I think I mismatched my infinite series, by starting the probability factor from 1 instead of from 1/2… Last Sunday I got 2 - ln(4) as an answer for a single family, now I get 1 - ln(2).
It’s hard to find mistakes since I have no intuition what the expected value of a ratio is going to mean.
It seems that people have voted for “greater than 1”, “exactly 1”, and “less than 1”.
Are you suggesting that the correct answer is “infinity” or “need more information / problem is unclear”?
I agree that the problem could be considered unclear, since there are various assumptions about how random births behave, gender assignments, practical limitations of family sizes, etc. Depending on what assumptions you make, I think it’s possible to have any of the top-three answers.
If you believe that the answer should be infinity instead, could you please explain why? It seems that you have already stumped everyone.
For any size country, there’s a finite chance that all families will have a boy first, then stop.
For example, with ten families, the probability of getting no girls would be 1 in 1024. If this happens, the ratio of boys to girls is 10 to 0, or infinity.
So the expected ratio, just being some weighted average involving infinity and a bunch of usual numbers, is also infinite!
The point of the first problem is that even though families can’t alter boy/girl birthrates, it’s not true that everything averages out to 50%. The point of this one is that it’s not just a minor detail; you can be infinitely far off if you make this wrong assumption.
I also need to start reading more carefully. I was trying to compute the ratio of boys to the country’s population.
To be honest, these questions really make little sense if you think about what’s actually being asked.
If someone told you you would never win a Go game again, would you stop playing?
0 voters
Just someone confused between his dreams and the reality.
I would refuse to believe that. If I actually believed that I will never win a game of go, then the sad truth is yes, I would probably stop playing.
I wonder. In go and other subjects stronger players and other people who walked the path advise beginners and such, and say “oh, this thing helped” and such. But I think there’s a great difference between how people learned and how people remember they learned. In hindsight the path looks very different. Some important things are forgotten, other trivial things get stuck in your memory. Additionally your current weaknesses and strengths change your worldview. In the end, you can’t make an accurate judgement about how you learned and what helped you.
After a number of years of learning go, I did many things, and at this point I’m certain I don’t remember what I did or what helped. Still won’t stop me from giving advice though.
And I would also doubt that teachers can make accurate judgement about whether one specific regimen is better than other.
What is worse?
0 voters
Is there a difference between “logically possible” things, and things which are “logically possible to know”? Is an ordered set containing every one of the infinite digits of pi logically possible? I think so, though I’m not mathematician. But is it logically possible to know that set? Certainly it’s not possible in a finite universe, but that’s not the question.
Or, what is the largest infinity which could logically exist? Certainly in order for it to be logically possible to know every digit of pi, the number of digits in pi must be a lower bound on the largest infinity which could possibly exist.
This has implications for what the omniscience of an MGB includes.
I would question whether a “largest infinity” exists. It’s like asking “What is the largest natural number?”.
There is a proof stating that, no matter which non-empty set “S” we start with, the power set P(S) has a larger cardinality, that is, there exist injective functions f: S → P(S), but there exist no surjective functions from S to P(S).
So if we assume that a set S has largest cardinality, then the above argument gives a contradiction.
The questions really hinges on how exactly we define the concept of “logically possible to know”. Since you go on to talk about the digits of pi, I think the type of definition that you are after would be the concept of Computability - Wikipedia. This is a reasonable concept for defining our ability to “know” something based on the possibility of mechanically computing it, which also depends on defining an abstract notion of what is possible for a computing machine (see Universal Turing machine - Wikipedia).
The infinite sequence of the decimal digits of pi is a well-defined concept, and we have various practical algorithms for computing to any arbitrary (finite) point in that sequence, while using a finite amount of execution time and space (memory). See also: Computable number - Wikipedia (which uses pi as an example).
On the other hand, there are uncomputable numbers and Undecidable problem - Wikipedia, which could be viewed as examples of things that are “logically impossible to know (compute)”.
From another angle, we can consider the question in a more fundamental, epistemic way by asking whether or not there are statements that cannot be logically verified. A profound discovery are Gödel's incompleteness theorems - Wikipedia, which imply that in any “useful” logical system (a formal and consistent axiomatic system that is able to support reasoning about arithmetic) there are logical statements that are true but unprovable within that system.
Since the digits of pi can be viewed as just a countably infinite sequence, the cardinality of this infinity is rather mundane, as it can be thought of as being the same size at the set of natural numbers. On the other hand, the cardinality of the set of real numbers is larger, since they are uncountable. And as @martin3141 presented, one cannot have a largest infinity, even in the perspective of set cardinality.
