Now you define a new concept INNFINITE, that is defined by:This concept INNFINITY behaves in exact the same way as the concept infinity in ordinary mathematics. So you do not need the concept infinity. Every conclusion you do with the concept infinity, you can do with the concept INNFINITY. So you will not lose anything, if you discard the concept infinity. Infinity is not needed in mathematics.
If you have a bijection from all visible numbers of a set S, to all visible numbers of a true subset of S, then you say that the set S in INNFINITE.
Then you can use this concept INNFINITE, and you will get a consistent theory with no contradictions, because you have a finite visualization of this theory.
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