Which of the following is NOT a property of real number system?*1 pointArchimede's propertyDedekind's theoremAbsolute value propertyCauchy's theorem

To identify which option is NOT a property of the real number system, let's briefly review the properties listed:

1. Archimede's Property: This states that for any two positive real numbers $a$ and $b$, there exists a natural number $n$ such that $n \cdot a > b$. This property ensures that the real numbers have no "infinitely large" or "infinitely small" elements.

2. Dedekind's Theorem: This theorem relates to the completeness of the real numbers, stating that a subset of the real numbers can be segmented into two non-empty sets such that every element of one set is less than every element of the other. This is a foundational property for the real number system.

3. Absolute Value Property: This typically refers to properties concerning the absolute values of real numbers, such as $|a| \geq 0$ for any real number $a$. This is indeed a property associated with real numbers.

4. Cauchy's Theorem: While there is a Cauchy criterion for convergence, there's not a specific "Cauchy's theorem" that is universally recognized as a property of the real number system.

Conclusion

Cauchy's theorem does not appear to be a standard property universally recognized as belonging to the real number system. Therefore, the correct answer is:

Cauchy's theorem.