[過去ﾛｸﾞ] 現代数学の系譜　カントル　超限集合論 (1002ﾚｽ)

現代数学の系譜　カントル　超限集合論 http://rio2016.5ch.net/test/read.cgi/math/1570237031/

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324: 現代数学の系譜 雑談 ◆e.a0E5TtKE [] 2019/10/13(日) 07:23:50.76 ID:sXrN/kYa >>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. But category theory also has its impact on order theory on a larger scale. Classes of posets with appropriate functions as discussed above form interesting categories. Often one can also state constructions of orders, like the product order, in terms of categories. Further insights result when categories of orders are found categorically equivalent to other categories, for example of topological spaces. This line of research leads to various representation theorems, often collected under the label of Stone duality. つづく http://rio2016.5ch.net/test/read.cgi/math/1570237031/324

230: １３２人目の素数さん [sage] 2019/10/10(木) 03:44:50.32 ID:64e05J/b >>324 違います。 Zermelo ordinal number なるものが何かまだ誰も定義していません。 Z(0)=0, Z(1)={0}, Z(2)={{}},‥‥ はいいでしょう。 そのように定義したいなら定義してもいいでしょう。 ただしコレもキチンと論理式で定義しないとだめなんですよ。 しかしココまでは難しいけどできるのは確認済みです。 問題になっているのはω番目以降です。 まだだれも Z(ω), Z(ω+1),‥‥ を定義した人はいません。 基数の全体cardinal numberについては x:cardinal number :⇔ x:ordinal number ∧ ∀y<x(#y≠#x) と定義され、 よつて整列順序クラスOrdの部分クラスなので自然に整列順序集合となり、 整列写像: ℵ:Ord→Cardが定義されます。 この対応からCardの超限帰納法を用いる定義 ℵ(0) :=0 ℵ(a+1) := min{x ∈Ord | #x>#a} ℵ(a) := min{x ∈Ord | #x>#a} (if a is a limit number) が誘導される事がわかります。 のでこれを定義に用いる事も出来ます。 どちらも大して難しい定義ではないのでどちらを定義に採用する事もあるとは思いますが、 ポイントは超限帰納法で定義するなら後者ℵ(a+1)をℵ(x) (x≦a)で表現するだけではダメでaがlimit numberのときのℵ(a)を定めないと超限帰納法は完成しません。 あなたはaがlimit numberの場合のΩ(a)を論理式を用いて定義しなければなりません。 http://rio2016.5ch.net/test/read.cgi/math/1570237031/230

325: 現代数学の系譜 雑談 ◆e.a0E5TtKE [] 2019/10/13(日) 07:24:42.49 ID:sXrN/kYa >>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] (引用終り) 以上 http://rio2016.5ch.net/test/read.cgi/math/1570237031/325

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