I’m not really sure what you guys are discussing, but it reminds me of the proof for the non-existence of a smallest integer number without interesting properties.
Boy, are we off topic!!!
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That seems like an easy one.
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Yeah, for every number n with some interesting property, number n + 1 has the interesting property of being 1 greater than number n with said interesting property. 0 has the interesting property of being the additive identity. Therefore all natural numbers have some interesting property
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Nah, you can define interesting as describable in some small number of words so that there are only finite possibilities. Then there must exist for example “the smallest integer which cannot be described in twelve words or less.”
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I’m sure one could wrangle some Latin and/or Greek prefixes in order to describe any natural number in only 1 word. Might have to come up with one’s own convention of their use, but if that convention can be described in 11 words you’re golden
And that’s assuming English. Ithkuil allows you to inflect any word with, among a thousand other things, an arbitrarily long phrase in a minilang with a vocabulary of about 500 lexemes. So you can definitely describe your convention there
Then change “words” to “letters from the Latin alphabet”, problem solved.
I guess there’s always tricks to get out of paradoxes, like “the number equal to the height in millimeters of the following letter A”
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I guess the intuition then would be that we have an imperfect definition of “interesting”, and a proper definition would not allow such things
Nope, it has nothing to do with the word “interesting”. The problem is rather with “definition”.
Take Berry’s paradox:
“The least natural number not definable in less than eighty letters in the English language.”
There’s nothing ambiguous about what it defines, yet it is paradoxical.
The usual way logicians would resolve this, is by stratifying logical language. You get symbols to describe stuff with, but you cannot include symbols in your language that describe your language itself. In order to talk about your language, you have to move to a meta-language, in which you can talk about what is definable in the first language (using the symbols of the first language). To talk about what is definable in the meta-language, you need to move to a meta-meta-language. And so on.
This is the same resolution for similar self-referential paradoxes, like the liar paradox.
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Assume that x is the smallest integer number without interesting properties. Then x has the interesting property of being the smallest integer number without interesting properties. That’s a contradiction, so x doesn’t exist.
Isn’t that a simple proof by contradiction? Where is the paradoxon?
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You didn’t define what “interesting” means.
Yes, I know, I’m just assuming that being the smallest uninteresting number would be an interesting property. What would change, if I gave a proper definition?
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This is basically the same as the induction proof from @Samraku above.
There naturally is no parodox if all numbers are declared interesting 
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I see, so saying “smallest uninteresting is interesting” leads to all being interesting, which is kind of boring. The definition of “interesting” would have to exclude “smallest uninteresting” to make it interesting.
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