In this paper, we show that for each forcing notion P in a transitive model M of ZFC, if P satisfies some conditions, there is an elementary embedding from M into a generic ultrapower contains a P-generic set. And, we also introduce the result that if we assume the existence of some large cardinals, the above generic ultrapower can be well-founded. Using this result, we prove some theorems on the problems of regularity properties of definable sets of reals.

Gaisi Takeuti extended Gentzen's work to higher-order case in 1950's–1960's and proved the consistency of impredicative subsystems of analysis. He has been chiefly known as a successor of Hilbert's school, but we pointed out in the previous paper that Takeuti's aimed to investigate the relationships between "minds" by carrying out his proof-theoretic project rather than proving the "reliability" of such impredicative subsystems of analysis. Moreover, as briefly explained there, his philosophical ideas can be traced back to Nishida's philosophy in Kyoto's (...) school. For the proving the consistency of such systems, it is crucial to prove the well-foundedness of ordinals called "ordinal diagrams" developed for it. Takeuti presented such arguments several times in order to show that they are admitted in his stand point. As a starting point of investigating his finitist stand point, we formulate the system of ordinal notations up to ε0 and reconstruct the well-foundedness arguments of them. (shrink)

Layunin ng pag-aaral na mailahad at matukoy ang kahalagahan ng sikolohiya, implikasyon, at mga danas at lakbayin ng “Sunod” sa pangkat ng Biga at mga karatig tribo nito sa Tabuk City, Kalinga. Ang nasabing pananaliksik ay sumailalim sa Kuwalitatibo at Deskriptibong pag-aaral na nilapatan ng Sikolohiyang pagdulog, Sosyolohiya, at Etnograpiya, tinalakay ang mga kulturang kinagisnan ng isang pangkat. Mga piling “Pangat” o pinuno at matatanda sa pangkat Biga ang mga kalahok sa pananaliksik na nagbigay ng kani-kanilang mga karunungan, karanasan at (...) lakbayin sa pagdalumat ng “Sunod” sa buhay. Ang resulta ng pag-aaral sa mga Ibiga at mga karatig tribo ay may maayos na ugnayan at mapayapang pamumuhay. Malakas ang tungkulin ng pamilya sa pagbibigay ng disiplina at pagtutulungan sa bawat miyebro. Lumitaw rin na hindi kailangang magkadugo ang maituturing na “sunod” kundi lahat ng katribo at mga karatig tribo. Dahil dito ay naiiwasan ang mga di-kanais- nais na kaganapan o mga kaguluhan sa pagitan ng bawat isa. Matatag na kanilang sinusunod ang kahalagahan ng relasyong “sunod” sa bawat araw ng kanilang pakikibaka sa buhay. Napatunayan din na malakas ang kapangyarihan ng mg “Pangat” o “Tribal elders” at ibang matatanda sa pagpapanatili ng kaayusan. (shrink)