numbers, which is itself an ordinal number
    Order types(what NFU distinguishes from sets)
    A way of describing the pattern or structure of how items are arranged in a sequence, independent of what those items actually are.
    Ordinal

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    (as a mathematical structure being preserved or not)
    A type of number used in mathematics to describe the order or position of things (like 1st, 2nd, 3rd), not just how many.
    Sets(as mathematical/philosophical objects being discussed)
    In mathematics and philosophy, a set is a collection of objects grouped together; for example, the set of all prime numbers or the set of all students in a classroom.
    Type-level distinctions(how NFU organizes sets)
    Differences between things based on what 'category' or 'level' they belong to; for example, treating individual numbers differently from collections of numbers.
    Type-shifted(how Omega functions in NFU)
    Changed or moved to a different category or level; here it means Omega's status as an ordinal changes depending on which logical level you're looking at.

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    The order type Omega of the natural order on ordinal numbers is itself one of th...

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