St. Patrick's Day Note a Month Late
Consider the following axiomatic system:
1. Leprechauns exist.
2. Leprechauns are non-human humanoids all of whom are three feet tall or less.
3. Leprechauns occur only in Ireland.
4. Leprechauns’ clothing and possessions consist only of things producible by the technology of the late Middle Ages.
5..Some leprechauns have hair and beards.
These axioms seem consistent. Their consistency is as nearly obvious as that of axioms in some other well known systems.
One can use these axioms to prove meaningful theorems such as
1. Every humanoid over five feet tall is not a leprechaun.
2. Leprechauns do not have cell phones.
3. The clothing of leprechauns is made from natural materials.
One can elaborate the system by adding another axiom such as
“6. A red unicorn lives in the same part of Ireland as the leprechauns do”,
and argue that both it and its denial likely are consistent with the first five. This axiom would be useful in proving theorems not provable by the first five alone.
Of course the system and its conclusions are false and have nothing to do with reality. There are not any leprechauns. One can create a very similar system by replacing the original axiom 1 with “Leprechauns are fictitious creatures of stories and legends.” The new system would be demonstrably consistent and also true, since one easily could display a storybook containing leprechauns satisfying all five axioms.
This fairly silly example illustrates the fact that in considering systems of axioms, one has to look at each axiom and decide whether it is really axiomatic, that is .whether it should be accepted as true about something. One cannot escape from the need to consider meaning. The same thing holds in cases that are not silly.
Labels: Axiomatic systems, jokes, logic