I made some posts about this in some subreddit here and received nothing but drivel from closeminded fools who think that there is no apparent contradiction. They held that "10" always equals "9+1", even when bases are being discussed. I said that's retarded and presumptuous and limiting. 10 means one in the tens digit, and zero in the ones digit. The size of the tens digit is defined by what base system is being used, and so saying "base 10" is meaningless. Once a base system is being used, it can be described as base 10. Base 10 is merely an identity, a definition; it doesn't mean anything without having an implicit base being used as the "actual base".
I've read the first 5/8 of Godel Escher Bach.
This explains things. I too have read about that much of that book. I got to where he started in on AI and felt like it was taking too much of a turn toward being a book about robotics. You are absolutely correct. Don't let others convince you that "Base 10" is not oxymoronic, since they're wrong, and they'll try just the same.
One answer is to describe bases just like how you did with the three Js and the 5. Base "ten" could be accurately described as Base "9+1." You could also do it in the very cumbersome form of "base 'ten in base ten'", or the noticeably less problematic "base 'eleven in base nine'" . The former allows description of the base number's base number with numbers higher than 10, which leads to this same original problem, as "Base ##" is being used. The latter form would refuse to use numbers higher than nine for the internal base, so it doesn't incur that problem.
A GED-focused explanation, not for 5-year-olds: This is all the result of the typographical rules we use. There is clearly a problem, since the format of two single-digit numerical characters smushed next to each other invokes the ordinary rule of our number system, which is that of normal decimal numbers (i.e., the form based on the identity 10 = 9+1 ). That typographical rule says that, whenever multiple numerical characters are connected to each other without symbols or spaces between them, each number's value is equal to the value of its numerical character multiplied by 10 to the power of that number's position in the sequence, counting from the right. I don't think an illustration is required, since I'm just describing how to read a number.
The problem is that the typographical rules when describing base numbers aren't as obviously clear and uniform as that rule. And since describing base numbers means that the ordinary assumption is being overridden, then it becomes less clear where the assumption is being overridden and where it isn't. Because of that effect, something like "Base 111" has no practical meaning without having to carry forward the presumption that Base 10 is being, even though bases are being used at that moment. This causes extreme limitations if complex calculations are being performed, as they must be rooted to base 10, and cannot exist the freer state which would be permitted by allowing more drastic departures from our normal boring base 10 system.