Fallacy of Completeness in Real Numbers

"Intuitively, completeness implies that there are not any “gaps” (in Dedekind's terminology) or “missing points” in the real number line. This contrasts with the rational numbers, whose corresponding number line has a “gap” at each irrational value. In the decimal number system, completeness is equivalent to the statement that any infinite string of decimal digits is actually a decimal representation for some real number."

https://en.wikipedia.org/wiki/Completen ... al_numbers

1. The decimal represention for some real number in itself is a rational number.

2. This necessitates a "gap" as the number of decimal points, as a rational number necessitating the real number, results in a gap between points.

3. This "gap" between points, where the point effectively act as rational number in itself, necessitates an irrational value between the points resulting in "fractal points" as a contradiction in terms.

## The Contradictory State of Real Numbers and Completeness

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