[過去ﾛｸﾞ] 現代数学の系譜 カントル 超限集合論 (1002ﾚｽ)
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321: 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/10/13(日)07:09 ID:sXrN/kYa(1/5) AAS
（>>313より）
おっさんずゼミ＝「どこのだれとも知れぬ”名無しさん”のおっさんたちとの、ゼミ」やる気ないです
但し、好きなときに好きなことを書かせてもらいます（＾＾
ちょっと思いついたので、下記をば
>>314
＞ωの一番右のΦってなんだよ？ｗ
じゃ、
省6
322(3): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/10/13(日)07:11 ID:sXrN/kYa(2/5) AAS
>>318
>無限公理によるωは、ノイマンのsuc(a)=a∪{a}の超限回繰り返しではない
>なぜなら
>ω=suc(a)=a∪{a}となるようなa（つまりωの一番右の元！）
>が存在しないから
そう！ その指摘は正しいね
ωは、下記の通り、”任意の自然数よりも大きい最小の超限順序数 ω”で、「 0 でも後続順序数でもない順序数」だ
省26
323(1): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/10/13(日)07:22 ID:sXrN/kYa(3/5) AAS
>>322
参考追加
外部ﾘﾝｸ:en.wikipedia.org
Order theory
(抜粋)
Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations.
It provides a formal framework for describing statements such as "this is less than that" or "this precedes that".
省20
324(2): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/10/13(日)07:23 ID:sXrN/kYa(4/5) AAS
>>323
つづき
Category theory
The visualization of orders with Hasse diagrams has a straightforward generalization: instead of displaying lesser elements below greater ones, the direction of the order can also be depicted by giving directions to the edges of a graph.
In this way, each order is seen to be equivalent to a directed acyclic graph, where the nodes are the elements of the poset and there is a directed path from a to b if and only if a ? b. Dropping the requirement of being acyclic, one can also obtain all preorders.
When equipped with all transitive edges, these graphs in turn are just special categories, where elements are objects and each set of morphisms between two elements is at most singleton. Functions between orders become functors between categories. Many ideas of order theory are just concepts of category theory in small. For example, an infimum is just a categorical product.
More generally, one can capture infima and suprema under the abstract notion of a categorical limit (or colimit, respectively). Another place where categorical ideas occur is the concept of a (monotone) Galois connection, which is just the same as a pair of adjoint functors.
省3
325: 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/10/13(日)07:24 ID:sXrN/kYa(5/5) AAS
>>324
つづき
History
As explained before, orders are ubiquitous in mathematics. However, earliest explicit mentionings of partial orders are probably to be found not before the 19th century.
In this context the works of George Boole are of great importance. Moreover, works of Charles Sanders Peirce, Richard Dedekind, and Ernst Schroder also consider concepts of order theory.
Certainly, there are others to be named in this context and surely there exists more detailed material on the history of order theory.
The term poset as an abbreviation for partially ordered set was coined by Garrett Birkhoff in the second edition of his influential book Lattice Theory.[2][3]
省2
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