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On the Wikipedia article for codomain, in the third paragraph, it roughly says:

When the domain of a function is a proper class X, in which case there is formally no such thing as a triple (X, Y, F). (?) With such a definition functions do not have a codomain.

As a proper class is a class that cannot be a member of some class, i.e. cannot be a set, I was wondering why a function with its domain being a proper class does not have a codomain?

Thanks and regards!

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1 Answer 1

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My guess is: proper classes cannot belong to sets, and triples are sets, so the triple in question does not make sense.

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Thanks! I was wondering why triples are sets? Is a triple a tuple? Here it says a tuple is not a set en.wikipedia.org/wiki/Tuple#Characteristic_properties –  Tim Feb 14 '11 at 23:17
For the fine details of what things are... you should probably consult something other than Wikipedia. In the most usual presentation of set theory, everything is a set, so there "$X$ is not a set" is always false! One can consider forms of set theory which have tupling as a primitive operation... and it is up to the particular for of the theory you consider whether classes can or cannot be components of tuples. –  Mariano Suárez-Alvarez Feb 14 '11 at 23:19
My humble opinion is, unless you are doing something which really requires you to think about this, you simply ignore Wikipedia and its tangential remark about the codomain of functions defined on proper classes: Wikipedia has a way of mixing absurdly unhelpful information with its generally useful information, specially for people who are starting on various subjects. –  Mariano Suárez-Alvarez Feb 14 '11 at 23:21
@Tim: The remark in Wikipedia is simply to indicate that in tuples the order matters, not just the elements being listed. Formally, the set $\{2,3,3\}$ is the same as the set $\{3,2,2\}$ and they are the same as $\{2,3\}$, while the three tuples $(2,3,3),(3,2,2),(2,3)$ are all different. What they meant to say is: The tuple $(a,b)$ is not just the set $\{a,b\}$. Actually, there is a way using sets of making sense of tuples. The point is that all we care about is that $(a,b)=(c,d)$ iff $a=c$ and $b=d$, so we can "identify" the order in which $a,b$ are listed. (Cont.) –  Andres Caicedo Feb 15 '11 at 7:30
The usual set theoretic way of defining tuples is by setting $(a,b)=\{\{a\},\{a,b\}\}$. One can then show that indeed $(a,b)=(c,d)$ iff $a=c$ and $b=d$. Triples are then defined by, say $(a,b,c)=((a,b),c)$, and similarly for other tuples. Of course, it really does not matter whether ordered tuples are defined as above, and there are several alternatives. For some history and other options, see "Reconsidering ordered pairs", by Dana Scott and Dominic McCarty, The bulletin of symbolic logic, vol 14 (2008), 379-397, available at math.ucla.edu/~asl/bsl/1403/1403-004.ps –  Andres Caicedo Feb 15 '11 at 7:37

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