Set theory: how is $\left\lbrace 1, 1 \right\rbrace$ different from $\left\lbrace 1 \right\rbrace$? What set theory differentiate  sets: $\left\lbrace 1, 1 \right\rbrace$ from $\left\lbrace  1 \right\rbrace$; order does not matter?

Thank you in advance.

Addition/clarification: Thank you very much, everybody, for the providing idea of multi sets - I have found few papers and I hope it will be useful. 

I am from microscopy/Imaging field without formal mathematical education, and I try to find the most basic mathematical object which can represent image. I was thinking that set would be the most basic/universal. It works well for just one measurement (or pixel), e.g. {1}; if we have two measurements (pixels), we can have two identical values {1, 1} from set of rational numbers. In classical set theory, it immediately collapses to {1}, so I lose my information. Practically it can happen if I measure number of photons from fixed point inside my object (cell) and I do not care about order of the measurements.

The idea of set is quite attractive as I can embed it in other sets, form universes, etc. On other side, I feel that image always requires some sort of order - except from the very basic measurements I have mentioned before. It can be ordered pair of set of rational numbers and structures, e.g. Cumulative hierarchy which represent pixels (or measurements) and order of the image. I am not sure how easy I can work with them. 

Meanwhile I appreciate all your answers, comments and suggestions.
People often refer to these sorts of objects (containers where the order of the elements within them doesn't matter, just how many times each appears) as "multisets".

From any theory of ordinary sets you get a corresponding theory of multisets, and vice versa; an ordinary set can be thought of as just a multiset where nothing appears more than once, and conversely, a multiset can be thought of as just an ordinary set along with a function from that ordinary set to positive cardinalities (representing the number of repetitions of each element).
In elementary set theory, we do not distinguish $\{1,1\}$ from $\{1\}$, we simply characterize a set by the elements it contains. In a similar vein, we say that $\{1,2\} = \{2,1\}$. In general, two sets are equal (by definition) if and only if they contain the same elements, i.e., $A=B \iff \left[x \in A \iff x\in B\right]$. I'll leave it to you to verify that this condition is true for the above two examples.