Extermination

There's a ringing in my ears and I can't see straight. I can't remember much of anything either. I can't remember when I got to the diner, or when I got so tired. I can't remember closing my eyes or waking up at all. It takes all my strength just to pick up my head and look around. Though cruel reality wasn't something I could cope with, not this time. Like waking from a dream and into a nightmare; I was horrified by the massacre that lay before me. The bodies of familiar faces, people I know from town. People I live and work with. So still, so silent. Even the old family diner looks the same amid the chaos. No, it doesn't. Things are different now. The scene of warm childhood memories has turned cold and gray like the rest of my world.

I sit alone in my booth shuddering, in too much pain to shout and call for help. But none of that mattered; they came for us. In their white, faceless hazmat suits to round up the dead. They start to go through my friends and family. Carefully and efficiently, they gathered the children, then the women. Yet it wasn't until they pulled my lifeless body from me did I realize how much I had lost. The cold-hearted men in white collected who they could and left. Never stopped, never looked back. I just sat there in shock and disbelief. Everything I knew was fading.

...

by Emily Schroder

All Rights Reserved

Unusual things with infinite sets

I was speaking with a coworker the other day about 'infinity' and it brought to my mind some interesting things I've learned in my studies of mathematics. Infinite sets don’t behave in a manner that we would expect.

The concept of infinity (∞) is often misinterpreted as representing a very big number like 10exp10exp52 (10 raised to the power of 10 raised to the power of 52). Infinity isn’t a number at all, but a concept to describe an ‘unbounded’ set of things. Such unbounded sets don’t behave at all like finite sets, regardless of how big.

For example if I take the set of non-negative integers from 0 to 100 {0, 1, 2, 3, ..., 100} and then get rid of all the odd integers {1, 3, 5, 7, ..., 99} it’s pretty clear that the set of even integers I’m left with {0, 2, 4, 6, ..., 100} is about half the size of the set I started with. But what if I start with the infinite set of integers, called the set of natural numbers, which is unbounded, and get rid of all the odd integers in this set? Even though both sets are infinite, is the remaining ‘infinite’ set around half the size of the ‘infinite’ set I started with?

Set of all natural numbers: {0, 1, 2, 3, 4, 5, ...}
Set of all even naturals: {0, 2, 4, 6, 8, 10, ...}

Our common sense seems to suggest that the second set is smaller than the first, because our common sense is trained to deal with finite things, not unbounded sets. The fact is, there is a mathematical way to check the ‘size’ of a set. If I can make a one-to-one mapping from an element of the first set to one and only one element of the second set, then clearly the two sets have the same number of elements. If, on the other hand, I end up with something ‘left over’ in either set, the two sets cannot be the same ‘size’. I can make just such a mapping from the set of all integers to the set of all even integers as follows: take an element of the set of integers, multiply it by 2, to obtain one, and only one, member of the set of even integers:

2  x2  4
3  x2  6
4  x2  8
5  x2 10
Etc.

I can take any arbitrary element of the first set, say 2345, and obtain its corresponding match in the second set, 4690, by multiplying by two. Similarly, I can take any element in the second set, say 12386, and easily obtain its unique match in the first set by dividing by two, 6193. Such a 1 to 1 mapping of elements in the first set to all members in the second leaves nothing ‘left over’ in either set. So for each element of the set “all natural numbers” there is one and only one, matching element of the set “all even integers”. So the sets are the same ‘size’ even though we obtained the second set by getting rid of every other element of the first. Strange yes?

It can also be shown that the set of all rational numbers, that is, all numbers that can be expressed as a fraction (a ratio of two integers) can also be mapped one to one to the set of natural numbers. So if we add in all the fractions to the set of integers, the two sets are still of the same ‘size’. Similarly if we add in all the negative rational numbers, the set can be mapped one to one to the set of all positive rationals. The mappings are more complex and I won’t go into them here, but they do exist and can be found in any book of set number theory. In fact, all of these sets can be mapped to the set of natural numbers and are said to have the same cardinality instead of ‘size’. Infinite sets having cardinality (size) the same as the natural numbers are symbolized by Aleph₀ (the Hebrew character).

