Every Infinity Paradox Explained

Hilbert's Hotel Hilbert's hotel is a thought experiment proposed by German mathematician David Hilbert in 1925 it involves a hotel with an infinite sequence of rooms 1 2 3 and so on to Infinity this is called countable Infinity since each room can be associated with a counting number the hotel starts out fully occupied however a new person shows up and wants a room surprisingly you can accommodate the new guest without removing any current guests the hotel simply moves a guest in room one to room two the one in room two to room 3 Etc in general

the guest in room n moves the room n plus one afterward room one is free for the new guest Hiller Brit's Hotel can also accept a countably infinite number of new guests the guest in room one moves to room two the one in room two to room four and in general the guest in room n to room 2times n this leaves the odd-numbered rooms free for the new guests the fact that Hilbert's Hotel can accommodate new guests despite being full is known as ht's Paradox this is an example of a veridical paradox a statement that

is true despite seeming false cantor's diagonal argument cantor's diagonal argument is a proof that the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers the cardinality of a set can be thought of as its size this proof was published by gor Cantor in 1891 to prove this fact it is sufficient to show that there are more real numbers between 0 and 1 one than there are natural numbers if this is the case then there are definitely more real numbers than natural numbers the proof is by contradiction

a proof by contradiction starts with an assumption and shows that the Assumption leads to an impossible conclusion thus proving that the assumption is wrong in cantor's diagonal argument We Begin by assuming that the set of real numbers between 0 and 1 and the set of natural numbers have the same cardinality if so that means each real number can be paired with a natural number 1: one with no numbers left over so let's do that and put the pairs in a list our list of pairs will have the natural numbers on the left and the real

numbers on the right making sure the digits of the real numbers are aligned with each other then we draw a downright diagonal through the digits of the real numbers now let's construct a new real number starting at the top left of the diagonal take the first digit of the first real number and change it to something else for example if it is nine subtract one from it otherwise add one we will put this digit in our new number now go down the diagonal repeating this process for all the real numbers in the list so is

our new number a part of the list Well it can't be the first number in the list since the first digits don't match it can't be the second number either since the second digits don't match our new number will always differ from any given number in the list since there will always be at least one digit that does not match the digit on the diagonal thus we have a real number that isn't part of the list meaning we haven't achieved a onetoone pairing which contradicts our earlier assumption even if we insert this new number into

the list we can just repeat this process to generate more and more new real numbers this implies that there are more real numbers than there are natural numbers the idea that some infinites are bigger than others is counterintuitive but it is a true fact Thompson's lamp Thompson's lamp is a hypothetical device proposed by British philosopher James F Thompson in 1954 it is a lamp that can be switched on and off as fast as you want let's say that you start a timer and turn the lamp on you wait 1 minute and then you turn the

lamp off again after another half a minute has passed you turn the lamp back on each time you wait for half of the previous amount of time and then you toggle the lamp because the lengths of these intervals add up to 2 minutes you will have toggled the lamp an infinite amount of times after 2 minutes have passed a task like this with an infinite amount of action in a finite amount of time is called a super task but when you are finished will the lamp be on or off each time you toggle the lamp

off it was immediately followed by toggling it back on and vice versa thus it seems like the question has no logical answer which creates a paradox mathematically this is related to the behavior of a certain infinite sum 1 - 1 + 1 - one and on and on forever and that's kind of a tongue twister an infinite sum like this is called a series and this particular one is called Grande series named after Italian mathematician Guido Kandi because the result just bounces back and forth between zero and one forever as you take more and more

terms the series doesn't approach a specific number or converge so we say that it diverges or has no finite value however certain methods can be used to assign the series a meaningful value for instance you can take the averages of the results as you go along known as Cesaro which gives a value of 1/2 Thompson had heard of Grande's series being given this value but his thought experiment did not allow the possibility of the lamp being halfon thus he rejected that grandi series and his thought experiment had any direct relation Gabrielle's horn Gabrielle's horn is

an infinitely long horn-shaped object with a finite volume but an infinite surface area it was first considered by Italian mathematician Evangelista torielli in a paper published in 1643 in order to form Gabriel's horn take the graph of the curve y = 1 /x in the XY plane with the domain of X being greater than or equal to 1 then rotate this curve in three dimensions around the x- axis and fill in all the points that the curve passes through forming a surface a three-dimensional surface created by rotating a curve around an axis like this is

known as a surface of Revolution now imagine closing off Gabriel's horn with a cal we can find the volume of the inside region using integration using slices perpendicular to the x-axis slice the inside region Into Thin discs that are approximately thin cylinders labeling the radius of each cylinder as R the base area of each of these cylinders is PK R 2 and the little thickness is DX so the volume is approximately p piun r^ 2 * DX the radius at each point is just the distance between the x- axxis and the graph of the function

