initial ordinal sentence in Hindi
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- Every regular ordinal is the initial ordinal of a cardinal.
- The least of these is its initial ordinal.
- The ?-th infinite initial ordinal is written \ omega _ \ alpha.
- If the axiom of choice holds, every cardinal number has an initial ordinal.
- The least ordinal of cardinality ( i . e ., the initial ordinal ) is.
- Infinite initial ordinals are limit ordinals.
- So as an ordinal, an infinite initial ordinal is an extremely strong kind of limit.
- A regular ordinal is always an initial ordinal, though some initial ordinals are not regular.
- A regular ordinal is always an initial ordinal, though some initial ordinals are not regular.
- An ordinal that is equal to its cofinality is called regular and it is always an initial ordinal.
- The least ordinal associated with a given cardinal is called the " initial ordinal " of that cardinal.
- Any limit of regular ordinals is a limit of initial ordinals and thus is also initial but need not be regular.
- Any limit of regular ordinals is a limit of initial ordinals and thus is also initial even if it is not regular, which it usually is not.
- The axiom of choice is equivalent to the statement that every set can be well-ordered, i . e . that every cardinal has an initial ordinal.
- In theories with the axiom of choice, the cardinal number of any set has an initial ordinal, and one may employ the Von Neumann cardinal assignment as the cardinal's representation.
- If that initial ordinal is \ omega _ { \ lambda } \,, then the cardinal number is of the form \ aleph _ \ lambda for the same ordinal subscript & lambda;.
- Then we can apply the axiom of replacement to replace each element of that powerset of " x " by the initial ordinal number of the same cardinality ( or zero, if there is no such ordinal ).
- In this case, the ordinals 0, 1, \ omega, \ omega _ 1, and \ omega _ 2 are regular, whereas 2, 3, \ omega _ \ omega, and ? ?? are initial ordinals which are not regular.
- In this case, the ordinals 0, 1, \ omega, \ omega _ 1, and \ omega _ 2 are regular, whereas 2, 3, \ omega _ \ omega, and ? ?? are initial ordinals that are not regular.
- ZFC set theory, which includes the axiom of choice, implies that every infinite set has an aleph number as its cardinality ( i . e . is equinumerous with its initial ordinal ), and thus the initial ordinals of the aleph numbers serve as a class of representatives for all possible infinite cardinal numbers.
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