So are all infinite sets the same ‘size’ in this manner? After our experience so far we might think so, but the answer is surprisingly, ‘no’. The sets we have been looking at, although they are infinite are still ‘denumerable’. In other words, one can find rules to list out the elements of these sets in an ordered array without leaving anything out, and then find rules for mapping the elements of the set to the elements of the natural numbers in a one-to-one way. In other words, the sets can be counted. But the set of Real numbers (rational numbers plus non-repeating decimals like √2) is NOT denumerable in this way. To see this, lets assume you make a list of all the Real numbers:

...901234.8767894847...
...901234.2937640974...
...901234.5564737222...
...
...000002.8878903489...
Etc.

Of course, any element of your set can have any number of digits to the left or right of the decimal, but we can always use leading zeros so the decimal points line up in this way. This is what we did for the third number in the list above. I just limited the number of digits shown before the decimal to seven and after the decimal to ten to make this illustration easier to put on screen (I left my infinite screen at work).

So let’s assume that you tell me that every single real number is on your list of numbers, not a single one is left out. Assuming you had unlimited room for such a list on your paper and an unlimited amount of time to make it, is your list complete? No. Because I can come up with a rule to make a real number not on your list, no matter how big it is! How? Well I simply choose the nth digit of my number to be different than the nth digit of the nth number on your list. Let’s just look at digits after the decimal. First, I make the first digit different from the ‘8’ in your first number so my number can’t match the first number in your list. I’ll pick ‘2’. Then I make the second digit after the decimal ‘3’ so it doesn’t match the second number in your list which has a ‘9’ in that position. I choose the third digit in my number different from the third digit in your third number, and so on ad infinitum.

Your List          Mine    Explanation
.8767894847... ... .2...   It won't match your 1st number
.2937640974... ... .23...  It won't match your 2nd number
.5564737222... ... .231... It won't match your 3rd number
Etc.

In this way I can make a number that doesn’t match any of the numbers on your list, no matter how long your list is. So no ‘list’ of real numbers can contain all the real numbers. Mathematicians say the set of real numbers is not denumerable. It is called the ‘continuum’ set because it represents the set of points on a continuous line. The continuum is an infinite set of higher cardinality than the set of natural numbers. There are many sets that map one-to-one to the continuum set and they are all represented as having cardinality represented by C.

There is at least one more set of cardinality even higher than the continuum, written 2^C, but it involves higher concepts in mathematical theory than I want to cover here. Theoretically there is no limit to the number of different cardinalities and as far as I know and have been able to discover from my research, there is no known ‘maximum’ cardinality. Sets of larger cardinality than 2^C seem to exist mostly in symbolic mathematics as “sets of subsets”, without necessarily corresponding to things more “concrete” like the set of natural or real numbers. Whether such infinities really exist as more than concepts, in other words, whether they correspond to real things ‘out there’ in the universe, is an open question.

Infinite numbers are interesting things. I can counter-intuitively take a subset of an infinite set that doesn’t include certain elements of the set, yet still end up with a set the same size I started with. Even so, not all ‘infinities’ are created equal and there are different sizes of ‘infinity’.

Lots to Update

In February I took a job with a company working at the Hanford reservation in the Richland-Pasco-Kennewick, WA area. I drove up in February and stayed in a corporate apartment until June while Crystal and the kids stayed in Austin, TX until the end of the school year.

I work a compressed work week, 9-hour days, with every other Friday off. This allowed me to see my other kids that live in Oregon every other weekend, which was nice. It's a five hour drive from Richland, WA to their house but it didn't seem so bad because a large portion of it is through the beautiful Columbia River Gorge.

Crystal flew up with the kids on 6/14. Originally we planned on having her drive the van up, but she had a car accident only five hours out from Austin in Pecos, TX. Fortunately nobody was hurt beyond a few bruises and no other vehicles were involved. Shelby wasn't in the car because we had planned on flying her up separately anyway so she was with her grandparents in San Antonio during the accident. Even so, the van was out of commission for the rest of the month, so Crystal decided to fly with the kids and have the van shipped once repairs were completed. I posted a pic of the van on my Facebook.

