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We study about the metamathematics of Zermelo-Fraenkel set theory with the axiom of choice. We use the validity of Addition and Multiplication. We provide an example that the two operations Addition and Multiplication do not commute with each other. All analyses are performed in a finite set of natural numbers.

Zermelo-Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. It has a single primitive ontological notion, that of a hereditary well-founded set, and a single ontological assumption, namely that all individuals in the universe of discourse are such sets. ZFC is a one-sorted theory in first-order logic. The signature has equality and a single primitive binary relation, set membership, which is usually denoted Î. The formula

We use the validity of Addition and Multiplication. Here we aim to provide an example that the two opera- tions Addition and Multiplication do not commute with each other. All analyses are performed in a finite set of natural numbers.

Assume all axioms of Zermelo-Fraenkel set theory with the axiom of choice is true.

Let us start with a singleton set

We treat here Addition. We have

Thus we obtain 2. By using the obtained 2, we have

Thus we obtain 3. By repeating this method for an even number time, we have

By repeating this method, we have

Thus we have the following finite set of natural numbers

By using the set (6), we discuss that the two operations Addition and Multiplication do not commute with each other.

We consider a value V which is the sum of the results of trials. Result of trials is 1 or 2. We assume the number of 2 is equal to the number of 1. The number of trials is 2 m. We have

We derive the possible value of the product

We assign the truth value “1” for the following proposition

We have

The value

We assume that the possible value of the actually happened results

The same value is given by

We only change the labels as

and

Here

The step (16) to (17) is OK. The step (17) to (18) is valid under the assumption that the two operations Addition and Multiplication commute with each other. The step (18) to (19) is true since we have only changed the label as

The above inequality (19) is saturated since

and

We derive a proposition concerning the value given in (11) under the assumption that the possible value of the actually happened results is 1 or 2, that is

We do not assign the truth value “1” for the two propositions (10) and (24) simultaneously. We are in a contradiction. Thus we have to give up the assumption that the two operations Addition and Multiplication commute with each other.

In conclusions, we have used the validity of Addition and Multiplication. We have provided an example that the two operations Addition and Multiplication do not commute with each other. All analyses have been performed in a finite set of natural numbers.

Koji Nagata,Tadao Nakamura, (2015) Do the Two Operations Addition and Multiplication Commute with Each Other?. Open Access Library Journal,02,1-4. doi: 10.4236/oalib.1101803