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Irrational numbers and infinity have made me come up with this problem: pi, for example, is an irrational number, which means that it doesn't terminate or repeat. Every new digit found in pi increases the value of the number, no matter the value of the digit (for example, 3.141 is larger than 3.14, and 3.1415 is larger than 3.141). If pi never ends, then that means that there is an infinite amount of digits that will increase the value of pi by a tiny fraction. Therefore, pi should be infinitely large. So, pi = infinity. But there is a problem: pi is between 3.13 < x

Irrational numbers and infinity have made me come up with this problem: pi, for example, is an irrational number, which means that it doesn't terminate or repeat. Every new digit found in pi increases the value of the number, no matter the value of the digit (for example, 3.141 is larger than 3.14, and 3.1415 is larger than 3.141). If pi never ends, then that means that there is an infinite amount of digits that will increase the value of pi by a tiny fraction. Therefore, pi should be infinitely large. So, pi = infinity. But there is a problem: pi is between 3.13 < x

Response from André Carus on :

This is actually a very good question to illustrate how everyday intuitions can lead you astray in thinking about philosophical problems -- or about any problems, really, that require finer and more sophisticated distinctions than needed in typical situations of everyday life.
Everyday intuitions, embedded in the categories of our inherited natural languages, are insensitive to certain fine distinctions made in the more finely-tuned artificial languages we've invented. From an everyday point of view, there is no reason to think that your argument about pi being infinitely large is flawed at all. But in mathematics, we've had to deal with the problem that that argument works in some cases and not in other cases -- some infinite series "converge," as mathematicians say, while others "diverge." The sum of all the reciprocals of the natural numbers (i.e. one half plus one third plus one fourth plus one fifth, and so on) diverges, i.e. it's infinite, as in your intuition. But the sum of all the...