God and Infinity

We who believe in God often call him infinite. Even those who do not believe often accept the infiniteness of God as part of the necessary definition of God. I’d like to examine this usage in more detail, and suggest that using “infinite” in referring to God needs to be done carefully.

The basic problem is how one defines “infinite”. In common usage it means “immeasurably great” or “unbounded”. There is nothing wrong with these ideas in and of themselves. The supreme being should be “immeasurably great” and “unbounded”. A bounded being certainly is less than whatever binds it and so can no longer be considered supreme. As a believer in the Bible, I also find that God is there often described as having immeasurably great power and unbounded mercy and love.

The problem is not with these concepts, but trying to extend beyond these to draw logical conclusions. As a mathematician I’ve learned that ideas about infinity do not follow our normal intuition.

Mathematical Infinity

For many centuries mathematicians were aware of infinity, but never really defined it. Some considered it so vague that it should not be part of proper mathematics. Even during my time at UCLA, long after infinity had been precisely defined, the professor who taught me the most about infinity and its definition personally disliked the idea of infinity and told me he thought we should avoid it. When mathematicians, who excel at precise definitions avoid something, we should take notice.

Sets and Size

Mathematicians have defined infinity by using the notion of a set. A set is a group of things. If a set contains the numbers 1, 2, 3, and 4 it would be shown like this: {1, 2, 3, 4}. A set can contain anything. For example, I could also have the set {cat, apple, computer, moon}.

To define infinity, we first address the question of how big a set is. More specifically, how do we tell if two sets are the same size? The obvious answer is to count. The set {1, 2, 3, 4} has four elements, as does the set {cat, apple, computer, moon}.

However, there is another way to compare sizes. We can tell if sets are the same size by matching the members. For example, we could make this match: 1 – cat, 2 – apple, 3 – computer, 4 – moon. The match shows the sets are the same size. Mathematicians call this a mapping, specifically a one-to-one mapping. Now say we have another set, {dog, fish, pen}. We could try the same sort of mapping: 1 – dog, 2 – fish, 3 – pen, 4 - ?. However there is a problem. The 4 has nothing left to match. This means the sets are different sizes. There is no one-to-one mapping between them.

Determining a set size by matching is useful when counting is hard. For example, if I have a few hundred pennies and want to count them, I might put them in stacks of ten. Rather than count each stack, I make one stack of ten, then just compare each new stack with the first stack to make it the right size. Doing all those comparisons is a lot faster and less prone to error than counting every penny. To get the final count I just count the stacks and multiply by ten.

This brings us to infinite sets and Georg Cantor. Infinite sets are not only hard to count, they are impossible to count. Cantor asked whether all infinite sets are the same size. This seems like a ridiculous question. Of course they’re all the same size – they’re infinite! But what about when you have an infinite set that contains another infinite set. Is the contained set smaller or the same size?

Take the set of all the counting number, the positive integers: {1, 2, 3, …}. (The “…” means they just keep going.) Also look at the set of even numbers: {2, 4, 6, …}. Which is bigger? One obvious answer is the counting number are bigger because even numbers are contained in (are a subset of) the counting numbers. However, another obvious answer is that they are the same size, because both are infinite. Which is correct?

Consider a mapping between the sets where the counting numbers are multiplied by two. So counting number “1” maps to even number “2”, counting number “2” maps to even number “4”, and so on. Every counting number can be mapped to an even number. In other words, given any counting number, just multiply by two to get its corresponding even number. If you have 324, its corresponding even number is 648. Likewise, given any even number, you can find its associated counting number by dividing by two. If you have the even number 486, its corresponding counting number is 243. The point is every counting number has a matching even number and every even number has a matching counting number. Thus the sets have a one-to-one mapping and they are the same size.

Infinity Defined

At this point many might be thinking, “It’s a trick. The even numbers must be smaller. They are a subset of the counting numbers.” All I can say to that is that it’s not a trick, but follows directly from the idea of a one-to-one mapping. In fact, it leads to the following definition of an infinite set.

An infinite set is a set which has subsets that are not equal to it but are the same size.

That definition probably does not make anyone happy at first glance. It seems again like a trick. Shouldn’t the definition be more like “An infinite set is one that goes on forever”? Trouble is, words like “forever” contain the idea of infinity, so such a definition is circular. The mathematical definition recognizes an infinite set as the only kind of set which can have elements removed and still remain the same size. As such it supports the notion that infinity remains infinite even tweaked a little. Removing some stuff does not reduce its size. Conversely, adding some stuff does not increase its size.

