ordinals

While you wait on the final word about that 750GeV excess, or the images from Juno, or the final tournament of the Euro 2016 between France and Portugal, or the return of James Corden after a two week break,
you may want to read the recent three blog posts of John Baez, {1}, {2}, {3}, to pass the time.

One begins with the empty set {}, then considers the set which contains the empty set {{}}, then the set which contains the empty set and the set which contains the empty set {{},{{}}}, and so on and so forth.
It turns out that the "and so on and so forth" is real magic and creates the ordinals and in particular large ordinals (*); This is truly mind-boggling stuff and as usual John explains it quite well.

But once you make it to the third blog post, keep in mind that "all the ordinals in this series of posts will be countable", which I find quite amazing to be honest.


(*) Initially one recreates the natural numbers as 0 = {}, 1 = {{}}, etc.
The first large ordinal is encountered as w = {0,1,2...} , i.e. the ordinal associated with the set of all natural numbers.
Then we get to w+1 = {1,2,3, ..., w} and the real fun begins...

14 comments:

Lee said...

Somehow Baez writes in such a way about mathematics that I feel l can mostly understand what he's saying. The feeling is probably false, but it's still a good feeling.

But I have a question that I'd like your thoughts about that is off topic. Since it is off topic, if you want to delete this comment or not respond, that would be fine with me.

Some years ago in this blog you asked something like, -- In a family with two children, what are the chances, knowing only that one of the children is a girl, that the other child is a boy? The answer is 2/3. Scott A. used to have a link to the same problem and I think it is a fairly well known example of conditional probability.

Recently a friend drew my attention to Leonard Mlodinov using it as an example in a book he wrote called, "The Drunkard's Walk." But then Mlodinov goes on to say that if you find out the girl's name is Florida, it changes the probability that the other child is a boy back to very close to 1/2. He listed the possibilities in the sample space (NF stands for girl not named Florida and F as girl named Florida) as: (girl-F, boy), (boy, girl-F), (girl-F, girl-NF), (girl-NF, girl-F), (girl-F, girl-F), (boy, girl-NF), (girl-NF, boy), (girl-NF, girl-NF), (boy, boy). The last 4 can be eliminated because they don't contain a girl named Florida. Mlodinov didn't actually work the example, but if I put in the occurrence of the name Florida in the population of girls as 1/30, I get a probability that the other child is a boy as 60/119. It turns out that 1/30 is a higher frequency than the most popular girl's name in the United States, so it is a conservative estimate.

So now come my questions that I don't know the answer to. Why can't you replace F in Mlodinov's listed possibilities with "arbitrary girl's name" and NF with "girls with different name?" Maybe there is a reason you can't do that, but if there is, I don't understand why. If you can do that and since every girl has a name that in the United States occurs with a frequency of less than 1/30, why isn't the answer to the original problem less than 60/119 and greater than 1/2, instead of 2/3. The answer 2/3 works for black and white balls, but why should it work for girl-boy combinations where it is known that they have other attributes besides just gender?

Anyway, if you or anybody else who reads this blog would like to enlighten me, I would appreciate it. I hope I've written this in a clear enough manner that it can be understood.

wolfgang said...

Well, I would say the short answer is that you use "Florida" and not "arbitrary girl's name" because this is the name you were actually told.

I guess the additional complication in this variant of the puzzle is that people usually don't give their kids the same name, so knowing that one is named "Florida" (almost) identifies her and this is what the original puzzle is all about.

The original puzzle hinges on the phrase "the other child", so from (G,G), (B,G), (G,B), (B,B) we eliminate only the (B,B) case and get the 2/3 answer.
Once we identify which girl we actually know about we would be back to
(G,G), (G,B) OR
(G,G), (B,G) and thus a probability of 1/2.
Knowing the name almost gets there ...

I am afraid this is all I can do to help your intuition.
Otherwise my recommendation is to simply write down all (equally likely) possibilities
and eliminate the impossible ones after new information arrives.



Lee said...

>> I am afraid this is all I can do to help your intuition.

Thanks Wolfgang. However in this particular case it was my reasoning that was failing me and not my intuition. If it weren't so embarrassing I'd tell you how I was thinking about it. The good thing about being old though is I can tell myself the lie that I wasn't always an idiot and am only an idiot now because I'm old 8-).

It's kind of strange how it works psychologically though. I was positive that I was making a simple mistake because Mlodinov and others would have mentioned what I was talking about otherwise. However for the life of me I couldn't figure out what I was doing wrong. But just reading the first line of what you wrote in your comment without reading any farther made me rethink what I was doing and I could easily see my error. So even though I knew the generic "they" were correct, it was only after reading I was wrong by a specific person that my brain could stop going down the same wrong road every time I thought about it. I wonder if that ever happens to other people?

wolfgang said...

