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nLab > Latest Changes: New article: [[axiom of pairing]]

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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeJan 8th 2011

    I created a link to axiom of pairing from constructible universe, then satisfied it.

    • CommentRowNumber2.
    • CommentAuthorSridharRamesh
    • CommentTimeJan 8th 2011
    • (edited Jan 8th 2011)
    I saw binary/nullary pair linked to from within that article, and took a stab at creating a very rough version of such an entry.
    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeJan 8th 2011

    Great, thanks!

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeJan 9th 2011

    Although I first redirected unordered pair to axiom of pairing, I decided that it deserved its own article.

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeJan 9th 2011
    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeJan 10th 2011

    Just a suggestion for adding a variant: I am used to hear axiom of pair, less than axiom of pairing.

    • CommentRowNumber7.
    • CommentAuthorTobyBartels
    • CommentTimeJan 11th 2011

    Grammatically, I think that this would have to be “axiom of pairs”. I put that in. Does that make sense?

    • CommentRowNumber8.
    • CommentAuthorRodMcGuire
    • CommentTimeJan 11th 2011
    • (edited Jan 11th 2011)

    added pair set terminology to unordered pair.

    I’ve heard this usage and it appears in Wikipedia: unordered pair, though I have no idea how common it is or what its provenance is.

    Below is what Hans Adler says from the Wiki talk page (I don’t seem to be able to get blockquote to work):

    I could think of four alternative names for sets with 2 elements and checked their frequency in Google Books by searching only in books that have “set theory” in the title to get rid of spurious results. Here is the result:

    • 79x unordered pair
    • 38x two-element set
    • 23x pair set
    • 3x binary set, but these hits referred to three different unrelated things (binary set operation, binary set theory, binary set-relation).
    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeJan 11th 2011

    Thanks for clarifications.

    • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeJan 11th 2011
    • (edited Jan 11th 2011)

    The term ‘two-element set’ isn’t really appropriate, since {x,x}\{x,x\} has only 11 element. (Although I see from the discussion by Adler that ‘unordered pair’ and even the Fraenkel’s version of the pairing axiom were originally restricted to this case.)

    • CommentRowNumber11.
    • CommentAuthorRodMcGuire
    • CommentTimeJan 14th 2011

    I edited binary/nullary pair to discuss the case when a nullary operation does not exist. In particular I discussed the example of computation on ℝ\mathbb{R} considered as unbounded lattice which lacks a nullary operation which can be handled by working in the extended reals ℝ ±∞\mathbb{R}_{\pm \infty}.

    This probably needs some cleanup by people that know better (and I hope I didn’t invert something!). Also I seriously lack decent chops in laying out thing in iTex so that too can be improved.

    My general question (which I left in my edit) is other than unbounded lattices are there any other examples where an associative binary operator or (co) product lacks an nullary-form or (co) terminal object? If such exist, does there always exist an extended context in which computations can be handled by nullary/binary operations?

    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeJan 14th 2011

    I am used to the weaker formulation of the axiom, I mean that there exists a set containing both x and y (and possibly some other elements), then using comprehension one derives this stricter version. This way is e.g. in my favorite Set theory book, by my favorite logics teacher, Kunen.

    • CommentRowNumber13.
    • CommentAuthorSridharRamesh
    • CommentTimeJan 14th 2011
    @Rod McGuire: Perhaps I misunderstand what you are asking, but, sure, there always exists an extended context in which the n-ary products of an associative binary operator can be handled by nullary/binary operations. Given an associative binary operation, just formally add an identity element (your new nullary operation), and you have a full-on monoid, in which any n-ary operation can be decomposed into nullary/binary operations in the usual way. In fact, the only thing you need the nullary operation for is the n = 0 case anyway; for positive n, just the binary operation already suffices (well, for n = 1, you just use the identity map, but that's part of the clone generated by the binary operation anyway).
    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJan 14th 2011

    @Zoran, interesting. I prefer the axiom that gives you the set you really want directly (namely {x,y}\{x,y\}) rather than requiring the intermediary of the comprehension axiom. But of course tastes differ. (-:

    • CommentRowNumber15.
    • CommentAuthorTobyBartels
    • CommentTimeJan 15th 2011
    • (edited Jan 15th 2011)

    @ Zoran:

    Like Mike, I prefer to get the unordered pair immediately. But as I began adding your variation to the article, I realised that the result that I really want not only gives this but also states its uniqueness (so as to justify introducing notation and terminology for the operation of unordered pairing). While trying to explain all of this, I decided that it was simpler just to start with your axiom and then remark what could be easily proved from it.

    As an axiom, your version also has the virtue of being shorter and weaker. That’s less important now than it used to be, but this is largely a historical matter and that was important once.

    I made a corresponding change to axiom of union.

    • CommentRowNumber16.
    • CommentAuthorTobyBartels
    • CommentTimeJan 15th 2011

    @ Rod:

    Another example of an associative binary operation with no identity element is addition of positive integers (the “natural numbers” in the original sense). Of course, we fix this by computing in the nonnegative integers (the “natural numbers” in the modern sense).

    I don’t really think that your material belongs on that page; if there’s no nullary operation, then there’s no pair, and the binary operation alone is sufficient. And conversely, binary/nullary pairs apply to more things than associative operations!

    However, that material does belong somewhere. For now, I’d like to put it on associative operation (which doesn’t actually exist but which has related material at associativity and at associative operad, from which I could cobble together an article).

    • CommentRowNumber17.
    • CommentAuthorzskoda
    • CommentTimeJan 15th 2011

    but also states its uniqueness

    The uniqueness is easy, by extensionality; I mean once we use comprehension to choose only xx and yy it does not matter from which bigger set I chose them.

    • CommentRowNumber18.
    • CommentAuthorTobyBartels
    • CommentTimeJan 15th 2011

    Yes, the uniqueness is easy by extensionality, just as the strictness (by which I mean the ability to choose PP such that a∈P⇒a=x∨a=ya \in P \;\Rightarrow\; a = x \;\vee\; a = y as well as the converse) is easy, by bounded separation. The question is how much of that to put into the axiom of pairing. (If it were the theorem of pairing, then there would be no question: put it all in! But for axioms one might want to do the reverse.)

    • CommentRowNumber19.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 20th 2014

    I finally moved binary/nullary pair to biased definition, where it could be expanded and generalized, but more importantly where it will catch incoming links.

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