/sci/ - Science & Math

What is an "element index"?

First of all, sets generally can't have an index unless you have the axiom of choice, say in the form of
http://en.wikipedia.org/wiki/Well-ordering_theorem

Further, what you seem to describe are just lists,
<span class="math">\bigcup_{n\ge 0}X^n=\{\}+X+X\times X+X\times\dots[/spoiler]

>>6758763
Something to do with tuples? Vectors?

>>6758816
I mean where an element resides inside the set. I know basic set theory doesn't care for order.

I've never heard of the formal use of 'lists', is there a Theory of Lists or are they constructions of a union series in set theory?

>>6758840
As I said, there isn't a "where" in common set theory without postulating the order. If you do that, then you're just in order theory.

I guess the construction I gave captures the notion of lists, right?
For a broader theory of "indexed data structures", lets say, like lists, see e.g.
http://en.wikipedia.org/wiki/F-algebra

>>6758840
As I said, there isn't a "where" in common set theory without postulating the order. If you do that, then you're just in order theory.

I guess the construction I gave captures the notion of lists, right?
For a broader theory of "indexed data structures", lets say, like lists, see e.g.
http://en.wikipedia.org/wiki/F-algebra

ad: Here are lists, formally (and not just algebraically "produced out of set theory", as I did it above)
http://en.wikipedia.org/wiki/Initial_algebra#Use_in_Computer_Science

>>6758853
Cheers, I'm new to Set Theory et al so sorry about the obvious questions.

>>6758862
It's okay. My answer actually removes you from set theory a bit.
Maybe this reference would have been less obscure:
http://en.wikipedia.org/wiki/Algebraic_data_type

Btw. for an incomprehensive list (badum tss) of set theories, see

http://en.wikipedia.org/wiki/List_of_first-order_theories#Set_theories

I'll take the chance and also point out a fun fact.
There exists this:
http://en.wikipedia.org/wiki/List_of_lists_of_lists

>>6758763
You could use set theory itself, and even use that to generate the different actions.

A = { {1,1}, {2,2}, {3,3} }
B = { {1,3}, {2,2}, {3,1} }

>>6758911
You'd have to use tuples, as without ordering
{ {1,1}, {2,3}, {3,4}, {4,2} } = { {1,1}, {2,4}, {3,2}, {4,3} }

and if you use tuples
{ <1,1>, <2,3>, <3,4>, <4,2> }
then what you got here is (literally, formally) a map f, with f:{1,2,3,4}->{1,2,3,4}.
And a sequence is also preciely defined as such a function, usually denoted not by f, but something like a_n

>>6758816
>infinite sets can't have an index unless you have the axiom of choice
ftfy

OP here, just to say what I wanted to do explicitly - I was hoping to construct sets in the form of number systems like decimal and binary that acted as such under arithmetic.

Hexadecimal examples:
16 = {1,0}
1 = {1} = {0,1} = {0,0,...,1}
182 = {B,6}
107 = {6,B}
282 = {1,1,A}

>>6759015
okay, but now that we know that you need tuples (otherwise 182 = {B,6} = {6,B} = 107), what's this thread still about?
Of course, you can easily define a function f from numbers to the list of their coefficients. E.g. in base 10, you have
1452 = 1·10^3+4·10^2+5·10^1+2·10^0 --> <1,4,5,2> or whatever information you want to project out.
You can then, for each arithmetic operation b, define a operation b_set on the sets which do the same.
b_set := f b f^{-1}

>>6758763
>Is there a version of set theory where element index
I know of at least discrete probability with such concepts.

>>6759029
Glad I said it then! Yeah n-tuples seem pretty nice, then having functions map things is all quite simple.

I'm not sure what I was expecting, I thought it would be Principia Mathematica levels of building up to the system. This solution seems really simple.