Talk:Equivalence of Definitions of Ordered Pair

The weakness of Kuratowski formalization shows up when $a = b$, and therefore the ordered pair equals the set $\left\{\left\{{a}\right\}\right\}$. --L.P. 00:58, 17 January 2012 (EST)


 * ... your point is? --prime mover 01:31, 17 January 2012 (EST)


 * There is no point. This will only demand caution when defining larger ordered tuples. One needs to be aware of this awkward side-effect, though. --Lord_Farin 03:19, 17 January 2012 (EST)

Uh, what does the corollary have to do with the theorem? They seem like apples and oranges. I mean, well, it may be proved as part of the proof for the main result, but as it's so distant from the exposition of the theorem I'm inclined to think it as its own creature. --GFauxPas 22:37, 4 April 2012 (EDT)


 * The article proves that the cartesian product of two non-empty sets is non-empty. I guess that this should be explicitly mentioned as part of the theorem statement. --abcxyz 22:39, 4 April 2012 (EDT)


 * How about a new page, Theorem: Product of Non-Empty Sets is Non-Empty. Proof: Base case, induction, etc. --GFauxPas 22:41, 4 April 2012 (EDT)
 * Eh, still feels like two theorems merged into one. I think you should create a separate page Product of Non-Empty Sets is Non-Empty, and put it there. --GFauxPas 22:47, 4 April 2012 (EDT)


 * All right, I'll do that. --abcxyz 23:06, 4 April 2012 (EDT)

Please note that products of arbitrary cardinality can be considered; in that respect, non-emptiness for all of these is equivalent to AoC. --Lord_Farin 02:58, 5 April 2012 (EDT)


 * Yeah but we aren't using AoC on this page. --abcxyz 09:57, 5 April 2012 (EDT)


 * My point was that the title Product of Non-Empty Sets is Non-Empty may be misleading, and that prefixing 'Finite' should be considered. --Lord_Farin 10:24, 5 April 2012 (EDT)


 * Yes, that's exactly what I did: Finite Cartesian Product of Non-Empty Sets is Non-Empty. --abcxyz 10:30, 5 April 2012 (EDT)


 * Good job. --Lord_Farin 10:44, 5 April 2012 (EDT)

Ad Questionable
What this page might be converted into is demonstration equivalence between a definition with a universal property (comparable, but not identical, to the product's universal property) and the Kuratowski formalisation. This abstract UP definition might aid in general, as that is what we're using the whole time (and probably what the informal definition is closest to). &mdash; Lord_Farin (talk) 11:17, 21 October 2017 (EDT)


 * Sounds good. You mean a UP of ordered pair? --barto (talk) 12:04, 21 October 2017 (EDT)


 * Yes, as stated here. &mdash; Lord_Farin (talk) 14:18, 4 November 2017 (EDT)