Final answer:
To demonstrate that the set of positive real numbers with the property that any finite subset sums to 2 or less must be finite or countable, we can argue using the concept of countability and the properties of real numbers. Countable subsets and the contradiction that arises from an uncountable set exceed the sum limit, confirming the countability of the set b.
Step-by-step explanation:
The question at hand deals with set theory in mathematics, specifically with the properties of a set b of positive real numbers where any finite subset summed together has a total of 2 or less. To show that this set b must be finite or countable, we can use the concept of countability and the properties of real numbers.
Assume for contradiction that the set b is uncountable. Then for any positive real number epsilon (ε), there must be infinitely many numbers in b that are greater than ε, otherwise, if there were only finitely many, the set would be countable. However, if b contains infinitely many numbers greater than a chosen ε > 0, then it is possible to select a finite subset whose sum exceeds 2, which contradicts the given property of set b. Therefore, b must be countable, as an uncountable set of positive real numbers with the given property cannot exist.
An alternative approach is to partition the set into subsets where each subset contains numbers between 1/n and 1/(n+1) for n being a natural number. Since each subset can contain at most n elements to keep the sum under 2, and there are countably many such subsets, the overall set b is at most a countable union of finite sets, which is countable itself.