Second generation principle
Before explaining what is the second generation principle, we must know what is called the first generation principle. Our lovely natural numbers 1, 2, 3, 4,... up to infinity, are generated by adding 1 every time (for example, 2 is generated by adding 1 to 1... ^ _ ^) This is the first generation principle. This is endless, and everyone feels very depressed, so we assume that it has an exhaustion (that is, the total number of elements in the natural number set, let's call it w, and ms is the last of the Greek letters)
Then continue to add w+1 and w+2 from this exhaustion until the next exhaustion. In this way, a new number set can be generated, and the elements are larger than the natural number set. We call this method the second generation principle.
The first generation principle tells us that numbers can be added continuously; The second generation principle tells us that we can add more when the number is exhausted... In this way, we can generate many, many out of limit ordinals
- Chinese name
- Second generation principle
- Nature
- Mathematical terminology
Let x be an ordinal number, then x 〈 {x} is also an ordinal number, marked as x+, called the successor of x; The ordinal number that is not a successor is called the limit ordinal number
For example, as mentioned above, Δ (empty set) is 0, {Δ} is 1, {Δ, {Δ}} is 2, and they are all successors
ω=0∪1∪2∪3∪...， It is the smallest limit ordinal number
Then ω+=ω →{ω}, and so on, until 2 ω, 3 ω, ω^2，ω^3，...，ω^ω，...