There are other notions of “larger and larger infinites”, such as Ordinal number - Wikipedia, but I see that @Vsotvep is also writing a reply, and I would be surprised if he does not mention it, and I’m certain he can explain and articulate this concept much better than me.
What is an MGB?
As someone specialised in logic and infinity, this is a lovely question! 
Allow me to geek out a bit.
Let’s start with the disclaimer: how to interpret “logically”. One needs to specify which logic to use. For example, there’s logics where contradictions are allowed (called paraconsistent, useful to talk about contradicting beliefs, for example), or logics where statements are neither true nor false (called paracomplete), or logics where truth value is a range of values (for example fuzzy logic), and so on.
I’ll assume for the sake of this question that we’re dealing with classical logic, which is by no means the “true” logic, but it’s the one accepted by most mathematicians. It’s the kind of logic that’s used in computers as well.
Yes, there’s a difference! First, let’s put another disclaimer up: epistemic logic is very broad and can be both highly philosophical, linguistical or mathematical (and any combination of those), thus it’s essential that we define carefully what is meant by something being known before trying to answer the question.
In context of this question, let’s say that knowing something means you have unconditional evidence that it is true, so in a way knowing something means you have proof. Now the question becomes whether there exist statements that are logically possible (hence true) of which it is impossible to know that they are logically possible (hence lack a proof).
Such statements necessarily exist by Gödel’s first incompleteness theorem: for any system strong enough to do arithmetic in, there are true statements that cannot be proved within the system. Hence, if we fix our framework, there are true statements about numbers out there that cannot be proved within the framework that we fixed; we would need stronger assumptions to prove it.
You’d say that the next step would be to prove that our framework is the “true” framework that is correct in our universe. But that’s impossible to decide by Gödel’s second incompleteness theorem: for any such framework, it is impossible to prove within the framework that it is consistent (i.e. that it does not contain any contradictions).
Therefore, we can’t prove what the correct framework is to build our knowledge in, and even if we assume a certain framework to be the correct one, then there will always be statements that we cannot know the truth of!
I also don’t know what an MGB is, but omniscience is a hard concept. I’d need more clarification to say something meaningful about this.
Again, this depends on the logical framework that one chooses. If we take the usual axioms that mathematics is currently based on (according to the majority of mathematicians), which are the axioms of ZF (for Zermelo-Fraenkel), then there is an axiom that tells us that there exist an infinite set.
Interestingly, as soon as we have an infinite set, we get infinitely many infinite sets and it is impossible to find a “largest infinity”. Mostly it’s comparable to how there is not a “largest number”: we can always find a larger number. This fundamental aspect of infinity is known as Cantor’s paradox: there exist infinities of arbitrarily large size, and indeed, the number of infinities is larger than any infinite size itself.
On the other hand, there’s a minority of mathematicians who believe mathematics should be done finitely. They reject the assumption that an infinite set exist. In this case, the question what the largest infinity could be, is of course meaningless: the answer is that there exists no infinity.
So to answer the question: there is no largest infinity that could logically exist, since we either have no infinity at all, or if we do, we have no “largest” infinity.
If it’s just the digits of pi, then this set would just be 0,1,2,3,4,5,6,7,8,9, since those are the digits that pi contains… But I assume you mean a set that contains, for example, all possible approximations of pi.
This is perfectly computable: there are algorithms that can compute what the N’th digit of pi is for any natural number N. The main obstacle is time, of course. If we have unlimited time, we can just let such an algorithm find each digit one by one.
However, I think there’s a more fundamental problem here. People are usually obsessed over the fact that pi has infinitely many digits, but we don’t use pi by looking at its digits: we know what pi is, because it’s defined as the ratio between the circumference and the diameter of a circle. That’s what pi is, thus in essence, we know pi completely.
Similarly, for any given accuracy we could approximate pi using algorithms.
However, this is not so different from what 1 is: 1 is that unique number that forms the multiplicative identity. Like pi, 1 also has infinitely many digits. It just turns out that all of them except one are equal to 0. This makes computing an approximation of 1 a lot more manageable than approximating pi, but in a sense, the numbers aren’t different.
I’d say that knowing each of the infinite digits of pi is one way to “know” pi, while the circle definition of pi is another way of knowing pi. The latter is a lot more useful.
However, this hinges on some other topic that’s quite similar but wholly different: is it possible to know every real number?
