Murat,
I see from the comments that sadly few have understood or seen the value of your proposition. I certainly do and commend you, as you'll see from my own essay which develops the theme by testing the power of continuous helical curves. I refer to Bayesian inverse distributions (quantum iPAD's) between cardinals as 'fuzzy subsets'.
Have you explored the parallel with Godel's 'n-valued'/fuzzy logic?
You also beautifully point out the issues with infinities, transcendental and irrational numbers, and the limitations of symbols, as I do with both. I loved your reference to "mathematical objects which can be calculated with arbitrary exactness." I suggest a position for a 'Dirac line' between and defining metaphysical math/derivatives, as separate from real physical nature as entities and interactions.
I'm pleased to have taken something more from your essay, including learning of Aristotle's great wisdom, and a greater confidence to reject the cardinalisation I too discuss and constrain. As an astronomer I have also found compelling argument for the essential dark energy condensate and foundation of nature as a continuum.
I believe your essay needs to be more widely read, and I'm pleased to be able to score it well. You may also be familiar with or wish to read that of Vladimir Rogozhin, but I greatly look forward to your views and comments on mine, particularly my suggestion regarding the foundations of the poor logic and mathematics.
Both our essays may perhaps be summed up as examining the; "existence of the intermediate cardinal between cardinal number and cardinal of continuum."
Very best of luck in the competition.
Peter