[gus_massa]

The proof doesn't assume that they are formed by not-too-pathological penstrokes.

<wrong>The idea is that you should measure the amount of black and white ink to change a symbol into another simbol, and if the total amount of ink is less than ε then they indistinguishable. (Where ε is some constant you must choose for the whole system.)</wrong>

I think that every horrible-totally-pathological ink splash has a nice indistinguishable version, but my real analysts is a bit rusty, and there are a few horrible things that I may have forgotten.

Edit: see comment below.

[Smaug123]

By "not-too-pathological" I intended at least to include the requirement "measurable".

[gus_massa]

I just notice that I used the wrong metric. The article uses the Hausdorf metric and I used the measure of the symetric difference.

And the article assume that the sets are compact, so they are measurable as you say. Anyway compact sets can be quite pathological (but not as pathological as non measurable sets).

[tel]

It also suggests the argument generalizes to symbols as non-compact sets.

[clintonc]

I made a comment elsewhere on this thread that explains that symbols themselves being compact isn't so important, but that the set of descriptions of the symbols must be compact. For example, if the description of the symbol is not the symbol itself as a set, but a map f:[0,1]^2 -> [0,1] that describes the "intensity" of ink at each point, then the natural conclusion is that the description of a symbol must be upper semicontinuous, which makes the set of descriptions compact.