By Michael Huemer (auth.)
Approaching Infinity addresses seventeen paradoxes of the endless, such a lot of that have no in most cases permitted ideas. The ebook addresses those paradoxes utilizing a brand new conception of infinity, which involves that an enormous sequence is uncompletable while it calls for anything to own an enormous in depth magnitude.
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And so on. In brief, the coin is flipped until the first time it comes up heads, whereupon you get a payoff of $2n, where n is the number of times the coin was flipped. The question is: how much is it worth to get a chance to play this game? 19 There is a standard way of addressing this kind of question: we calculate the expected payoff of the game. This is done by adding up, for every possible outcome of the game, the probability of the outcome occurring multiplied by the payoff you receive if it occurs.
I start with some simple mathematical puzzles before moving on to more exotic paradoxes. I will not attempt to solve any of the paradoxes in this chapter; solutions will be considered later. A word about the term ‘paradox’. Some people take the word to refer to a situation in which a contradiction is true. On this interpretation, there is of course no such thing as a paradox. None of my ‘paradoxes’ would count as true paradoxes, because in fact a true paradox would just be impossible by definition.
For the modern English translation, see Bernoulli 1954, 31. Seventeen Paradoxes of the Infinite 35 Second, it seems almost contradictory to hold that the value of the game is infinite, since if one plays the game, one will never gain an infinite payout. All the possible payouts are finite (though they increase without bound), so there is zero chance that one will gain $∞. ). More generally, it is strange that there should be a variable whose actual value is known with certainty to be less than its expected value.