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As a matter of the fact, these are called von Neumann ordinals and are one way of implementing it in set theory. To make it a true arithmetic, we need some operations, e.g.: These definitions assume, that just as you are able to wrap things up, you can also unwrap, then and check if they match some pattern.
In functional programming, it is called a pattern matching.
Now, if we looked at it from the other way round, it means that if the product is non-empty ).
The axiom of choice is sometimes replaced by well-ordering theorem or Kuratowski–Zorn lemma. Minimal requirements for building up natural numbers were defined in the 19th century by Giuseppe Peano.
To prevent that, we require that each set as well as each of its elements (if there are some; such elements are also sets) would be disjoint (share no common elements).
An inner set containing two elements defines our 2 elements and inner set with one element shows which one is first.
Let’s say we have a set ) of cardinalities of multiplied sets.
And each of these sets originates from an empty set wrapped in different ways.It has addition, multiplication, and comparison, but it doesn’t have subtraction.At least one that would be defined for all possible arguments.Because they are the same (in a way), we might not distinguish between them, and even identify the whole set with them. Yet, we need them to be able to describe positions and sizes in (among others) Euclidean spaces.And (as long as we won’t violate the property used to partition them), we can translate the operation on elements of the set to the operation on a whole set. if we What we might have not paid attention so far, is that from the very beginning such definition defines classes of equivalence for all possible pairs - even those, that were not handled by the original definition of subtraction. There are 2 popular ways to define real numbers using rationals: Dedekind’s cuts and Cauchy’s sequences.
If we define it like: What would be the meaning of that?