Are set theory functions sets too?

AI Thread Summary
Mathematical functions can be represented as sets of ordered tuples, linking elements from a domain A to a range B. Set theory also includes functions, raising the question of whether all functions can be defined as sets. While many functions can be expressed as sets, the power set function poses a challenge due to Russell's paradox, as its domain is not a set but a class of all sets. Consequently, the power set function cannot be described as a set and must instead be represented as a logical formula. This highlights the limitations of defining certain functions within set theory.
The UPC P
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I read somewhere that mathematical functions can be implemented as sets by making a set of ordered tuples <a,b> where a is a member of A and b is a member of B. That should create a function that goes from the domain A to the range B.

But set theory has functions too, could they be sets too?

For example the Power function would just be the set witht eh tuples <{},{{}}> and <{{}},{{}{{}}}> and so on. And the union and the pair function could be made into sets as well.

So what I want to ask is can all functions in set theory be defined as sets themselves?
 
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The UPC P said:
So what I want to ask is can all functions in set theory be defined as sets themselves?

Yes, if the domain of the function is a set. The power set function has as domain the class of all sets, this is not a set due to Russel's paradox. So the power set function can not be described as a set. Rather, it must be described as a logical formula.
 
OK thanks!
 

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