## Lucky Duck Induction

Zero is a lucky duck.

Each lucky duck is a mom lucky duck.

Zero is not a baby lucky duck.

Each baby lucky duck has exactly one mom lucky duck.

If a farm has the zero lucky duck and the baby lucky duck of
each mom lucky duck on the farm, then the farm has all the lucky ducks.

Being a baby luck duck means it has a mom lucky duck.

A summary is:

Zero is a lucky duck.  Every lucky duck is a mom duck with one baby, but zero is not a baby, each baby has one mom and a farm with zero and the baby of each mom on the farm has all the lucky ducks.

Zero is a duck, not a baby. Each duck is a single mom with a single baby. A farm with zero and the baby of each mom on farm has all ducks.

Alternatively, we could write

Zero is a lucky duck.

Each lucky duck is a mom lucky duck.

Zero has no mom lucky duck.

Each lucky duck other than zero has exactly one mom lucky duck.

If a farm has the zero lucky duck and the baby lucky duck of
each mom lucky duck on the farm, then the farm has all the lucky ducks.

The first version works better.