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Background: Explaining the confusion

Set

In difference between class, set , family and collection the named differences where discussed, but only to some extend. I am not even sure what counts as a duplicate. Let's take a clinical trial for example: If I have a data collection, as in a set of vectors, then I might have two samples that have the same feature values - however they might originate from two different patients. Is my data a set in the mathematical sense?

I read that sets also have no order in their elements. However, in my publications I regularly write things like $o_i\in O$ where $O$ is a set. Seems like an order to me. (I read the same by other publications.)

Multiset and Tuple

On the one hand difference between class, set , family and collection gave me the impression that the set is the only defined collection. On the other hand, I read that there are other collections like the multiset, or the tuple.

Tuple and Vector

From my perspective a vector is also a collection - a collection of scalars. They are both ordered and both can contain duplicates: $(1,1,2)$ is a valid three dimensional vector. I am not quite sure what the difference is? As I understand the elements of the vector need to have the same type, while the tuple has not this requirement. Does this mean a tuple can contain anything as elements (e.g. sets)? Why have this restriction on vectors and thus so many different collections? Is the vector a special type of vector?

Finite and infinite sets

It seems to me that this is a worthwhile distinction. An example would be a set of data objects (finite) vs. the real numbers (infinite). Are there names for these? Aren't vector spaces infinite sets? Aren't finite sets in the case of datasets actually multisets? (As discussed before.)

Questions

In order to get order from this chaos (in my head), basically I want to know

  1. what collections exist and
  2. what differentiates them from each other.
  3. Also I would like to know whether they are actually used outside of set theory.

To be honest, I have not seen anything but sets, vectors and vector spaces being used in my fields (statistics, machine learning, data mining and analytics, artificial intelligence, signal processing).

Further subquestions have been asked in my previous sections.

Further reading:

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  • A tuple $( o_i)$, with $o_i \in O$ is a function from ${ 0, 1, \ldots n-1 }$ to $O$. Functions a sets, and thus: YES, a tuple is a set. – Mauro ALLEGRANZA Apr 25 '17 at 13:37
  • A set does not have multiple elements, i.e ${ 1,2 }$ and ${1, 2, 1 }$ are the same set, while multises have. But a multiset is not a tuple, i.e. it has no order: ${ 1,2,1 }$ and ${ 1,1,2 }$ are the same multiset. – Mauro ALLEGRANZA Apr 25 '17 at 13:39
  • Finite vs infinite: worthwile. Correct; the theory of sets (Cantor) was developed to "manage" the infinite. – Mauro ALLEGRANZA Apr 25 '17 at 13:40
  • See The Early Development of Set Theory for the original motivations that lead to Cantor's theory and the discovery of paradoxes, that lead to the Axiomatization of Set Theory with the basic distinction between: collection and set. – Mauro ALLEGRANZA Apr 25 '17 at 13:42
  • In a nutshell, a collection is defined by a predicate or property: the collection of humans or the collection of natural numbers, while sets are those objects that the mathematical theory of sets assumes/proves that they exists. The Pardoxes has showed that some "reasonable" properties that we can describe with language produce unacceptable results; thus any well behaving theory of sets must leave the "natural" expectation that every propertys define a set ... – Mauro ALLEGRANZA Apr 25 '17 at 13:47
  • ... (see comprehension). The usual distinction is: every property describe a collection but not avery colelction is a set (in the precise sense of math set theory). – Mauro ALLEGRANZA Apr 25 '17 at 13:48

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