The whole family stayed in a small corporate apartment from Jun-Jul: 2 bedroom and a loft. My kids stayed with us for several weeks but it was broken up because of their summer camps which occur at different times. So we had from 5 to 8 people in the that tiny apartment. We diligently househunted most weekday evenings and on Saturdays. We finally found a house and just closed on it last Friday. It's on the other side of Kennewick so I'll have a 35 minute commute but that is what I was used to in Austin. The house is adequate to our needs and it will be nice when we move in this Friday to be out of this cramped apartment.

The scenery here is breathtaking with canyons carved out by large glacier flows and massive floods over the past few millenia as well as mountains inc. snow capped volcanos, lakes and rivers, forest and desert. The kids have never been outside of Texas, so the scenery is quite an experience for them. Most people think of lots of rain when they think of Oregon or Washington, but we are east of the Cascade Mountains, in desert country. There is little to no rain only 7 inches a year. The locals are complaining this is an unusually wet year - I think we had maybe fifteen days of rain total since I've been here since Feb 1. It was cool in the evenings through June, in the low 70s, and warm in the day in the 80s or 90s, and no humidity so as soon as you step into the shade the temperature drops 7 degrees. Now that August is here the temperatures have climbed to high 90s or low 100s in midday, but still much cooler at night.

So far this summer we've done the following:

(1) "Jump-Off Joe's" - an overlook at 2500 ft with a great view of the tricities and the Columbia basin. There are large wind machines up there, calling them "wind mills" doesn't seem right because they are huge metallic monsters. There was one that the kids could go right up to and look up at the huge turning blade.

(2) "Wallula Gap" along the Columbia River east of here - a canyon carved out by a massive flood one or two ice ages ago.

(3) Greenwood Renaissance Faire in Richland - very small, like a craft fair, but considering Richland is a small town it wasn't bad. We miss the ones at Scarborough near Dallas, or Renfest between Plantersville and Magnolia in east Texas (which is the largest in the country), but we are still looking for others in Washington and there is a Midsummer's Renaissance Faire near Tacoma in August we are planning on checking out.

(4) Gingko Petrified Forest State Park in Vantage, WA on the Columbia River north of here where we watched a small rattlesnake cross the footpath and curl up under a petrified log. I lived in Texas for 14 years and never once saw an armadillo or a rattlesnake in spite of the fact that I hiked the Hill Country all the time - I did see a coyote once when I was driving early in the AM, and also in Texas I saw a snapping turtle, racoons, and lots of opossums and deer, but for some reason the rattlesnakes and armadillos were elusive. In the past month here I've seen several coyotes and now a rattlesnake (there are no armadillos this far north). People tell me of Elk and several coworkers have seen them around here, but so far I've not seen any of those.

(5) Battle Ground WA - 4th of July at my brother's house. The whole family turned out, as well as many friends, probably 80 or so people. My brother put on a very large and elaborate computer controlled and choreagraphed fireworks show and we slept in a tent on his property.

(6) Seattle Aquarium and Pike Street Market. We had to go to Seattle Children's hospital for Shelby and decided to spend the day at the aquarium and Pike Street market.

Shelby continues to improve daily. She walks around the apartment, and can manage short walks in public, but generally anything longer than 15 minutes she needs to use her wheel chair.

Shelby Update 01/18/2010

One of our daughters, Shelby, had scoliosis surgery in early November. The intent of the surgery was to install metal rods and screws to straighten her spine and prevent further deformation. Because our daughter has congenital scoliosis, the risks were high and spinal cord monitoring was performed during the surgery. During and immediately after the surgery, all of us, including her surgeon (one of the top surgeons in Texas), believed everything went fantastic. The initial report was that her scoliosis was 90% corrected, which was far better than any of us could have hoped for; then things went terribly wrong. When she awoke in the recovery room she could not move her legs. She was rushed back into surgery, the rods and screws were removed, and she was in intensive care for two weeks in treatment for a severe spinal cord injury. She was completely paralyzed from the waist down for several weeks.

In December, about two weeks before Christmas, Shelby was moved to Our Children’s House at Baylor, a pediatric rehabilitation center, in Dallas, TX. She has since recovered motor control in both legs, but cannot yet walk independently, and is confined to a wheelchair most of the time.

