Ncfavier

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Ordered pair - Kuratowski, Function

You wrote in [comment]: "not being a function isn't something you can prove; you'd need to define functions first, which is usually done using ordered pairs, so this is circular. "

I don't agree that we have a vicious circle here. We have two commonly accepted definitions of functions: Bourbaki (a function is a triplet f=(X,Y,G) where X is the domain, Y is the codomain, and G is a graph, i.e. a set of argument-value pairs) and the second definition, probably Peano's, earlier, reducing Bourbaki's definition only to a graph, i.e. f=G (it is often added that it is also a certain binary relation at the same time).

The definition of G is based on the Cartesian product, i.e. a set of pairs (G is some subset of XxY). The way we define a pair does not matter for G.

Since we are considering the Kuratowski case, if we assume a pair as (a,b)={{a},{a,b}} we must consistently use this definition in the definition of G. Kamil Kielczewski (talk) 12:53, 30 July 2024 (UTC)

Even so, the statement is bogus because $\pi _{1}$ is not of the right type to even consider whether or not it's a function in the set-of-pairs sense: it is not even a set. What you wrote looks like a formal proof but really isn't one, and a short sentence conveys the same information without needlessly confusing the reader. — ncfavier 13:22, 30 July 2024 (UTC)

@Ncfavier What I wrote shows that if the function pi existed (as a regular function within ZFC) it would lead to a contradiction. Which does not mean that I insist that this proof should remain - your correction in the article is also ok and clarifies the situation. Kamil Kielczewski (talk) 18:40, 30 July 2024 (UTC)