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nLab > Latest Changes: Nelson arithmetic

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  1. Page created, but author did not leave any comments.

    Anonymous

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2022

    hyperlinked fragment and first order

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2022

    I suppose the line

    consisting of the predicate 𝒞\mathcal{C} and the following axioms

    should read:

    consisting of a predicate 𝒞\mathcal{C} satisfying the following axioms

    ?

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorNikolajK
    • CommentTimeMay 8th 2022

    The original and the current variant sound strange, since it makes it sound like a mere fragment of the known axioms. The alteration is probably also not the full correction, since only two of the axioms involve that predicate symbol.

  5. first section sounds weird, so I modified it

    Anonymous

    diff, v3, current

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 9th 2022

    I presume the predicate 𝒞\mathcal{C} is meant to capture the “feasible” numbers. If so, this should be mentioned, and also some mention of ultrafinitism.

  7. explaining what the predicate 𝒞\mathcal{C} is supposed to mean

    Anonymous

    diff, v4, current

  8. added section on the categorification of Nelson arithmetics.

    Anonymous

    diff, v7, current

  9. adding paragraph on how Nelson arithmetic relates to ultrafinitism in the foundations of mathematics

    Anonymous

    diff, v9, current

    • CommentRowNumber10.
    • CommentAuthorJem Lord
    • CommentTimeMay 17th 2022

    The reference defines the arithmetic to be an extension of Peano arithmetic, and includes induction for all formulae not containing the predicate 𝒞\mathcal{C}. The definition has been changed to correspond to the definition used in the paper (which extends Peano arithmetic, not just a fragment thereof), and the content that relies on not having induction now refers to the theory “Nelson arithmetic without induction”.

    Simplified the presentation of some models, and fixed some typos in the definitions of additionable and multiplicable numbers. A couple of “models” didn’t satisfy all of the axioms, so were modified or removed.

    A bunch of other miscellaneous edits too.

    diff, v11, current

1 to 10 of 10

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