A simplified version would go like this: IN a hypothetical world all girls are either named "Florida" or "Other" with equal probability and all boys are either named "Jim" or "Tom" with equal probability.
Initially we only know that the family has 2 kids, so there are 4x4 = 16 cases.
Now we know that one of them is named "Florida" so this gets us to 4/7 , which is between
2/3 and 1/2.
How did it happen? I would say the term "the other child" now contains more information ...

wolfgang said...

>> I wonder if that ever happens to other people?
I think everybody who does enough math and/or physics has experienced something similar.
One gets stuck in the wrong way and then something small all of a sudden gets us out ...

A similar experience happens to programmers, trying to fix a bug.
It is obvious that something is wrong (the compiler or debugger tells you so) but you cannot see it.
When a colleague looks at it he immediately sees the problem and then it is usually a slap on the forehead ...combined with "how could I not see it" ...

wolfgang said...

Btw I think this also happens in daily life, e.g. when people misplace e.g. their key.
The usual search/thinking goes like this: I put it on the table, but now it is not on the table, so what happened ...? And people cannot get out of this loop and narrow their search
to the area around the table, where the key is not ...

In this situation it helps to consider the possibility that the main assumption is wrong and the key was eventually *not* put on the table and one should retrace steps from the last time the person had the key. This then makes it possible to widen the search and eventually find the keys, e.g. in the fridge ...

Lee said...

Mlodinov made kind of a big deal out of the fact that Florida is an very uncommon girls name and that is why the result is 1/2. I, like you, calculated what the probability would be with 2 girls names equally distributed and got 4/7. Actually, realizing that the probability of the other child being a boy is 4/7 in the 2 names case for both F and NF is what led me down the path of idiocy. In any case, Mlodinov wouldn't have needed to choose an uncommon girl's name to have made his point.

Getting back on topic, there is something kind of awesome about the Feferman–Schütte ordinal. I had to read that several times to make sure I understood (or at least understood to the best of my present ability). I wonder if many things that are self referential are somehow related to each other in a way we don't understand?

Lee said...

I'm still bothered by something in Mlodinov's example. When you find out that the one child is a girl, more information than just gender comes with finding that out. For example you know that she has a name and you know that the frequency of that name in the general population of girls must be less than 1/30. That's not as much information as knowing the exact name and being able to look up the exact frequency of that name, but it's still a lot of similar information. I haven't found a way though of using that information to reduce the possibilities in the sample space to obtain an answer other than 2/3. So I guess my question is what makes some acquired information useful in updating probabilities and other similar information not useful? I imagine you're getting quite tired of me bringing up the same thing over and over. I apologize.

wolfgang said...

>> Feferman–Schütte

I have already a problem to wrap my brain around the fact that "Peano arithmetic is powerful enough to work with ordinals up to but not including epsilon0".
The problem I have is that the ordinals beyond epsilon0 are still countable, so in some sense one still talks about patterns of the natural numbers - but no longer describable within Peano.

wolfgang said...

>> still bothered

All I can suggest is to consider the case of 3 girl names next, which gets you even closer to 1/2 - and so on and so forth.

My intuition here is that the meaning of "the other child" changes.
Once you have 1 out of N possible names , with N very large, "the other child" is relative to 1 specified child.
But without a name, "the other child" is relative to 2 possible girls.
Increasing N gets you from one case to the other ...

That is all I can say about that, except repeat my advice to write down all possible
initial cases and then eliminate the impossible ones after you receive new information.

Lee said...

Thanks Wolfgang for taking the time to give me some of your thoughts on a completely off topic subject. It was gracious of you and I appreciate it.

>> The problem I have is that the ordinals beyond epsilon0 are still countable, so in some sense one still talks about patterns of the natural numbers - but no longer describable within Peano.

Do you know if there is any link between the Busy Beaver numbers and epsilon0? I wonder if all the non-computable Busy Beaver numbers lie beyond epsilon0?

wolfgang said...

>> if all the non-computable Busy Beaver numbers lie beyond epsilon0?
Well, the BBs should be natural numbers and as such are well below e0, even if some are not knowable.

But there are lots of comments to part 3 and I have yet to read through them and make sense of it all ... including comments about computability of large ordinals.
As John writes at one point, large ordinals are linked to proof theory which is a pretty tough topic ...

wolfgang said...

Just a quick summary of what little I know so far:
w (small omega) is the first ordinal after the natural numbers, so you have
1,2,3,...
w, w+1, w+2, ...
w*w , ... w^w ... etc.
which are still countable (so there is a one-to-one correspondence with the natural numbers) until you reach Omega. Beyond Omega are the non-countable ordinals, however
John only considers the case of large countable ordinals.
e0 is the first ordinal beyond the Peano axioms, which means one could not use
PA to proof that e0 and ordinals beyond e0 are well-ordered.
Of course there is a whole zoo of increasingly weirder large ordinals beyond e0 ...

wolfgang said...

Btw in the comments there is talk of transfinite Turing machines - whatever this is ...