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.