How to define a function in first order logic that return different values each time?

How to represent f() in first order logic so that following scenario holds?

M=f() N=f()

where M $!=$ N

• You don't. Axioms of equality deman that $M=N$ if $M=f()$ and $N=f()$. – Hagen von Eitzen May 19 '17 at 6:25
• how to represent a function f so that $M != N$ – Tom May 19 '17 at 6:34
• You cannot: a function in mathematics is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. – Mauro ALLEGRANZA May 19 '17 at 6:42
• But the symbol M=f( ) is meaningless: what is the input value ? When you say "a function that return different values each time?" what do you mean with "each time"? different input values? are there instant of time ? – Mauro ALLEGRANZA May 19 '17 at 6:43
• Yeah, by definition a function takes exactly one value on each argument. And there is no such notion as "each time" in logic. What you can do however is for instance to consider a function with one more argument, $t$, express that $t$ ranges over a linearly ordered set, and think of $f(x,t)$ as being the value "at time $t$" of a "function that depends on time". – Régis May 19 '17 at 7:07