Question

What the meaning of "
`\left \langle a_(\alpha : \alpha \in Ord) \right \rangle`"?

Answer 1

Refer to the step-by-step explanation.

Explanation:

The notation "<a_α: α ∈ Ord >" represents a sequence or collection of elements. Each element in the sequence is labeled or indexed by an ordinal number.

Breaking it down:

• The "<...>" notation suggests that it represents a sequence or function.

• a_α: This refers to the element of the sequence at the ordinal position α. Each ordinal α corresponds to a specific term in the sequence or function.

• The expression "α ∈ Ord" indicates that α is an element belonging to the class of all ordinals, Ord. This class consists of well-ordered sets that represent different levels of ordering.
• The ":" symbol separates the term from the indexing set.
• The notation "<a_α: α ∈ Ord>" as a whole represents the entire sequence or function, where each term is indexed by an ordinal α.

Therefore, "<a_α: α ∈ Ord>" signifies a sequence or function that is defined on the class of all ordinals, with each term indexed by an ordinal α. It represents a mapping from the class of ordinals, Ord, to a range of values, describing the order and arrangement of elements in the sequence or function.

For example:

If we have a sequence of elements {a_1, a_2, a_3, ...}, the statement <a_α: α ∈ Ord > suggests a more general form where the index can take on any ordinal number, not just the usual finite natural numbers.

It's important to note that this notation might be used in specific mathematical contexts where ordinal numbers are relevant, such as set theory or the study of well-ordered sets.

`\hrulefill`

What is an ordinal number?

An ordinal number is a type of number used to describe the position or order of an element in a well-ordered set. Unlike cardinal numbers, which represent the quantity or size of a set, ordinal numbers indicate the relative position or rank of an element within a sequence.

Here's an example to illustrate ordinal numbers:

Consider a set of five elements: {apple, banana, cherry, durian, elderberry}. We can assign ordinal numbers to each element to indicate their position in the set.

Using ordinal numbers, we can say:

1st element: apple

2nd element: banana

3rd element: cherry

4th element: durian

5th element: elderberry

In this example, the ordinal numbers "1st," "2nd," "3rd," "4th," and "5th" represent the positions of the elements within the set. The ordinal numbers provide a way to describe the order or ranking of the elements, emphasizing their positions relative to each other.

It's worth noting that ordinal numbers extend beyond finite sets. For example, if we have an infinite set of natural numbers, we can assign ordinal numbers like ω (omega), ω+1, ω+2, and so on to represent their positions within the set. This allows us to discuss the order of elements in infinite sequences as well.