intentional and extensional set definition so I'm HUGE noob in math so please be patient.
I have a question about set definition. They thought us in class that there are two ways to define a set of elements:


*

*Extensional:
Basically just write the elements of the set. For example: 
$$A= \{ 1,\ 2,\ 3,\ 4, \ 5\}$$

*Intensional: 
Defines a set by stating the unique property which characterizes  the ones and only elements of the collection, without ambiguity, like: 
$$A = \{x\ | \ x\in\ \mathbb{N} \quad \land\quad 1\le x \le 5\}$$
Now, this works fine for set of elements where the elements are numbers, but how can I intensionally define a set like this:
$$TV= \{ fox,\ cnn,\ sky,\ msnbc\}$$
The professor gave us this exercise in class and some students gave answers like
$$TV= \{x\ | \ x\in \text {TV_CHANNELS} \quad \land\quad \text{daily_views}\ge100k\}$$
But it looks wrong to me, I mean if one day msnbc daily_views are $\le100k$ then $\ msnbc\  \notin TV$ which means I have a different set... 
So my question is how do I intensionally describe a set of things, different from numbers, without ambiguity?
 A: The ambiguity arises because the real world (in which CNN, Fox, etc. operate) is fundamentally ambiguous.  This is in contrast to the Platonic world of mathematics, where objects, operations, and ideas can be defined in an unambiguous manner.
In this case, we would want to unambiguously define what is meant by "daily_views".  Is this some kind of average?  Is it a measurement taken at only one point in time?  Or does the set TV depend on some variable (such as time)?  We would also need an unambiguous definition of the set TV_CHANNELS, and we would likely want to specify that "100k" means 100000.
A: $B=\{1,2,3\}$  is an  abbreviation for $\forall x\; (x\in B\iff (x=1\lor x=3\lor x=3))$. 
$B=\{x:x\in \Bbb N \land 1\leq x\leq 3\}$ is an abbreviation for $\forall x\; (x\in B\iff (x\in \Bbb N \land 1\leq x\leq 3)).$
A: You need to introduce properties that characterise the individuals you want to describe. Say, they are all fs, i.e:
f(cnn) is true
f(foxnes) is true 
etc
and f needs to be such that:
for all x, if f(x) then (x = foxnews or x = cnn or ...)
It's nothing to do with the inherent ambiguity of the world. It's to do with your ability to formalise the properties you these things have. 
An intensional definition is then forthcoming, rather trivially, 
A is the set of x such that f(x).
You always work in a domain of discourse, with a theory, with a language that allows you to account more or less adequately for the nature of things -- in reality or in 'the Platonic world' of numbers if you believe in this interpretation. 
You may be well interested in looking at the use of formal logic in something that has everyday applications and that is called 'knowledge representation'. This is a discipline in AI that makes intensive use of the intensional approach in order to articulate knowledge about real world things. 
It is a legitimate question to ask what counts as a number of view, within which time frame and so on. Note, however, that when you are asking that, you are trying to understand the nature of these things. 
You might find this intersting: https://en.wikipedia.org/wiki/Knowledge_representation_and_reasoning
For a real world application in which people wonder what a TV channel is and how to categorise them, see for example: http://dbpedia.org/ontology/TelevisionStation
I find it's good that your teacher would make the bold move, although it is a bit difficult to appreciate the simplicity of the applicability of the concept in a context where you expect things to be very abstract and make little room for intuition. 
Have fun!
