## Friday, February 6, 2015

### Foundation Paradox and the Last Digit

Foundation Paradox and the Last Digit

The Foundation Paradox comes in various flavors, including the First Cause and the Munchausen Trilemma. To wit:

The First Cause:
Either: the chain of causation goes back infinitely, and there is no first cause;
Or: there is a first cause, itself uncaused;
Or; there is a first cause, caused by another cause, itself caused by the first cause, so causation loops.
And in none of these three cases is causation well-founded.

Munchausen Trilemma:
Any explanation is either:
infinite (it goes back forever);
incomplete (it starts with an unexplained explanation);
or circular (it loops).
And none of these are properly explanatory.

These two are clearly analogous; and they have many variants. (For instance, does the meaning of life have a meaning?)
The difficulty is often posed as theological, or scientific; but really the problem is mathematical. For consider the decimal expansion of a real number. Does it have a last digit?
Either: the decimal expansion is infinite, and there is no last digit;
Or: the decimal expansion terminates with a last digit and nothing but zeros thereafter;
Or: the decimal expansion loops in a cycle.
That is; a real number is either irrational, dyadic rational, or non-dyadic rational.

In general a sequence is either infinite or terminating or cyclic. This uncontroversial mathematical proposition implies theological and philosophical confusion.