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>But there are uncountably many finite collections of symbols

How do you mean? Because there are uncountably many symbols? I'm not really sure what a symbol means in this context.

I just picked on one of the points in the post that make no sense without further explanation. Just take each real number as a symbol.

It's normally assumed that the number of symbols is finite.

The number of symbols in the language may countable, but the number of sets of symbols will be indeed uncountable, i.e. AFAIR the number of sets of natural numbers is uncountable — you can even construct real numbers as sets on rational (still countable) numbers.

The set of all sets of natural numbers is uncountable, but the set of all finite sets of natural numbers, is countable.

True, that's why I haven't used 'finite' in my comment.

Apparently I've missed that the parent referred to the finite collections and in fact was wrong making my previous comment misplaced. Thanks for pointing it out :)

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