# Talk:Enumeration

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Field:  Foundations, logic, and set theory

This is a circular reference for enum in computer science. Can some one who is knowlegable about enum in computer science fix this?

The pages have been split and the disambiguation page has been fixed. The CS enum page has moved to enumerated type. TooMuchMath 05:44, 14 April 2006 (UTC)

Could it be argued that in a more abstract sense, enumeration is the application of not just natural numbers but any non-repeating set of symbols to a list? The required usage of natural numbers appears overly restrictive as relying on some formal number theory as an underlying definition of enumeration. It appears that without relaxing the natural number requirement, then a listing such as "a. thing, b. different thing, c. something else" is catagorically not an enumeration, so that enumeration is not applicable to listing and outlining in general. Hotfeba 18:10, 2 April 2007 (UTC)

First there's the tricky problem about what you mean by a "set of symbols". For instance, if I take my set of symbols to be lines of arbitrary length, where lines of different length are different symbols, then my set of symbols is in 1-1 correspondence with the positive real numbers. This set is uncountable and, in particular, not enumerable. (Yes, at the atomic level this eventually breaks down, but I'm a mathematician, not a physicist.)
Perhaps your objection is more in regards to the fact that we could enumerate a set with, say, binary numbers, or Roman numerals. This is just a labelling issue - the set of natural numbers is independent of these particular representations. We can also throw away certain "aspects" of the integers such as addition and multiplication since these do not (in most cases) have anything to do with the enumerations.
However the natural numbers are ordered, and this "set of symbols" may not be naturally ordered. This doesn't matter so much for cardinality considerations but makes a big difference if the enumeration is being used to iterate over the set, or to identify items in a set. For instance in computer programming an (integer indexed) array is an enumeration of the elements of the array. The ordering of the elements in this array can become quite important due to locality of reference. More generally, enumerating an arbitrary set can be thought of as the process of ordering the elements. Of course if your set of symbols S is enumerable and infinite (f:N->S bijective), then an enumeration of another set X (g:N->X surjective) would lead to an "enumeration by symbols" given by h:S->X h = f^-1 * g. TooMuchMath 02:47, 5 April 2007 (UTC)
Sorry, I may have been sloppy in my query. I did not consider sets of countably infinite size or larger because the practicality of human usage does limit the labelling of items in a list to sets of symbols that have one-to-one correspondence to subsets of the natural numbers and not reals, without the worry about ordinal arithmatic. I am intrigued by your example of lines of different length, as this points to sets of symbols of arbitrary complexity as uncountable and therefore not enumerable as well. Your statements on the labelling issue and disregarding math operations on integers appear to satisfy what I was looking for. Thanx! Hotfeba 23:35, 9 April 2007 (UTC)

## Very unbalanced article

Hi, sorry to drop in somewhat rudely, but being mostly interested in enumerative combinatorics I find this article shockingly unbalanced. For most ordinary human beings enumeration and counting has to do with finite sets, and mathematically this is also by far the most interesting case, (since once one hits infinite cardinality there are zillions of variations that are no longer distinguished by cardinality; I mean of course uncountably many (ordinals)). However this article hardly seems interested in that case at all. Finite sets only get marginal mention in the section on countable versus uncountable context. (The non-mathematically trained reader might also be confused by the idea of counting in an uncountable context.)

Also the following quote put me off a lot

If one works in Zermelo-Fraenkel set theory without the axiom of choice, one may want to impose the additional restriction that an enumeration must also be injective (without repetition) since in this theory, the existence of a surjection from I onto S need not imply the existence of an injection from S into I

First I though this was complete nonsense because I had not noticed the reversal of the arrow. But in any case everybody knows proper counting requires a bijection, not injections or surjections, so it is unclear to me why this point is pertinent at all, and the remark remains utterly confusing.

Again excuse for the somewhat rude tone, but I really think this article needs a thorough rewrite for encyclopedic style and serving general public before aiming at some specialism. Marc van Leeuwen (talk) 08:04, 5 March 2010 (UTC)

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