User talk:Ncfavier
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Hello, Ncfavier, and welcome to Wikipedia! Thank you for your contributions. I hope you like this place and decide to stay.
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- If you have a question that is not one of the frequently asked questions below, check out the Teahouse, ask me on my talk page, or click the button below. Happy editing and again, welcome! Rasnaboy (talk) 07:55, 9 October 2022 (UTC)
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Ordered pair - Kuratowski, Function
[edit]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 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)