All infinities are equal, but some infinities are more equal than others.
Although it is not immediately obvious, there is a radical closure of a field and it is a field. What is not obvious is that if you take two complicated radicals and add them, multiply, divide them, the result can still be expressed as aa complicated radical, but it is true. And the solution to x^5 - x - 1 =0 is not in the radical closure of the rationals.
I have no comment on infinities.
I am impressed that after a 7 year hiatus the OP returns within 10 minutes of this thread being bumped by a regular. Talk about bird-dogging your handiwork. Wow.
All the more impressive when you notice the OP’s sole contribution to SDMB other than his posts in this thread was, a few years ago, to resurrect a different 6 year-old zombie: http://boards.straightdope.com/sdmb/showthread.php?p=13795647#post13795647
Understand I’m not complaining; I’m just remarking that it’s unusual and noteworthy for that.
We can always use more mathematicians around here. Why not stick around, and take part in other discussions, too?
Aside: I just noticed that two of the current participants in this discussion are named after fictional mathematicians. Who’s smarter, Heinlein’s supergenius or Asimov’s supergenius?
I was also impressed with Big Ed’s math chops from his comments back in 2010. That’s more substantive commentary than I’ve heard from him on any topic during my 14 year* membership. Obviously a bright dude. I wonder how Cecil found him?
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- I just noticed this month is my Doperversary. Yaay me.
That’s the beauty of it!
t is obviously ≤ c, once you see the definitions. You are correct that mathematicians essentially were surprised by the fact that ZFC decided the question of whether t = p, and that it was long clear that ZFC could not (if consistent) prove p < t.
For what it’s worth, it’s not that hard to define p and t. Given two subsets f and g of the naturals, let’s say f <= g if there are at most finitely many things in f which aren’t in g. We’ll say a collection S of subsets of the naturals is “inconsistent” if there is no infinite set which is <= each thing in S. We’ll say S is “finitely consistent” if no finite subset of S is inconsistent. And we’ll say S is a “tower” if for each f and g in S, either f <= g or g <= f.
p is the minimum size of a finitely consistent but overall inconsistent collection. t is the minimum size of a finitely consistent but overall inconsistent tower.
Since the latter condition is more stringent than the former, we have that p <= t automatically. Furthermore, since these are collections of subsets of naturals, we also have that p and t are both <= c automatically.
Now, I’ve been a bit glib, in that to know that these values are well-defined at all, we need to show that finitely consistent but overall inconsistent collections/towers actually exist at all, but this can be done without too much difficulty. We can also show that no countable collection will work, and thus find that p and t are both > aleph_0.
In fact, let’s go ahead and show why no countable collection will work: Suppose you had A_1, A_2, A_3, … . Let’s turn this into B_1, B_2, B_3, where each B_n is the intersection of A_1, A_2, through A_n. Clearly, each B_n is a superset of the following B_{n + 1}. What’s more, the As are finitely consistent iff the Bs are all infinite sets, and the As are overall inconsistent if and only if the Bs are overall inconsistent.
So let’s presume the Bs are all infinite sets, and show that they cannot be overall inconsistent (i.e., that there is some infinite set which is <= each B_n).
Note that our sequence B_1, B_2, B_3, …, may be eventually constant. If it’s eventually constant, then its eventual constant value is some infinite set <= each of the Bs, and thus the Bs are not overall inconsistent.
Otherwise, there are infinitely many occasions on which B_{n +1} has shed some of the elements in the prior B_n. On each such occasion, pick one element which was shed; then group all of these choices of shed elements into some infinite set Shed. Note that Shed <= each B_n (since by the time of B_n, at most n of the elements in Shed have already been shed, which is to say, there are most finitely many elements of Shed not in B_n). Thus, the Bs are not overall inconsistent.
This concludes the proof that there can be no countable collection which is finitely consistent but overall inconsistent.
Now let’s go ahead and show why there CAN be a finitely consistent but overall inconsistent collection at all, and indeed even such a tower.
Given an infinite set of naturals X, define f(X) as any infinite set you like which is obtained from X by shedding infinitely many elements.
Now let A_1 be the set of all naturals, let A_2 be f(A_1), let A_3 be f(A_2), and so on. Then define A_{omega} as some infinite set which is <= each A_n for finite n. Then define A_{omega + 1} as f(A_{omega}), define A_{omega + 2} as f(A_{omega + 1}), and so on. Then define A_{omega * 2} as some infinite set which is <= each An for n < omega * 2. Keep going in this way until, at some point, at some limit ordinal, you’re unable to find an infinite set <= each previous A_n. [This must happen eventually, because the A values keep getting distinctly smaller in the <= ordering, and eventually run out of distinct possibilities to take on]. At that point, you’ve constructed a finitely consistent but overall inconsistent tower. Hooray!
I assume that David Marcus had “subscribed” to the thread so that he got an email message when CIB posted his reply. I didn’t know that was a feature the SDMB had, but there it is down under “Additional Options.”
Agree that’s almost certainly how it happened. I use the other thread notification features myself.
It was just a bit surprising to me for somebody then a newbie to post a thread, care enough to set up notifications, and then 7 years later, having all but ignored SDMB the whole time, have the notification trigger, and then have him be at the ready to read the bump, compose an update, and post it. All in 10 minutes flat.
It just struck me as an odd / incongruous combination of utter disinterest and total interest. It’s not like SDMB hasn’t had a few other choice math threads in the last almost-decade he might have chosen to contribute to. Yet he did not.
Odd is not bad; it’s merely noteworthy.