Equal sets with different symbols?

[bahamagreen]

TL;DR

Where or how is the intention of the "type" or "category" of the membership of a set defined such that one may determine whether something is or is not a member?

{1, 2 ,3} = {1, 2, 3, 3, III}?
{1, 2 ,3} = {one, dos, three}?
{Tom, Dick, Harry} = {Thomas, Richard, Harrison}?

Seems to me, these are undetermined until the set's "type" or "category" definition of its members is defined so as to determine what elements are members of the set... whether membership fails from a difference in typeface, difference in symbol, difference in language, or in this last case difference in membership based on whether members are individuals, or just their names.

Does set theory have this "type" or "category" definition of membership?
If so, what is it called? What form does it take (I haven't seen it)?

 

[mitochan]

Hello. I am not good at set theory at all but let me give you a thought about your problem.

Let say the set A = {x|all the symbols and words meaning 1 or 2 or 3}
A ⊃ {1,2,3}
A ⊃ {1, 2, 3, 3, III}
A ⊃ { one, uno, ein }
The relation among the right hand subsets depends on members they contain,e.g.
{1,2,3} ∩ {1, 2, 3, 3, III}={1,2}

 

[bahamagreen]

Thanks, that goes right to the heart of what I'm asking.

If set A is "all the symbols and words meaning 1 or 2 or 3", then the intersection you show excludes those different symbols 3, 3, 3, and III that mean 3 because the intersection considers the different shaped symbols themselves as different, despite being equal and all meaning the same thing.

If I were writing with chalk on a black board, my symbols for 3 would not be identical, yet my intent would be the meaning of 3. Even page printed 3s if examined closely will not be identical. Some people write 4 like the way it looks there with a closed top vertex, but others write it with an open top, some put a bar on the stem of their 7s, etc.

What if I do this?

Let say the set A = {x|equal to the first three positive integers}
A ⊃ {1,2,3}
A ⊃ {1, 2, 3, 3, III}
A ⊃ { one, uno, ein }
Now the relation among the right hand subsets depends on the equality of members they contain, or still just identity of the specific symbols?

Well, if I try to do it by equality {1,2,3} ∩ {1, 2, 3, 3, III}={1,2,"3"}, but now which member representation(s) equaling "3" might be in the intersection?

Are there no operations which work using meaning of the sets' members vs their various symbols?

 

[mitochan]

If you want to understand set theory well, I strongly recommend you to read good introductory texts.

After stating so, let me add something I thought. Not all the relations form sets. Only the relations that give definite statement for everything "yes, it belong to me" or "no, it does not belong to me", form sets.
Is each #3, three, tres, this dirty hand writing figure on black board, the teacher, my mother, etc. equal to 3 ? When everybody inovloved has common definite yes or no answer for each, the set {x|equal to 3} is formed. If you find any ambiguity on the belongings, you have to solve them first to make use of set theory.

 

[fresh_42]

Summary: Where or how is the intention of the "type" or "category" of the membership of a set defined such that one may determine whether something is or is not a member?

Seems to me, these are undetermined until the set's "type" or "category" definition of its members is defined so as to determine what elements are members of the set... whether membership fails from a difference in typeface, difference in symbol, difference in language, or in this last case difference in membership based on whether members are individuals, or just their names.

No, this is not true. $\{\,1,2,3\,\}$ and $\{\,1,2,III\,\}$ are two different sets. $\mathbf{3}$ or $\underline{3}$ would have worked as well, but I like to keep the typesetting discussion out. If elements are not equal, then the sets containing these elements can't be equal. It is immediately clear, that there are two different questions unanswered:

1. What is the universal set elements can be chosen from to build a set?
2. Are there equivalences to be considered, e.g. the number of elements?

As long as the first question is not answered, any thinkable set of elements can be considered, which makes $3$ and $III$ distinguishable and so its sets. If we only consider e.g, natural numbers, then we set $3=III$ which is already an equivalence relation which makes $3$ and $III$ two representatives of the same equivalence class.

