ω上非主算术超滤存在性的一个定理

每个非主算术超滤p∈βω-ω都可用来形成一个简单的不可数算术模型Np= {f(p):f∈ωω}N．用这个模型中的超滤(代替自然数)作成的有限分数的等价类便得到所有正实数．用分数表示实数,这正是古希腊人曾有的想法．人们已经知道,Martin 公理的较弱形式MAcountable蕴涵着下面的命题Q: (Q)若B■P(ω)具有sfip(强有限交性质)且|B|<2ω,则存在Q点qB．本文证明命题Q蕴涵着结论:ω上非主算术超滤存在．

A Theorem on the Existence of Nonprincipal Arithmetical Ultrafilters on ω

Each nonprincipal arithmetical ultrafilter p∈βω-ω is associated with a simple arithmetical model Np = {f(p) : f∈ω) N, and a positive real number is just an equivalence class of finite fractions made by arithmetical ultrafilters (instead of natural numbers), just like the ancient Greek's idea. It is well known that MA countable implies the statement (Q): (Q) If B■p(ω) has the sfip (strong finite intersection property) and |B| < 2ω, then there is a Q-point q  B. In this paper we will prove that the statement (Q) implies our Hypothesis: There exist nonprincipal arithmetical ultrafilters on ω.