# Forward referencing in mathematics

• V0ODO0CH1LD
In summary, the conversation discusses the concept of "forward referencing" in mathematics, where an object is mentioned within its own definition. The question is whether this is allowed and if there are ways to define objects without mentioning them. It is possible to make valid recursive definitions, but caution must be taken to avoid contradictions.
V0ODO0CH1LD
In most programming languages, mentioning an object inside it's definition can cause a lot of trouble at compile time because what's being mentioned doesn't technically exist yet.

But in mathematics is "forward referencing" allowed? Can I, for instance, define a set and use the set being defined in the definition??

For example, using set builder notation:

S = {x in X : P(x)}.

Where the formula P makes mention of S? Like,

S = {(a,b) in AxB : for all b' [((a,b) in S and (a,b') in S) => (b = b')]}.

Is that allowed in mathematics? If not, is there always a way to define things not mentioning them? In the case of the functional relation above, how would I do it?

V0ODO0CH1LD said:
For example, using set builder notation:

S = {x in X : P(x)}.

Where the formula P makes mention of S? Like,

S = {(a,b) in AxB : for all b' [((a,b) in S and (a,b') in S) => (b = b')]}.

Is that allowed in mathematics? If not, is there always a way to define things not mentioning them? In the case of the functional relation above, how would I do it?

Unless B is either empty or a singleton your definition of S makes no sense anyway. So I would work on fixing that first before worrying about the whole "forward referencing" issue.

My second definition of S does not matter, it was just an example.. You can forget about it completely if it helps. The actual question has to do with referencing objects within their definition. I'm just wondering if that is allowed or if it raises some contradiction.

V0ODO0CH1LD said:
I'm just wondering if that is allowed or if it raises some contradiction.

If people are careless with their writing, then it happens sometimes. In principle one should be able to avoid this however.

I don't see any reason why you can't make a valid recursive definition, in a similar way to defining a recursive function in a programming language. But you need at least one alternative in the definition that is not recursive.

But be careful - if you try to define "the set of all sets that are not members of themselves", bad stuff happens

## 1. What is forward referencing in mathematics?

Forward referencing in mathematics refers to the use of information or variables that have not yet been defined or calculated. This can occur when solving equations or problems that involve multiple steps, where one step relies on the results of a previous step that has not yet been completed.

## 2. Why is forward referencing important in mathematics?

Forward referencing is important in mathematics because it allows for the solving of complex problems by breaking them down into smaller, more manageable steps. It also allows for the use of variables and unknowns in equations, making it easier to solve for a specific value.

## 3. How do you handle forward referencing in mathematical equations?

To handle forward referencing in mathematical equations, you must first identify and label all known and unknown variables. Then, you can use algebraic operations to rearrange the equation and isolate the unknown variable. This will allow you to solve for the unknown value using the information provided in the equation.

## 4. What are some common mistakes when dealing with forward referencing in mathematics?

One common mistake when dealing with forward referencing in mathematics is forgetting to label or identify all variables in an equation or problem. This can lead to confusion and incorrect solutions. Another mistake is not properly rearranging the equation to isolate the unknown variable, which can also result in incorrect solutions.

## 5. Can forward referencing be used in all branches of mathematics?

Yes, forward referencing can be used in all branches of mathematics, including algebra, geometry, calculus, and more. It is a fundamental concept in problem-solving and is essential for solving complex equations and problems.

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