How can I express mathematically this indetermination? I'm trying to express mathematically this idea, where ka=$t_1$,kb=$t_2$,kc=$t_3$ ...,kn=$t_n$, n is a type of product(i.e.apple, book,...), k is the number of it, and $t_n$ is the unkown value of it. To know the value of a product it must be compared with another product. For example, $t_1$=kc. However, we don't know the value of kc, so the expression doesn't apply. In order to escape from this vicious circle, the owners of all products decide to make kc a general value. Therefore, kc has a concrete value since it is born, and the expression $t_n$=kc becomes logical. 
However, I think that even in this case the vicious circle remains. The value of kc has to have some economical basis and cannot be decided by the will of the owners. To become a general value, the original value of kc must be known, but in this case we fall again into the same dilemma.
Is there any way to express mathematically this indermination?
Sorry for the long text, and excuse my lack of mathematical knowledge.
 A: You could say that the set $V$ of values of products is a one-dimensional totally ordered real vector space. That is, $V$ supports the following operations:


*

*There is a distinguished zero value in $V$, the value of nothing.

*You can add two values to get the value of a basket of both products. If $t_1$ is the value of an apple and $t_2$ is the value of a book, then $t_1+t_2$ is the value of an apple and a book.

*You can negate a value: $-t_1$ is the value of owing someone an apple.

*You can multiply or divide values by real numbers: $\frac52 t_1$ is the value of two and a half apples; $t_2/3$ is the value of a third of a book.

*You can divide two values and get a real number: If $t_2/t_1=10$, then a book is ten times as valuable as an apple.

*You can decide which of any two values is the greater: in this case, $t_2>t_1$, and both are greater than the value of nothing.


However, since we are formalizing $V$ as an abstract vector space, and we are not setting $V=\mathbb R$, the following operations don't make sense:


*

*We cannot multiply two values and get another value.

*We cannot take the cosine or the cube root of a value and get another value.

*We cannot ask whether $t_1>5$, or whether $t_2$ is a prime number.


Now, although $V$ is not the same as $\mathbb R$, $V$ is isomorphic to $\mathbb R$. Given that $t_3>0$, you can agree to express every value in $V$ as a real multiple of $t_3$, which becomes your unit of currency, aka basis vector. In this way, you might say that $t_1=3t_3$ and $t_2=30t_3$ and so on, and feel that you "know" the value of $t_1$ and $t_2$ better than before. And although you'll write $t_3=1t_3$, you still won't have to commit to identifying $t_3$ with a real number.