So the answer to your question lies in the undetermined question itself. It vanishes as soon as you post the question rigorously.

 

[bahamagreen]

I'm not clear what you are saying. If we answer the first question and consider just the natural numbers where 3 and III are two representatives of the same equivalence class, are you saying {1,2,3} and {1,2,III} are no longer two different sets?

edit - Does set theory have a method where a symbol stands for all representations of an equivalence class, so that one may substitute the equivalence class symbol for the corresponding elements in operations like intersection?

 

[fresh_42]

I'm not clear what you are saying. If we answer the first question and consider just the natural numbers where 3 and III are two representatives of the same equivalence class, are you saying {1,2,3} and {1,2,III} are no longer two different sets?

If we only allow natural numbers, then we have either to identify $3=III$ or $\{\,1,2,III\,\}$ isn't allowed anymore. In the first case we collect all representations of a certain number in one equivalence class, which can be represented by any of its elements, or the second "set" does not occur anymore since it is undefined.
In the first case, it would be better to note the fact that we took equivalence classes, e.g. $\{\,[1],[2],[3]\,\}=\{\,[1],[2],[III]\,\}$. However, we usually do not consider various representations at the same time, so the question about equivalence classes does not come up and we can simply write $\{\,1,2,3\,\}\,.$

edit - Does set theory have a method where a symbol stands for all representations of an equivalence class, so that one may substitute the equivalence class symbol for the corresponding elements in operations like intersection?

See above. It usually occurs in group theory where the elements of $\mathbb{Z}/2\mathbb{Z}$, which are the two equivalence classes $\{\,\text{ odd numbers }\,\}$ and $\{\,\text{ even numbers }\,\}$ are identified with $\mathbb{Z}_2=\{\,0,1\,\}\,.$ The same again: if it is clear that we operate in $\mathbb{Z}_2$, then $0,1$ as element notation will do. If we want to stress that they stand for either even or odd numbers as equivalence classes, we can write
$$
0=[0]=\underline{0}=\mathbf{0}=0+2\mathbb{Z}\, , \,1=[1]=\underline{1}=\mathbf{1}=1+2\mathbb{Z}
$$
It's a matter of taste. You could even write $0=\{\,\ldots,-4,-2,0,2,4,\ldots\,\}$ and $1=\{\,\ldots,-3,-1,1,3,\ldots\,\}$.

 

[Stephen Tashi]

Summary: Where or how is the intention of the "type" or "category" of the membership of a set defined such that one may determine whether something is or is not a member?

We have to distinguish between questions about mathematics versus questions about the application of mathematics to particular problems.

If you are going to apply set theory to a particular problem, it must be the case that, for each "element" you consider, you can determine if that element is in a set that you wish to consider. So you can't proceed to apply set theory until you specify what your notation means. If you are applying set theory to typographical symbols, you might wish to say that 3 $\notin \{1,2,3\}$. If your application does not distinguish a difference in meaning between "3" and "$3$", you can say 3 $\in \{1,2,3\}$.

In all situations where mathematics is applied to particular problems, it is necessary to specify unambiguously how the relevant mathematical structures correspond to structures in the problem. I don't know if there is a technical terminology for establishing this correspondence - or whether "type" or "category" are terms used in describing the process. Set theory doesn't cover the topic of how set theory is to be applied.

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The above explanation is an easy answer because it considers applying mathematics to things outside of mathematics. We can ask a much harder question (which I don't know how to answer!). What happens when we apply mathematics to mathematics? For example, if we begin to discuss set theory in terms of a relation denoted as "$\in$", are we using a term ("relation") whose definition uses the concept of "set"? If we develop set theory by using logical quantifiers and make statements with notation like "$\forall x$", are we implicity assuming "$x$" denotes something in a particular set of things?