Shelby is scheduled to be released from the rehabilitation center this coming weekend and will continue outpatient rehabilitation locally. She is excited to return home and to return back to school but will require accommodations due to her handicap. In spite of everything, the prognosis is good, and it is generally believed that she will be walking again sometime within the next year. She still faces a long and painful recovery and she still requires further surgery to re-install the rods to correct her scoliosis and stabilize her spine.

Shelby’s grandparents had been providing a great deal of help to our family during this difficult time and then two weeks ago; her grandmother had a heart attack. She recently underwent open heart surgery and is now herself in rehabilitation. We ask that you please pray not only for Shelby but also for the extended family during this extremely difficult time. We have had more than our share of tragedy in the past three months, but it is because of the support of family and friends that we have gotten through this. A website has been set up by Shelby’s teachers and schoolmates, located on Facebook, here. We welcome your well wishes and continued support and thank everybody for all of your kind words, your cards and gifts, and most importantly, your prayers. We ask that you please add Shelby's grandmother to your prayer lists.

September 19, 2009

Wow, I've been terrible about updating the blog. A lot has happened since my last post. The kids returned to school in Texas on August 27th and in Oregon on September 8th. Laurel is very busy with the dance team. She is excited but has been practicing 2.5 hours each night, and with her homework load she's a little concerned about balancing school work and extracurricular activities. Emily and Shelby are both in choir this year. All three girls had their first pep-rally/half-time events this evening. We celebrated Emily's birthday at the beginning of the month and Shelby's is coming up this weekend. Laurel's will then be in November.

The kid's have already given us Xmas lists, but that just seems too far away to think about. Lionel is still looking hard to further his career. Crystal hasn't had any new substitute opportunities yet this year, but is expecting the flood gates to open soon.

We've been getting into the Heroes series. We had never seen it before when it was on TV. We don't have cable, but we have Netflix. We found it pretty exciting and have had a marathon on weekends to cover the first three seasons. It's proven to be too complicated for the kids to really enjoy, but we like it. Lionel is getting back into classical music. Coincidentally, some of the music he was listening to recently Holst's the Planets was played by the Texas Tech band tonight at the Georgetown ISD event.

We're gearing up for Shelby's back surgery for her scoliosis. She has a myelogram scheduled for Monday. Doctors say she will be out of school for at least a month or two post-op. The surgery is planned for November so keep her in your prayers/incantations.

We've finally had a good week of rain, bringing much needed relief from what has been the worst drought in Texas in 50 years.

Almost 100%

Well, I'm finally feeling much better today - first time I've been out of bed in 3 days! I don't know what Em and I caught, but it was pretty horrific. I have lingering congestion, but beyond that feel pretty good now. The kids this week are with their grandparents and we want to take advantage of this time to get some things done around the house we've been wanting to do for some time. First day back at it and I had major dental work, fixed a flat tire, had the Civic inspected / oil changed, framed and hung the family portraits, mailed some items my kid's left behind. I already miss them. Laurel continues to text daily and that helps a lot, I feel closer somehow always in touch like that. Crystal and I went out to Ichiban for Sushi for dinner tonight. It was nice to be out on a real date again. I updated our family website. Check it out. I'm hoping to post some summer pics of the kids before the week is out. I'll let everyone know when they're up via twitter.

Short Update

Lionel's three children were down from Oregon for only 2 1/2 weeks this year. This is because our oldest daughter went on a field trip to Washington, DC, Philadelphia, and NYC in the beginning of June and dance camp in the middle of July, forcing us to a shorter time period.

We had a great time with all of the kids here just the same. We went to Hamilton Pool. Spent a lot of time going to the rec center. Enjoyed the 4th of July celebration at San Gabriel park. All in all it was a good time. At grandpa's urging we got Laurel a cell phone and she has been texting regularly since the kids got back to Oregon.

Right after the kid's left Emily and then Lionel got sick - soar throat, slight fever, congestion. Lionel's been out of it for a few days. It works out because the three kids are with their grandparents this week, giving him time to recover.