What is the No Free Lunch Axiom? I have tried to know what exactly is the no free lunch axiom.

I got multiple links but none of them explain (a bit rigorously, with examples and mathematical formulation) what is this axiom and how I can formulate it in a problem.

Any explanation/formulation is very much welcome. Thanks.
First, to give a little more background to 123's answer, a production set $Y$ is the set of all feasible production values. With $y \in Y$, $y$ is a vector in $\mathbb{R}^L$, where positive elements indicate outputs, negative values indicate inputs. The inputs you put in that get you outputs can be expressed with either a transformation or production function.

123 is right when stating the formal microeconomic axiom:


  If that $y \in Y$ and $y \geq 0$, then $y = 0$.


However, no free lunch axiom needs more than merely "passing through the origin."

The first image shows a production set (two dimensions) that satisfies no free lunch. The second image shows a production set that does not, even though it passes through the origin.

(yes this image is a bad rendition of the one from Mas-Colell)



Interestingly though, the author has not tagged this with a microeconomics tag, but with a macro and financial econ tag. The closest thing I can thing of to a no free lunch axiom here is some sort of transversality condition. For example, in simple money market models with borrowing with households, you might have a condition that you have to pay off all debt eventually (or just pay the interest for the rest of your life). So that would look something like,

$$\lim_{t \rightarrow \infty} \frac{b_t}{(1+r)^t} = 0$$

where $b$ is the amount of borrowing in a period. I do not hear conditions like these called "no free lunch" however. Something more like "no ponzi scheme condition" or "borrowing constraints" (snoree) are more apt.