# How can you translate the mathematical statement 5=5 into a second order symbolic logic statement?

Hi I'm trying to describe the statement 5=5 using symbolic logic. I initially attempted to describe this into English as:

There is such a thing x and such a thing y, such that x is equivalent to y, iif for all x, x equals {a,b,c,d,e} AND for all y, y equals {a,b,c,d,e}.

and then more formally as:

(x)(y)((x=y)≡(∀x(x={ a, b, c, d, e}) ∧ ∀y(y={ a, b, c, d, e}))

Is this correct? Does it even make sense?

• It follows directly from the axioms of set theory. Mar 28, 2015 at 17:29

## 1 Answer

`(5=5)` is already an atomic sentence in FOL=. It evaluates to true. If you wanted, you could get unnecessarily complicated and say something like `∀x∀y(((x=5)∧(y=5))→(x=y))` but this idea is already incorporated into what `=` does.

Equality in FOL (or SOL for that matter) is defined outside of the logic (in metalogical terms as an equivalence class, yadda yadda) so we can just use it out of the box. Set theory does something a bit funkier with equality (of sets), but vanilla FOL/SOL doesn't know anything about sets.