We define Set as: " The collection of well defined and distinct objects", but empty set has no object. By what reason it is called a Set...???
Shah Jamal Wazir
3 years, 11 months ago
A collection of nothing is still a collection.
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said the troll.
Is not zero a well-defined and distinct number? If we define numbers to be well-defined distinct quantities, it could be argued that zero is not a number, because nothing is not a quantity. The argument is trivial; whether or not zero is considered a number, it doesn't affect anything we do. 1 - 1 would still be nothing. In the same vein, whether or not the null set is a set is a meaningless question, because it doesn't affect our calculations in any way.
It is not a meaningless question; it is a question of existence. In a set theory, can one prove that there exists a set without any elements? The answer is: yes, one can. Furthermore, one can generally prove uniqueness too, so we can talk about "the" null set, as opposed to "a" null set.
The key idea is that sets are not defined (and neither are numbers, as you've touched upon). They are constructed, and their properties are stated, but they are not defined.
In formal set theory (i.e. ZF), "set" is an undefined term. It ends up having to be, similar to how "number" and "point" also remain undefined.
Hence, the empty set is not hampered by definition. In set theories, it is generally one of the easier theorems to prove that a set without elements exists and is unique, and thus one has an empty set.
Give answer with proof