Video instructions and help with filling out and completing What Form 2220 Amounts

Instructions and Help about What Form 2220 Amounts

Hey Vsauce Michael here what is the biggest number you can think of a Google a googolplex a milli million all Plex well in reality the biggest number is 40 covering more than 12,000 square meters of Earth this 40 made out of strategically planted trees in Russia is larger than the battalion markers on Signal Hill in Calgary the six found on the above and badges in England even the mile of pie Brady unrolled on numberphile 40 is the biggest number on earth in terms of surface area but in terms of amount of things which is normally what we mean by a number view big 40 probably isn't the biggest for example there's 41 now and then there's 42 and 43 a billion a trillion you know no matter how big of a number you can think of you could always go higher so there is no biggest last number except infinity infinity is not a number instead it's a kind of number you need infinite numbers to talk about and compare amounts that are unending but some unending amounts some infinities are literally bigger than others let's visit some of them and count past them first things first when a number refers to how many things there are it is called a cardinal number for example for bananas twelve flags twenty dots twenty is the cardinality of this set of dots now two sets have the same cardinality when they contain the same number of things we can demonstrate this equality by pairing each member of one set one-to-one with each member of the other same cardinality pretty simple we use the natural numbers that is 0 1 2 3 4 5 and so on as Cardinals whenever we talk about how many things there are but how many natural numbers are there it can't be some number in the naturals because there'd always be one plus that number after it instead there's a unique named for this amount aleph-null Aleph is the first letter of the Hebrew alphabet and Aleph null is the first smallest infinity it's how many natural numbers there are it's also how many even numbers there are how many odd numbers there are it's also how many rational numbers that is fractions there are that may sound surprising since fractions appear more numerous on the number line but as Cantor showed there's a way to arrange every single possible rational such that the naturals can be put into a one-to-one correspondence with they have the same Cardinale point is aleph-null is a big amount bigger than any finite amount a googol a googolplex a googolplex factorial a googolplex - a googolplex squared times Graham's number Aleph Nolan is bigger but we can count past it now well let's use our old friend the super task if we draw a bunch of lines and make each next line a fraction of the size and a fraction of the distance from each last line well we can fit an unending number of lines into a finite space the number of lines here is equal to the number of natural numbers that there are the two can be matched one two one there's always a next natural but there's also always a next line both sets have the cardinality aleph-null but what happens when I do this now how many lines are there aleph-null plus one no unending amounts aren't like finite amounts there are still only aleph-null lines here because I can match the naturals one-to-one just like before I just start here and then continue from the beginning clearly the amount of lines hasn't changed I can even add two more lines three more four more I always end up with only aleph-null things I can even add another infinite aleph-null of lines and still not change the quantity every even number can pair with these in every odd number with these there is still a line for every natural another cool way to see that these lines don't add to the total is to show that you can make this same sequence without drawing new lines at all just take every other line and move them all together to the end it's the same thing what hold on a second this and this may have the same number of things in them but clearly there's something different about them right I mean if it's not how many things they're made of what is it well let's go back to having just one line after an Aleph null sized collection what if instead of matching the Naturals one to one we insist on numbering each line according to the order it was drawn in so we have to start here and number left to right now what number does this line get in the realm of the infinite labeling things in order is pretty different than counting them you see this line doesn't contribute to the total but in order to label it according to the order it appeared in well we need a set of labels of numbers that extends past the naturals we need ordinal numbers the first trans by night ordinal is omega the lowercase Greek letter Omega this isn't a joke or a trick it's literally just the next label you'll need after using up the infinite collection of every single counting number first if you got omega 'the place in a race that would mean that an infinite number of people finished the race and then you did after Omega comes Omega plus 1 which doesn't really look like a number but it is just like 2 or 12 or 800 then comes Omega plus 2 Omega plus 3 ordinal numbers label things in order ordinals aren't about how many things there are instead they tell us how those things are arranged their order type the order type of a set is just the first ordinal