Every Infinity Paradox Explained

To try everything Brilliant has to offer—free—for a full 30 days, visit brilliant.org/ThoughtThrill . You’ll also get 20% off an annual premium subscription

No intro, no outro, lots of examples given to help understand, slow pace speaking but gets through all the concepts fast, too good.

The Ross-little wood paradox isn´t a paradox at all. Assume, the vase actually CAN get empty. Then there has to exist a step "n", after which only one ball is left in the vase (somehow), because in every step only one ball is removed. So step "n+1" could theoretically empty the vase. But in this step, ten balls are put into the vase. So the vase can never get empty, because there does not exist a step, where only one ball is taken out, but no balls are put in.

My long-time statistics hot take is that the "expected value" operation is given undue significance. Its name is very suggestive of it being the inherently correct way to judge risk and reward, but the fact of the matter is that it is fairly uncommon for it to actually yield a value which you can, in any meaningful way, expect. The St Petersburg paradox shows this well. There is less than a 7% chance of getting $32 or more from the game. A person simply cannot, in the ordinary sense of the word, expect more than $32, regardless of the value of the operation which has been named "expected value". Pascal's mugging is another "paradox" that just comes down to the expectation operator being treated as gospel. A mugger says if you give them your money, they will give you x amount back later. No matter how improbable you believe it to be that they will uphold the deal, they can give an x for which the "expected value" of taking the deal is positive, giving muggers a surefire strategy against people who treat the "expected value" operation as the ultimate arbiter. "Expected value" is only genuinely accurate to its name when the risk is taken infinitely many times. Humans aren't immortal. The risks also often have an ante of some kind, and if the risk failed enough times, you can't afford the ante anymore and can't try again. Even with large corporations, which can absorb many more and much larger failed risks than people can, risk assessment goes beyond expected value -- they qualify or quantify the level of risk itself and compare that to a risk tolerance

Your production quality is getting better and better! keep up good work :)

Infinity is a renaming of an operation involving 1. proportion (division) and continuity (something continuous) under 2. empirically induced abstraction of identity and therefore quantity and discreteness, which is known as counting. Infinity is nothing more or less than an irrational number in relation to the standard under which we intuit its irrationality, that standard is what we call a metric. The proof of the irrationality of the number will never add any new concept to what we already intuit in two overlapping spheres where neither is rendered from the other. There simply is not enough reality for us to move outside of the spheres of dualities such as the continuous and the discrete, and this duality is involved necessarily in every instance where something non-zero x is not a product of something non-zero y. Mathematical Paradox is always the expectation that an x that is not rendered from y should non the less go hand in and with the metric we induce from y or by which we construct y.

Regarding the St. Petersburg Paradox, I think the reason for why people would not pay an infinite amount of money to play this game is that if the probability of tails is even slightly less than 50%, then the expected value immedeatly becomes finite. For example if the probability of tails was 47.9%, then paying 25 Dollars would already put you at a disadvantage as a player.

I find the St. Petersburg paradox very interesting, it highlights (as many other paradoxes here do) how "right" results are wrong when we ignore other variables applicable to that context. For example looking at the variance. While the EV is theoretically infinite we can plot every price of the game to the odds of turning a profit, and how much total capital we would need to ensure a profit in the long-run.

I'm always frustrated with the diagonal argument, because it is not immediately obvious why the same argument doesn't work for rational numbers.

I think the saint petersburg paradox happens because the 1/4*4 depends on the 1/2*2 so it just converges to a total of 2 But IDK tho I'm not a mathematician

Note to self never stay at Halbert's hotel. I'm not tryna keep switching rooms everytime a new schmuck shows up

The ones that are said to be "task done in half the time the previous one" stop at plank time or go theoretically infinite times(in a finite time)?

The lamp is half as bright as it can be. This is true for ever-so-fast frames, and so it must be true for continuous perception.

cantor's diagonal can be counter argued too. the digits of an irrational number are countable infinite (evident, since we can count them) if we assume there are uncountable irrational numbers that means the list contains more rows than columns. the number we form using the diagonal can be below the countable infinite first numbers. that means, the number on cantor's diagonal IS on the list. which destroys cantor's proof that the cardinality of the real numbers is greater than the cardinality of the naturals. maybe it is indeed larger but cantor's diagonal doesn't prove it. thomson's lamp can be gaslighted by taking into account plank time. once you reach it, you can't meaningfully reduce it more. that means there's a finite amount of times you can half the time. but, of course, that's just me being a SoB.

The Ross-little wood paradox blew my mind lol idk how I’ve never heard it before

This video is look more entertaining then your other videos. I liked the sarcasm in this video and also the animation you put in which make video easy to understand. Kudos to you 👏

Some more infinity paradoxes Consider "the set of everything." By the diagonal argument (on subsets rather than on digits) one can prove that it doesn't coontain the power set of "the set of everything", hence "the set of everything" doesn't contain everything. Solution: modern set theory gives axioms that give a restriction to what can be a set, denying "the class of everything" as a set. So it has no power set. According to (modern) set theory, cardinal numbers and ordinal numbers are both sets. A cardinal number is defined to be an ordinal number alpha that is equal to the lowest ordinal number beta such that alpha can be mapped injectively into beta. Then the smalles transfinite ordinal number often is denoted w (in fact we call it omega.) It is also a cardinal number, and as a cardinal number it will often be denoted Aleph_0. But, w^w is countable while Aleph_0^Aleph_0 is uncountable. Explanation/solution: when doing ordinal numbers, alpha^beta is an ordered set. But when doing cardinal numbers, alpha^beta is just a set (and we forget about order.) So w^w differs from Aleph_0^Aleph_0. Banach - Tarski Paradox. It is possible to 'cut' a solid ball into a finite amount of pieces and reassemble those pieces such that you assemble two balls, each being a perfect copy of the original ball. However, each of those 'pieces' is in fact an infinite scattering of points, rather than a solid piece of the ball. Achilles Paradox. Consider a race between Achilles (who was able to run very fast) and a turtoise (who was quite slow), with the turtoise starting ahead. Each time Achilles covers the distance between him and the turtoise, the turtoise also moved a little bit. Hence Achilles has to cover another distance to catch up the turtoise. This repeats infinitely many times. "So Achilles can't win the race." It took quite the time before this paradox was finally solved. First we had to define a notion of "converging limits" before finally reason well why Achilles is able to come ahead of the turtoise. The point where Achilles will pass the turtoise is the limit point of the set of points where either is at any of the mentioned moments. These are the moment the race started, the moment Achilles has reached the starting point of the turtoise, the moment Achilles has reached the point where the turtoise was at the moment Achilles reached the starting point of the turtoise, etc.

The Ross-Littlewood paradox should result in 0 balls remaining regardless of the order they are removed in. The number of balls removed is countable infinite, as is the number of balls placed inside. Just because the balls going in is ten times more dense, doesn’t make it uncountable. Because the two “sizes/types” of infinity are the same then after infinitely many steps all the balls have been removed. You can just number them after the fact too…

Some people need to realize that paradoxes aren't necessarily formulated to 'prove' anything, but rather to show a gap in our understanding of the subject on hand.

How does this affect the local trout population