FUNDAMENTAL... OVERSIGHTS
      IN MATHEMATICS

SECTIONS

PSYCHOLOGY of MATHEMATICIANS

Let us start with a humorous insight into the psychology of mathematicians:

• ALL mathematicians believe that it is theoretically possible that set theory is inconsistent, but NO mathematicians believe it is actually possible that set theory is inconsistent.

and a serious one:

• If mathematicians refrain — either “formally” or informally — from allowing or accepting as valid any derivations that yield contradictions, then it is literally impossible for any theory to be found by them to be inconsistent.

There are 2 common fundamental... oversights that afflict mathematicians when the subject of the inconsistency of set theory is raised. Mathematicians tend to forget at least 2 essentials:

• the formal definition of “inconsistency” as regards a mathematical theory

As absurd or terrifying as it may seem, mathematicians feel that if they avoid deriving a result that contradicts a standard theorem, then they maintain the consistency of e.g. set theory.

• that any defined entities must be replaceable by their original definitions-constructions-meanings

The classic example of both the above is when finite arithmetic (including the rule of inference that equal quantities may be subtracted or added to both sides of an equation and the result will still be an equation) is used to derive transfinite arithmetic. The most fundamental result of the transfinite arithmetic of set theory is a₀ + 1 = a₀. We quickly note that if we subtract a₀ from both sides we get 1 = 0, and the inconsistency of set theory. Therefore, even though transfinite arithmetic is defined in terms of finite arithmetic and its axioms and rules of inference, we must abandon not only the formal definition of the inconsistency of a theory (that a theory is inconsistent if it is even possible to derive a contradiction, independent of whether we perform the derivation or even notice or acknowledge that it could be performed), but one of the most fundamental operating principles in mathematics (the rule of inference regarding subtracting equal quantities etc.).

Mathematicians will sometimes offer the lame excuse that transfinite numbers and transfinite arithmetic are “different” (from finite numbers and arithmetic). Well, yes, they are, but not in a good way, not in a way that should be mathematically acceptable.

These considerations start to answer the question of how mathematicians have failed to properly appreciate the relationship between paradox and inconsistency, a fact that many have bewailed over the last century and a half.

It is worth taking a look at:

Induction... Oversights

before looking at

Fundamental... Oversights in More Detail