From researchgate: https://www.researchgate.net/post/What_is_the_actual_difference_between_1st_order_and_higher_order_logic What is the actual difference between 1st order and higher order logic? Yes, I know. They say, the 2nd order logic is more expressive, but it is really hard to me to see why. If we have a domain X, why can't we define the domain X' = X u 2^X and for elements of x in X' define predicates: SET(x) ELEMENT(x) BELONGS_TO(x, y) - undefined (or false) when ELEMENT(y) etc. Now, we can express sentences about subsets of X in the 1st-order logic! Similarly we can define FUNCTION(x), etc. and... we can express all 2nd-order sentences in the 1st order logic! I'm obviously overlooking something, but what actually? Where have I made a mistake?

## First and second-order logic formalizations

## First and second-order logic formalizations

## First and second-order logic formalizations

From researchgate: https://www.researchgate.net/post/What_is_the_actual_difference_between_1st_order_and_higher_order_logic What is the actual difference between 1st order and higher order logic? Yes, I know. They say, the 2nd order logic is more expressive, but it is really hard to me to see why. If we have a domain X, why can't we define the domain X' = X u 2^X and for elements of x in X' define predicates: SET(x) ELEMENT(x) BELONGS_TO(x, y) - undefined (or false) when ELEMENT(y) etc. Now, we can express sentences about subsets of X in the 1st-order logic! Similarly we can define FUNCTION(x), etc. and... we can express all 2nd-order sentences in the 1st order logic! I'm obviously overlooking something, but what actually? Where have I made a mistake?