Cartesian products and the definition of a map

[wisvuze]

Hello, I was wondering if there were alternative definitions to a "function" ( alternative to the standard f is a subset of A X B if f : A -> B ).

I was introduced to the "general" definition of a cartesian product ( with respect to an indexing set H ) , it is weird to me because the general cartesian product is defined as a set of mappings, so it doesn't "quite" sync up with the standard "tuple" cartesian product of sets ( indexed by natural numbers ) as a generalization.. I thought I could get around this if the definition of a mapping doesn't rely on the original definition of a cartesian product, but I cannot think of another way to define a map.

However, is it unreasonable to leave a mapping as an undefined object? I am not too familiar with the depths of set theory, so I do not know if it would lead to disastrous results..
thanks :)

 

[micromass]

However, is it unreasonable to leave a mapping as an undefined object? I am not too familiar with the depths of set theory, so I do not know if it would lead to disastrous results..
thanks :)

Hello wisvuze!

That's not unreasonable at all! That's quite a valuable insight actually. Leaving mappings as undefined objects is basically one of the philosophies of category theory and topos theory. However, it will probably not come as a surprise that the resulting theory in radically different than the usual set theory! For example, the axioms that we must put on the functions might look weird, at first glance.

I recommend that you read the book "Set for mathematics" by Lawvere and Rosebrugh. It starts from the very basical things that you mentioned: seeing sets and functions as undefined objects (and in the beginning, there is not such thing as \in, this has to be defined). Gradually, the book imposes axioms to let the theory ressemble the usual set theory. In the end we end up with a(n elementary) topos, which is (roughly) the set theory that we know.

 

[wisvuze]

Thank you! I will check out these recommendations :)

 

[Hurkyl]

A less drastic line of study would just be type theory, or some sort of many-sorted logic.

e.g. when working with real numbers, you might have one type "R" (the type of real numbers), and another type "R -> R" (the type of real-valued functions of real numbers).

Then if f is an expression of type R -> R and x is an expression of type R, then we define the "evaluation" expression f(x) to be an expression of type R.

Like other logical operators (such as "and" or "implies" or "ordered pair of"), evaluation satisfies some properties that reflect what it means to be a function.




Really, the usual set-theoretic definition as a subset of the Cartesian product is simply an interpretation -- we interpret functions as being such things because it is a convenient way to treat functions in a theory that tells us about sets. The fact we represent things that way isn't really all that important for actually using functions; it's really just a technical tool to allow us to use a theory of sets to work with functions.