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roystgnr comments on The Number Choosing Game: Against the existence of perfect theoretical rationality - Less Wrong
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It's not a problem with the math system. It is part of the situation that you aren't allowed to say 100 minus delta where delta is infinitesimally small. In fact, we can restrict it further and rule that the gamemaker will only accept the number if you list out the digits (and the decimal point if there is one). What's wrong with perfect rationality not existing? On the other side of the question, on what basis do we believe that perfect rationality does exist?
There exists an irrational number which is 100 minus delta where delta is infinitesimally small. In my celestial language we call it "Bob". I choose Bob. Also I name the person who recognizes that the increase in utility between a 9 in the googleplex decimal place and a 9 in the googleplex+1 decimal place is not worth the time it takes to consider its value, and who therefore goes out to spend his utility on blackjack and hookers displays greater rationality than the person who does not.
Seriously, though, isn't this more of an infinity paradox rather than an indictment on perfect rationality? There are areas where the ability to mathematically calculate breaks down, ie naked singularities, Uncertainty Principle, as well as infinity. Isn't this more the issue at hand: that we can't be perfectly rational where we can't calculate precisely?
Just as an aside, no there isn't. Infinitesimal non-zero numbers can be defined, but they're "hyperreals", not irrationals.