Kuratowski's Definition of Ordered Pairs

[CRGreathouse]

b>a, c>b (antisymmetric); c>a (transitive); c>b>a (total). Just say I want to extend this process through the whole alphabet. I can obviously do this by typing a simple command on the computer, but I want a formal and concise expression like (a<b<c<...<y<z)

So write that, and say that you have a total order. Depending on what the individual variables are you could represent it more efficiently if needed (say in binary), but since no one on this thread except possibly you knows what you want it's hard to say more.

For example, if you have n numbered variables and a total order amongst them, you could store it as about n ceil(log_10(n) + 5) bytes in ASCII ("a_1413 < a_2208 < ..."), or n ceil(log_256 n) bytes in a binary format, or ceil(log_2(n!)) in factoradic, and so on. But none of this has anything to do with ZFC or Kuratowski.

 

[CRGreathouse]

No, it's not wrong. Expressions like that are needed in foundations theory, but Set Theory, to me at least, to me, is also of practical value.

Practical, yes, but there's no reason to only use one formalization. Use whatever is convenient.

You don't write the Pythagorean theorem as "=+^a2^b2^2c", do you?

My Harper Collins Dictionary of Mathematics EJ Borowski and JM Borwein eds, 1991 p421 defines an 'ordered set' as "a sequence of elements that is distinguished by both identity and by the order of elements, so that &lt;a,b&gt; is not identical to &lt;b,a&gt; unless a=b." I'm not sure how well established this use of angle brackets is as part of Set Theory. I assume they are not formal symbols under ZFC.

The definition is standard. The notation is not -- () is more common. But you can extend just about any set theory you want with this definition without strengthening it

 

[SW VandeCarr]

Again, it sounds like you want a data structure rather than set theory of any kind.

Yes, but it's very useful to define ST concepts like union, intersection, subset, set membership and set elements for data sets. It also relates to the distinction I make between equality and identity which was the topic of another post in this forum.

 

[CRGreathouse]

Yes, but it's very useful to define ST concepts like union, intersection, subset, set membership and set elements for data sets.

OK, now I'm convinced you want a data structure. Computer science spends a lot of time working out new data structures that are faster at certain combinations of such operations. See for example
http://www.cs.sunysb.edu/~algorith/files/set-data-structures.shtml
for a list of implementations.

 

[SW VandeCarr]

OK, now I'm convinced you want a data structure. Computer science spends a lot of time working out new data structures that are faster at certain combinations of such operations.

Thanks. I'm interested in combining concepts from Set Theory with object oriented programming among other things.

http://books.google.com/books?id=4x...esult&ct=result&resnum=4#v=onepage&q=&f=false

 

[CRGreathouse]

Got it. So yes, you really don't want the be expressing things in first-order logic directly -- you want efficient data structures resembling the set-theoretic objects.

An important consideration here is how you want to use things. For example, here are two ways to code a multiset:
* As an expandable array, containing multiple copies of an object if it's in the multiset more than once.
* As a fixed-length unsigned integer array, with each uint representing the number of times the element is in the multiset.

The second is good when you have many copies of each element and few distinct elements. The first is good when you have few copies of each element or many distinct objects.

 

[SW VandeCarr]

Got it. So yes, you really don't want the be expressing things in first-order logic directly -- you want efficient data structures resembling the set-theoretic objects.

An important consideration here is how you want to use things. For example, here are two ways to code a multiset:
* As an expandable array, containing multiple copies of an object if it's in the multiset more than once.
* As a fixed-length unsigned integer array, with each uint representing the number of times the element is in the multiset.

The second is good when you have many copies of each element and few distinct elements. The first is good when you have few copies of each element or many distinct objects.

Thanks GR. The first item is what I'm doing now I'm not using actual data now but working on a general model for organizing data for subsequent analysis. My background is in medical epidemiology and I've worked with large data sets. I'm now retired (not too old yet though)and I'm free to think about the general principles of organizing data prior to statistical analysis. The latter is straightforward once the objects and methods are defined.

 

[CRGreathouse]

Yes, but you should have an idea of what the data would look like. So if you're measuring, say, the expected spread of a disease you have billions of nodes, each of which will connect with very few others; you'd want to use a sparse representation of a matrix rather than a dense one.

I don't know what epidemiological application you have for tuples (so many possibilities!) so I won't hazard a guess as to what sort of representation would be best there.

 

[SW VandeCarr]

Yes, but you should have an idea of what the data would look like. So if you're measuring, say, the expected spread of a disease you have billions of nodes, each of which will connect with very few others; you'd want to use a sparse representation of a matrix rather than a dense one.

I don't know what epidemiological application you have for tuples (so many possibilities!) so I won't hazard a guess as to what sort of representation would be best there.

Epidemiology has become much more mathematical in the last 30-40 years, and it isn't just concerned with "epidemics". It's really concerned with outcome likelihoods given the relevant data. It can involve very large numbers of data points with many variables. Vectors in vector spaces with a large number of dimensions require large tuples.

 

[CRGreathouse]

Epidemiology has become much more mathematical in the last 30-40 years, and it isn't just concerned with "epidemics".

I have a friend who recently got his Master's in the field. I'm not terribly familiar with it, but I know at least that much. :)

It's really concerned with outcome likelihoods given the relevant data. It can involve very large numbers of data points with many variables. Vectors in vector spaces with a large number of dimensions require large tuples.

This still doesn't tell me what kind of data structure you need.

 

[SW VandeCarr]

This still doesn't tell me what kind of data structure you need.

There are several structures that could be used, but I think I indicated that an expandable array (by row and/or column) is the one I'm investigating. These arrays would be stacked in temporal sequence. My interest is defining meaningful ways to construct sets in terms of the hypothesis being evaluated using a fixed but expanding data base. That is, data already stored doesn't change or repeat. Because time series analysis can be done, simulations are also possible.

The basic structure is relational but the stacking provides a number of table profiles: cuts along columns, cuts along rows, and "horizontal" point in time cuts. SQL (I'm told) can be used, probably in conjunction with an object query language. Column order is considered fixed (hence ordered tuples), while row order need not be fixed, but in practice, probably would be.

http://www.agiledata.org/essays/mappingObjects.html

 

[informatimago]

1. Why is an ordered pair defined in terms of unordered pairs? Doesn't {{a}, {a, b}} = {{a, b}, {a}} = {{b, a}, {a}}, and if so, how does this in any way become ordered

There are different order relationships. The inclusion relationship between two sets is an order. So we can say that R<S ⇔ R⊂S.

Therefore it is easy to distinguish, and order, the two sets given in this set:
{a}<{a,b}={b,a} ⇔ {a}⊂{a,b} which is true.

We can define L(X) as e where {e}=y such as y∈X and ∀z∈X, y⊂z.
Similarly, R(X) is e where {e}=z∖y such as y∈X a, z∈X and y⊂z.

(However, these definitions have a difficulty with (a,a) which is represented by {{a},{a,a}}={{a}}; we would need a more complex definition R for this case).

2. How was this definition arrived at? Where I've looked, it's usually just stated without any context for why or how it emerged,

I guess it occurred during the axiomatization of mathematics and set theory. Once set theory had been axiomatized, logicians wanted to define the rest of mathematics in terms of sets.