Peano celebrations are in order

Every foo has a bar.
There is a foo, called fod.
No foo has fod as its bar.
Distinct foo have distinct bar; thus if ed and joe are foo, then ed is joe iff bar ed is bar joe.
If a statement is true for fod, and if being true for some foo means it is true for the bar, then it is true for all foo.

Oh, and: bar michiexile.