下記PDFで ”This new system would be constructed in a manner similar to Cauchy’s construction of the real numbers” ”Let us consider the factor ring R~^N = R^N/ 〜Fr where 〜Fr is the equivalence relation defined by (an)〜Fr(bn) if and only if {n : an = bn} ∈ Fr. This is no different to saying that (an) is equivalent to (bn) if and only if an = bn for all sufficiently large n. ”
ここに、Frは、フレシェ・フィルターです。 なるほど、なるほど、フレシェ・フィルターを使って、”similar to Cauchy’s construction of the real numbers”をやる ”where 〜Fr is the equivalence relation defined by (an)〜Fr(bn) if and only if {n : an = bn} ∈ Fr.” 数列のシッポの同値を使ってね
そうすると、”Non-standard Analysis 3.1 Construction of the Hyperreals *R ” が出る！
https://arxiv.org/pdf/1212.5740.pdf Filters and Ultrafilters in Real Analysis 2012 Max Garcia Mathematics Department California Polytechnic State University (抜粋) P12 2.4 Remarks Regarding the Fr´echet Filter
This new system would be constructed in a manner similar to Cauchy’s construction of the real numbers from rational sequences. The elements in this new system would be equivalence classes of real numbered sequences, which take into account sequence convergence (divergence) as well as the rate of convergence (divergence). Ideally, the resulting system will contain elements that can be used to characterize convergence in such a manner that we can do away with the limits of standard analysis or the set constructions from the Fr´echet approach. Let us consider the factor ring R~^N = R^N/ 〜Fr where 〜Fr is the equivalence relation defined by (an)〜Fr(bn) if and only if {n : an = bn} ∈ Fr.