1 /x so we can replace r with 1 /x so the volume of each cylinder is approximately piun * 1 /x^2 * DX and we just have to integrate this from 1 to Infinity which gives us Pi so the volume of this region is pi but what if we just find the area under the curve y = 1 /x or X from 1 to Infinity this is the same as this which becomes this as B grows bigger and bigger the natural logarithm of B also grows bigger and bigger without bound so the area under the

curve is actually infinite this is despite the fact that when we rotate this around the x-axis the volume of the solid we get has a finite value of pi due to the counterintuitiveness of this fact mathematicians at the time of its Discovery perceived it as a paradox Ross littlewood Paradox the Ross littlewood Paradox invented by John E littlewood in 195 53 and further explored by Sheldon Ross in 1988 involves a infinitely large empty vase and an infinite amount of balls a super task is performed at each step 10 balls are put in the vase and

then one ball is taken out each step takes half the amount of time as a previous one to ensure that the task is completed in a finite amount of time now at the end how many balls does a vase contain the intuitive answer appears to be infinitely many each step seems to involve a net increase of 9 balls in the vase so the number of balls just grows towards Infinity however there is a sense in which the vase may be empty imagine that at the start all of the balls are numbered at step one you

put in balls 1 through 10 and remove ball one at step two you put in balls 11 through 20 and remove ball two and so on for each step number n the ball numbered n is removed from the vase eventually ball number 1,000 will be removed then the ball number 1 million and so on since there are no balls that don't eventually get removed by the end all of the balls must have come out there is no General agreement on the solution to this Paradox besides the previous Solutions some people think it depends on exactly

which balls you take out of the vase others think that the problem does not give us enough information to State the answer or that it isn't well defined dartboard Paradox the dartboard Paradox is a paradox in probability let's say you have a dartboard and you randomly pick some point on it now let's say a dart hits the dart board at a random point is it possible for the dart to exactly hit the point you picked and if so what is the probability that this occurs this is a mathematical scenario so we will assume that the

point is infinitely small the answer to the first question is obviously yes the dart can hit any point on the dartboard after all however since your point is only one point out of infinitely many it seems that the probability of hitting that point is actually zero indeed in a rigorous mathematical sense this is actually the case but you can also use the same logic for any other point on the dartboard so the probability of the dart hitting any given point on the dartboard is zero as well however this seems counterintuitive since we said that a

dart must hit the dartboard the problem is that probability zero does not mean impossible in this case what we have is known as a continuous random variable because the space of possible contact points for the dart is continuous this is as opposed osed to a discret random variable where there are only a finite number of possible outcomes such as the role of a dice continuous random variables are common in probability for example the normal distribution models a continuous random variable in this case it is often more useful to talk about the probability that a given

event Falls within a specific region of the possibilities for example you can ask about the probability that the dart will land on the dart board's upper right quadrant in this case the probability is 1 and 4 or 25% when using a continuous distribution such as the normal distribution the probability of an event happening in a certain region is equal to the area between the graph of the distribution and the horizontal axis within that region which can be calculated using integration as we've seen paradoxes often take skilled mathematical Minds to resolve if you want to build

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build a strong base in Geometry which can be used as a launching off point for a variety of different subjects to try everything brilliant has to offer for free for a full 30 days visit brilliant.org thought thrill or click the link in the description you'll also get 20% off on annual premium subscription Petersburg Paradox the St Petersburg Paradox is another probability based Paradox named for the City St Petersburg where Daniel ber newly studied it imagine a casino game beginning with a stake of $2 the stake is the amount the player will be paid at the

end the player flips a coin and if it lands on tails the stake doubles otherwise the game ends and the player gets the stake we can calculate the expected value of the player's earnings the expected value is average value that you would predict from a random process there is a 1 half chance that the player lands on heads immediately earning $2 given the 1 half chance of Landing on tails to start with there is then a 1/ half chance of landing on heads on the next flip and earning $4 giving an overall chance of 1/4

to earn $4 this logic continues for all remaining flips we can calculate the expected value by multiplying each payout by its probability and adding the results together 1/2 * 2 + 1/4 * 4 + 1/8 * 8 is the same as 1 + 1 + 1 and this is the same as Infinity since it will keep going on now what would be a fair cost to play this game hypothetically no matter the cost even if it was $1 billion the casino always loses however according to Ian hacking few people would pay even $25 to play

this game which is in a paradox reman series theorem the reman series theorem named for and rigorously proved by German mathematician Bernard rean involves conditionally convergent series a conditionally convergent series is an infinite sum that converges to a particular finite value but would not converge if you added up the absolute values of the terms instead in other words a conditionally convergent series converges only under the condition that the sign of each term is taken into account this is as opposed to an absolutely convergent series which converges in both cases or according to the rean series

theorem any given conditionally convergent series can be rearranged to produce any real number this runs counter to how finite sums work in a finite sum rearrangement of the terms doesn't affect the result to prove the rean series theorem consider grouping all the terms of a conditionally convergent series into two groups a positive and negative in descending order of magnitude since the series is convergent the magnitudes of the terms in each group must get arbitrarily small however remember that the series is only conditionally convergent that means that each group of terms in individually doesn't sum to

a finite value if the positive terms were a sum to some positive value a and the negative terms were a sum to some negative value minus B then the overall sum of the absolute values of the terms would be a plus b implying that the series would be absolutely convergent not conditionally convergent it's also impossible that the positive term sum to positive Infinity while the negative term sum to a finite negative value otherwise the original series would diverge towards positive Infinity not converge this is also true the other way around thus the only possibility is

that the positive term sum to positive infinity and the negative term sum to negative Infinity due to the fact that these are unbounded we can combine terms to go as high or low as we want in order to test this out let's pick any real number X and try to get to it by adding our terms we start at zero if x is above us we add the largest positive term we haven't used used if x is below us we add the negative term of greatest magnitude that we haven't used if we are currently at

X we can add either a positive or A negative term we can repeat this process indefinitely going as far up or down as we need and gradually converging on X as our terms get smaller and smaller also another Paradox is that if you hit that subscribe button it magically turns gray anyways I'll catch you guys in the next video