First of all, it’s impossible to list all the real numbers. This is a result of Cantor as well, and the essential idea behind the fact that there are infinitely many infinities. Given any real number, we can write it as a decimal expansion. Now suppose that we have a list of all of the real numbers:
| # | decimal expansion |
|---|---|
| 1 | 0.00000000... |
| 2 | 3.14159265... |
| 3 | 2.46578467... |
| 4 | 0.33333333... |
| 5 | 0.57721566... |
| : | : : : : : : |
Now I can create a new number that cannot be found on the list: for this number I take the N’th digit of the N’th number in the list. If this digit is a 1, the N’th digit of my new number will be 2, and else it will be 1.
If we isolate the N’th digit in our list of numbers:
| # | decimal expansion |
|---|---|
| 1 | 0. 0 0000000... |
| 2 | 3.1 4 159265... |
| 3 | 2.46 5 78467... |
| 4 | 0.333 3 3333... |
| 5 | 0.5772 1 566... |
| : | : : : : : : |
Then we see that our new number is something like 0.11112.... Now, the claim is that this number does not exist on the list, because if it exists on the list, it’s equal to the N’th number on the list for some N, but by construction the N’th number and the new number are different in their N’th digit!
(footnote: you have to be a little careful, since there can be multiple representations of the same number, e.g. 0.9999999… is the same number as 1.00000…, so you don’t want to go changing 0’s in 9’s, since that may turn out to give a number that’s already in your list)
Hence, there are more real numbers than can be put in a list.
The proof that there is no largest infinity uses essentially the same method, which is called diagonalisation. Say we have a possibly infinite set of things, A, and we have the set of all subsets of A, usually called the power of A, or P(A). If the set A is the largest infinite set, then P(A) cannot be larger than A. Saying that one set is larger than another, means that we can use the larger set to “label” each of the elements of the smaller sets. So suppose we label each subset in P(A) with (at least) one of the members of A. So, if Y is a subset of A, let Y be labelled by y in A.
I can now find subset X of A that cannot have any label: namely, for any element x of A, I let x be an element of X if and only if x is not an element of the subset Y of A that is labelled by x.
Now what would the label of X be? If x is the label of X, then x is not an element of X by how we defined which elements are part of X. But then, since x is the label of X, we see that x should be an element of X. But then, we see it shouldn’t be an element of X. So it should be an element of X. etc.
Clearly x has to be both an element of X and not an element of X, which is a contradiction.
Now the real question (pun not intended) is of course whether we can know all the reals. This turns out to be rather tricky. We don’t even know how many reals there are!
Under the assumptions that the axioms of ZF are true, there is this technique called forcing that can be used to create more real numbers. It sounds weird, but if we assume a universe that forms a model of ZF (which means, a universe in which all the axioms of ZF are true), then we can extend this universe by adding more real numbers. The technique of forcing guarantees us that this new universe also is a model of ZF, so as far as mathematics is concerned, generally all of it is provable in this universe.
What’s interesting, is that forcing can be applied to add any infinite number of real numbers to our universe. That is, if we have the set of reals having a certain infinite size L, and we take another (much larger) infinite size K, then we can find a model of ZF where there are K many reals. The other way around works as well: if we have a large number of reals, we can collapse our concept of cardinality to get a new universe where there are only a few reals.
The problem then remains to decide which of these universes is the actual universe. This is once again impossible to know for sure, due to Gödel’s incompleteness theorem: we can never prove that there aren’t any inconsistencies in our universe, which would be a reasonable requirement to make.
Just for the sheer size of your comment, I already liked it 
Let me just make a cup of coffee, then I’ll be reading 
Lately this thread has become boringly serious. This was certainly not the intention of the original poster @Starline. Important was used in an ironic way. Check out the first hundreds of posts. Stop being so serious.
Or should I start to flag you all for being off topic?
I won’t read it, but I’ll like it on principle, lest you feel less motivated to geek out.
My mind went here, but obviously not.
MGB means may god bless.
I clicked on that link expecting and hoping that it would be a K-pop group.
I think people should ask and discuss whatever interesting philosophical questions they wish, whether they be “serious” or “light”. Even with the most technical discussions, there is still humor that can be found.
Further, if you wish to see the topic of discussion to go somewhere else or take a different tone, you are always free to post something yourself…
Maybe stop trying to act as forum police. Contributions are so often limited to “there’s a similar thread there”, “why are you asking this here”, “this doesn’t go with that topic”.
Yebellz is kinder than me as always, but seriously.