Theorem:

If two sets of statements, such as those of the teachings of two religions, contradict one another on one statement of truth, then both religions cannot have a correspondence with the universal set of truths of reality and they are contradictory to one another as religions.

Proof:

We will denote a statement using a signed variable such as “+s” for affirmation of the statement or “-s” for denial of the statement. Not all possible statements need to be defined in a religion (the religion is indifferent on that topic).

We will denote the universal set of true/false statements of reality as the set R. It contains all of the possible statements about reality with the correct disposition to them. (e.g. One goal of science is to find what is true based on reality, or in other words, if a statement is true according to set R.)

Let set X be a set of beliefs for religion X.

Let set Y be a set of beliefs for religion Y.

Suppose that there exists a contradiction between a statement s that is present in both set X and set Y such that the sign of s in religion X is opposite that of the s of religion Y. (e.g. +s is in X and –s is in Y) It is obvious that there could not be unity based on this statement in both religion X and religion Y since the statement is either true or false. Furthermore, if any statement was to be added or removed that is different from those that exist in set X, then set X would no longer have the same beliefs and therefore it would no longer be religion X. This is also the case for religion Y.

Since a religion is only the same if all of the statements it makes remain the same, any contradiction with another religion in one statement causes a contradiction to exist in the religions themselves. If X and Y have at least one contradictory statement in their comparison, then their union would contain a contradiction such that X U Y = {…+s, –s,…}.

Now consider religion X and Y related to the set R, the set of all correct statements about reality. Set R cannot be contradictory with itself by definition of reality. The set R must make a claim for all possible statements so there is no chance for indifference when set R contains all possible knowledge. If sets X and Y contain a contradiction between themselves, then it would be impossible for them to both be correct since there is only one set R and each statement can only be in either an affirmative or denial state. (e.g. if +s is in X and –s is in Y and +s is in set R, then religion Y must be wrong in that respect.)

In a practical sense, examine each religion in question using resources such as Wikipedia's "Major Religious Groups" article, http://en.wikipedia.org/wiki/Religions, and apply the above logic to the statements that they make.

QED