An intuitive interpretation of what was described in the previous article in more mathematical terms. The diagonalization operation described in the previous article is an example of a process that goes beyond one formal system, yet if you try to integrate it into the formal system, you get a limited formal system again and you can apply it again from the outside.
Each formal system, be it an algorithm, a formal theory made up of axioms and rules of inference, a formal grammar describing a set of strings of characters, or whatever kind of formalism, is a finite length text, so it contains only a finite amount of information. So even if a very large, infinite or at least unlimited amount of data can be generated from it or can be parsed by it, that data can only be structured according to a limited range of patterns. The formal system can then be viewed as a compressed version of all that data. The compression is only possible because the data contains redundancy, regularity or patterns.
Any data structured according to other patterns is not covered by the particular formal system. Since it is always possible to construct data following different patterns, each single formal system is limited. It has blind spots. It…
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