ΠΡΠΈΠΌΠ΅Π½Π° ΡΠΎΡΡΠΈΠ½Π³ ΠΌΠ΅ΡΠΎΠ΄Π΅ Π½Π° Π΄ΠΎΠΊΠ°Π·ΠΈΠ²Π°ΡΠ΅ ΠΊΠΎΠΌΠ±ΠΈΠ½Π°ΡΠΎΡΠ½ΠΈΡ ΡΠ²ΡΡΠ΅ΡΠ°
Paul Cohen constructed a model (Cohen’s symmetric model) for which he proved that the axioms of ZF hold in it, but not the Axiom of Choice. Nevertheless, Halpern and Levy proved in that in the Cohen’s symmetric model BPI (the statement that every Boolean algebra has an ultrafilter) is true. That proof was based on the Halpern-LΓ€uchli theorem, a Ramsey-Theoretic proposition about the partitioning of tree products. Later, Harrington found another proof of the Halpern-LΓ€uchli theorem using forcing. Later, many other applications of the Halpern-LΓ€uchli theorem were found cite{rs}. Here, Halpern-LΓ€uchli theorem is derived from the fact that BPI holds in the Cohen’s symmetric model, and it is shown how that model can be used instead of the Halpern-LΓ€uchli theorem to prove statements which are otherwise proved by applying the theorem. In other words, a mathematical method is given. That method is an alternative to the Halpern-LΓ€uchli theorem in its applications, whereby that method has the same power as the theorem itself. This method is abstracted by ZFC theorem without metamathematical notions in formulation. Also, a relative consistency with ZFC, of some principles which are stronger than the Halpern-LΓ€uchli theorem, is given. This provides a new proof of the Halpern-LΓ€uchli theorem from the axioms of ZFC. Corresponded ZFC theorem without metamathematical notions in formulation is also given.
https://www.cris.uns.ac.rs/DownloadFileServlet/Disertacija169078985470998.pdf?controlNumber=(BISIS)