1. 0.999…= 1
Although incredulity regarding this claim is what motivated this thread, I haven’t generally found that people can’t be convinced that it is true fairly quickly (of course, most people I’ve demonstrated this or similar claims to are math students, so I’m dealing with a biased sample). Also, I already explained why it is true in various ways, so here I’m just going to post two youtube clips. The first shows how/why 0.999…= 1 in a few ways, and the second (by the same individual) is a joke clip containing illogical (but perhaps intuitive!) reasons why this claim is false.
2. My go-to example.
I have often said that any past, present, or future mental health issues I had, have, or will have are due to the fact that my father decided to show me a proof that some infinities are larger than others when I was 6 years old. Cantor, the mathematician who proved this, actually wrote to a fellow mathematician (Dedekind) saying of his own results “Je le vois, mais je ne le crois pas!” (“I see it, but I don’t believe it!”). But I like to combine this result with a property of the real numbers to get something I think even more seemingly paradoxical.
It’s not that hard to show that between any two rational numbers, there exist infinitely many rational numbers: just recall that rational numbers can be written in decimal form, so 0.1 is rational, as is 0.01, 0.001, 0.0001, & 0.00…1. So, for example, even though 0.00000001 is really close to 0.00000002, there are infinitely many rational numbers between them of the form 0.0000000001, 0.0000000001, 0.00…1 (alternatively, I can say that between e.g. one thousandth and two thousandths exist infinitely many rational numbers like one ten thousandth, one millionth, one trillionth, etc.). The mathy way to describe this property is to say that the rationals are “dense” in the real number line.
In other words, intuitively the rational numbers have no “gaps”. If we imagine them as points on a line, no matter how close we “zoomed in” there would be no spaces because the rational numbers are infinitely close to one another.
However, we all know that there are irrational numbers such as pi. Not only that, but Cantor proved that there are more irrational numbers in the interval [0,1]
then there are rational numbers all together! So in the interval [0,1], even though there are infinitely many rational numbers infinitely close to one another, the “amount” of “space” they take up (their “length”, or technically speaking their measure) is 0. In this and in every interval, almost all numbers are irrational, such that if you imagine picking a number from any interval at random, the probability that the number would be rational is 0.
To conclude, if we pick any interval—say, [0,1], we have to fit an infinite set of irrational numbers greater than all rational numbers into infinitely small gaps between rational numbers.
3.
Double AND Nothing: The Return of The Cantor Set (a blog post I wrote)
4.
Hilbert’s hotel
5. Logically irrational
A University student was out for a stroll in the city. We'll call him James (his full name is James James Morrison Morrison Weatherby George Dupree, and at a young age his mother, despite James' instructions, went to a certain area of the town she oughtn’t have and has been missing ever since). Unbeknownst to James, a local protest rally has turned into a riot. Making matters worse, the local police are out arresting people left and right in order to stop the destruction which happened after the last protest rally some weeks ago.
James just happens to walk into the area in which the chaos is occurring. Although he was initially curious to find the source of all the tumult he heard, now that he realizes what's happening he decides the best course of action would be to leave quickly. As he is turning around, however, he sees a rock on the ground. But this is no ordinary rock (and James would know, as he's been collecting rocks since he was three, initially to cope with the loss of his mother). So he picks it up. Unfortunately, it is at that moment that a police officer sees him. Thinking that James is about to throw the rock into the window of a nearby building or car, the officer arrests James.
A few days later, James finds himself in front of the Judge. He has already explained that he picked up the rock because embedded in it was a rather large specimen of a type of quartz not common to the area. The police officer has likewise given his testimony. The judge doesn't feel there is any evidence, so he is inclined to rule not guilty. But he realizes (being the clever individual he is) that James never actually said he
wasn't going to throw the rock. So before he pronounces James innocent, he asks "If you weren't arrested, then would you have thrown the rock through a window?"
James is a philosophy student, and having read Frege's
Begriffsschrift, the
Principia Mathematica, and several other books which all contain systems of propositional and predicate logic, he knows exactly how to answer. After all, the judge has asked a conditional-an "if" question. As any intro to logic student would know, thinks James, a conditional, If a then b, only comes out "false" under one condition (he briefly draws a truth table just to double check; this isn't the time to make mistakes). If the antecedent is true, the "if" part, and the consequent is false, the "then" part, the conditional is false. Otherwise, the conditional is true.
James realizes that the antecedent here is false. He was in fact arrested. Therefore, he reasons, his commitment to logic allows only one answer, as the truth value of the whole conditional is clearly "true." So he says "yes." And is given a death sentence with the possibility of parole given good behavior after the execution.
6. You are presented with three boxes, A, B, & C, one of which contains a check for a million dollars. You are asked to pick box A. Rather than open the box, the game show host opens box C, which is empty. So the check is either in box A, or box B. Should you change your decision, or not? If so, why should you? If not, why not?
7. Imagine a "fair coin" is tossed a number of times (i.e. one that is equally likely to land either heads or tails but will not land on its side). Let's say H is heads and T is tails. Given, say, over a dozen tosses, we would expect a bunch of heads and a bunch of tails. We probably won't get a perfect division, but we would expect something like the following:
HTTTHHHTHTHTHHTTHTHTTHHHHTHTHTTHHTHHHTTTHTHTHTHHHTHHTHT
What we would would not expect, and rightly so, is to get all heads or all tails, like so:
HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
In other words, the first coin toss above is far more likely to occur than coin toss #two. Right? Wrong.
