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<p>Methods are known for estimating infinite subsets of natural numbers using the cardinality aleph-zero and the ordinal type omega. With these estimates, any infinite subset of the natural numbers has the cardinality aleph-zero or the ordinal type omega. A&#xa0;method for measuring countable subsets of the set of natural numbers as a&#xa0;fraction of the natural numbers has been developed. With this method of measurement, for example, the fraction of the infinite set of even numbers is half the natural numbers, which corresponds to mathematical intuition. On the other hand, the set of all positive powers of any natural number is a&#xa0;fraction equal to zero of the natural numbers. Other examples are given, and some subsets and their measurements are indicated. It is proven that the set for measuring the fraction of a&#xa0;countable subset is everywhere dense on the interval measuring the fraction. It is also proven that any interval with endpoints from 0&#xa0;to 1&#xa0;can be a&#xa0;fraction of some subset of the natural numbers.</p>

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A method for measuring the number of elements of subsets of natural numbers

  • 期刊论文

  • 发表日期：

    2026年5月21日

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  • Sergei A. Rudakov

摘要

Methods are known for estimating infinite subsets of natural numbers using the cardinality aleph-zero and the ordinal type omega. With these estimates, any infinite subset of the natural numbers has the cardinality aleph-zero or the ordinal type omega. A method for measuring countable subsets of the set of natural numbers as a fraction of the natural numbers has been developed. With this method of measurement, for example, the fraction of the infinite set of even numbers is half the natural numbers, which corresponds to mathematical intuition. On the other hand, the set of all positive powers of any natural number is a fraction equal to zero of the natural numbers. Other examples are given, and some subsets and their measurements are indicated. It is proven that the set for measuring the fraction of a countable subset is everywhere dense on the interval measuring the fraction. It is also proven that any interval with endpoints from 0 to 1 can be a fraction of some subset of the natural numbers.