Working with Nonstandard Models
Most of the research in foundations of mathematics that I do in some way or another involves the use of nonstandard models. I will give a few examples, and indicate what is involved.
Compact covering and game determinacy
All spaces are separable and metrizable. Suppose that the continuous and onto mapping tf : X → Y is compact covering. Under the axiom of ∑11-determinacy, we prove that tf is inductively perfect whenever X is Borel, and it follows then that Y is also Borel. Under the axiom ℵ1L = ℵ1 we construct examples showing that the conclusion might fail if “X is Borel” is replaced by “X is coanalytic”. If we suppose that both X and Y are Borel, then we prove (in ZFC) the weaker conclusion that tf has a Borel (in fact a Baire-1) section g : Y → X. We also prove (in ZFC) that if we suppose only X to be Borel but of some “low” class, then Y is also Borel of the same class. Other related problems are discussed.
Study of External Parasites Infesting Carcharhinus Melanopterus Fish From The Red Sea In Jeddah, Saudi Arabia
Parasitic diseases are harmful and limiting factors in breeding and rearing ornamental fish industry, such as the Monogenea ( Dactylogyrus sp and Erpocotyletiburonis ) and copepod ( Pandarus sinuatus  and Anchistrotostangi ). The copepod and the monogenean are gill parasites found on the shark; CarcharhinusMelanopterus . The primary purpose of this study was to determine the parasitic infections of sharks in red sea in Jeddah. For this purpose, 19 female C. Melanopterus and 14 male C. Melanopterus were first examined for ectoparasites using wet mount under a light microscope. All male sharks had Monogenea and/or copepod in their gill and only 3 females haven’t infections. Then, we aimed to find a combination of copepod and monogenea infection in C. Melanopterus gills. 50% of male and 42% of female were infected by copepod and monogenea. Our findings revealed that the protozoan infections are very common among sharks. Although, no gross pathology was observed among infected shark, but it is likely that in case of any changes in the environment, then parasitic infections could be harmful.