Comparison Of Zermelo's Axiom Of Choice

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Axiom of choice was formulated by Zermelo in 1904 and is known as Zermelo’s axiom of choice. This axiom stated that we can choose a member from each set in a collection of nonempty disjoint sets, C (Schechter, 2009). Additionally, according to Janes (2008), there exists a set X which consisting of one element taken from each set belonging to Y given that any nonempty set Y members are pairwise disjoint sets. Taking a role as a basic assumption used in many parts of mathematics as stated by Schechter (2009), there are numbers of theorems which must use the application of axiom of choice in order for them to work smoothly

To illustrate the concept, we can use a physical analogy as follows which was mentioned by Kelly et al., (2012). Assume there
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Then, we can say that both of them are in the same class from the division of real number into an uncountable number of disjoint classes. Any number in [0,1] from each class will form a new set of V⊆[0,1] which consist of the selected numbers (Saucer, 2016)

The connections between those three are
Both constructions of Banach-Tarski Paradox and Vitali sets depend on the axiom of choice in order for them to function well.
Banach-Tarski Paradox and Vitali sets are non-measurable sets and they are two of the consequences of the axiom of choice

We know that the set of real numbers R is uncountable although set is actually a collection of elements. Thus, we cannot measure the size of the size directly. However, we can measure the size using Borel set that we define its measure as a length. From this length, we can simply generalize the set to become Lebesgue measure

Borel measures on R, B(R) is the smallest σ-algebra that contains the open intervals of R as stated in Wikipedia (2016). The Borel measure on real number, R is the choice of Borel measure which assigns μ ((a,b])= b-a for every half-open interval (a,b]. Besides, Weir (1974) mentioned that a Borel measure is any measure μ defined on the σ-algebra of Borel

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