论文标题
欧几里得整数，欧几里得超滤器和欧几里得数字
Euclidean integers, Euclidean ultrafilters, and Euclidean numerosities
论文作者
论文摘要
我们公理地介绍了欧几里得整数的环$ \ bf {z}_κ$，可以将其视为[4]的欧几里得数量的欧巴文$ \ mathbb {e} $ [4]的``整体部分''的``整体部分''，在其中，有限制的$ bounal $ bine of-κ$ - $ - $ - $ - $ - $ - $ - 派出了任何派别的固定级别。其特征函数的transfiite总和可以保留所谓的自然操作。欧几里得整数是\ IT {阳性}时，仅当它是\ IT \ IT {自然数的跨常规总和。}此属性需要使用称为Euclidean的特殊Ultrafilters，此处引入了Ths End。 环$ \ bf {z}_κ$允许分配``Euclidean''尺寸（\ it {numerosity}），向``oldinal punktmengen''，即，序列的一组，即其特征函数的次数：因此，每个集合的次数都在平衡的范围内，以均一的命令，以平等的范围，以平等的范围{维护和乘法}，而欧几里得原理``整体大于部分'（\ it {一个集合（严格）（严格地）大于其适当的子集}）。
We introduce axiomatically the ring $\bf{Z}_κ$ of the Euclidean integers, that can be viewed as the ``integral part" of the field $\mathbb{E}$ of Euclidean numbers of [4], where the transfinite sum of ordinal indexed $κ$-sequences of integers is well defined. In particular any ordinal might be identified with the transfiite sum of its characteristic function, preserving the so called natural operations. The ordered ring $\bf{Z}_κ$ may be obtained as an ultrapower of $\mathbb{Z}$ modulo suitable ultrafilters, thus constituting a \it{ring of nonstandard integers.} Most relevant is the \it{algebraic} characterization of the ordering: a Euclidean integer is \it{positive} if and only if it is \it{the transfinite sum of natural numbers.} This property requires the use of special ultrafilters called Euclidean, here introduced to ths end. The ring $\bf{Z}_κ$ allows to assign a ``Euclidean" size (\it{numerosity}) to ``ordinal Punktmengen", i.e. sets of tuples of ordinals, as the transfinite sum of their characteristic functions: so every set becomes equinumerous to a set of ordinals, the Cantorian defiitions of \it{order, addition and multiplication} are maintained, while the Euclidean principle ``the whole is greater than the part" (\it{a set is (strictly) larger than its proper subsets}) is fulfilled.
