You can do that, but it's not compatible with the standard Peano axioms of N or extensions such as R.
Many Heyting algebras have those properties, and for any finite number n, there typically exist many such algebras.
There, is taken as \top element , and
In the logical semantics, a statement x AND a true statement has the same truth value as x (where there might be 2 or more truth values overall).
This top is also called "infinity", as the cardinalities of limit ordinals have this property w.r.t. the cardinality. As in