Different Sized Infinities

Despite what has been shown so far, despite the definition of an infinite set that allows some stuff to be added or subtracted from a set and that set remains the same size, there are different sizes of infinity. I feel I’ve already provided enough confusion. In exchange for not confusing you further, I ask that you accept the possibility that infinite sets can be different sizes. In fact, Cantor showed that not only are there there infinite sets of different sizes, he showed that there are an infinite number of different sizes of infinite sets! For those that wish a better discussion I recommend reading A Brief Introduction to Infinity.

Are These Infinities Real?

Do mathematical infinities correspond to God or any other reality? Are there physical (or spiritual) infinities of different sizes? I don’t know. I do know that the mathematical definition of infinity is the only rigorous one I know. It also seems to be about as simple as possible. Despite its relative simplicity, it leads to bizarre consequences like subsets that are the same size as their contained set and infinities of different sizes. Can we expect real-life infinities, if they exist, to be any less bizarre?

Infinity in Theology

Now that we’ve looked at a rigorous definition of infinity, let’s examine some theological concepts in light of the mathematical definition of infinity.

First, consider a common argument against the idea of God existing. “If God exists and is all-powerful, can he make a rock that he can’t lift?” If he can’t make the rock, the argument says, then he is limited and not all powerful. If, on the other hand, God makes such a rock, then the existence of the rock is a limitation of God’s power. Again we have a God who is limited. Quite some time ago I came to the conclusion that the problem with this argument is defining “a rock God cannot lift.” It is a nonsensical statement, like saying can God make a true lie? The sentence is correct grammar but a nonsensical combination words, as nonsensical as Jabberwocky.

However, now that I look at this argument from an infinity point-of-view, how do I know it contains a nonsensical definition? I can only know that if I understand infinity. Perhaps the real problem with this argument is assuming too much knowledge about “all-powerful”. Perhaps my objection to it and the argument itself both assume too much about infinity. Given that a mathematically infinite set can lose something and still be infinite, maybe God can lose part of his power and still be infinitely powerful. So maybe a rock God cannot lift is not a nonsensical concept. But by the same token, if God does create such a rock, maybe he is still all-powerful.

A related concept has to do with God being all-powerful and people having free will. Maybe an infinitely powerful God can give people some of his power and still remain infinitely powerful. Some say that God cannot allow free will in people because doing so reduces his sovereignty. Maybe these people can now relax because in the world of infinite it is possible to give power away and still retain the same amount of power.

We’ve talked about God’s infinite power, what about his boundless love and mercy? People often say that God cannot be loving because there is so much evil in the world, or so much evil in their own lives. But given how counter-intuitive even relatively simple mathematically infinity is, how can we expect to understand God’s infinite love? Are questions of God’s love valid? Certainly. Will giving a mathematical proof of his love comfort someone who is hurting? Certainly not. However, in our times of calm reflection and rational thought, perhaps we can at least realize how limited our understanding is of an infinite God.

Finally, let’s consider the Christian doctrine of the trinity, or triune nature of God. How can God be one and three at the same time? Again, I don’t know. But perhaps the mathematically concept of infinity can be illustrative. Earlier we saw how the even numbers are the same size as the counting numbers. Likewise the odd numbers are also the same size. We are able to split the counting numbers into two sets, odds and evens. Those two sets are not half as big, but are each as big as the original. A three-way split could just as easily be accomplished (multiples of three, numbers that divide by three with remainder of one, numbers that divide by three with remainder of two). Could not an infinite God also be divided into three parts, each of which is as big and the original? Does this prove or even fully explain the trinity? Of course not. But at least it is food for thought and opens new avenues of thinking.

Infinite Possibilities

As I said above I make no claims that mathematical infinity matches any reality. In fact, making such a claim can lead to other problems. For example, as we mentioned above mathematical infinities have infinitely many different sizes. If we claim God fits the mathematical definition of infinite, we must immediately ask what size is God’s infinity. Or is God somehow more infinite than all mathematical infinities (which, I think, was roughly Cantor’s view)? Rather than making such a strong claim, what I hope I’ve done is convince you to be careful in your handling of concepts that include infinity and to open you up to new thought possibilities in your search for truth and contemplation of the infinite.

Creation Science Destroys the Beauty of Genesis

I was in Sunday School last week when our teacher, the brilliant author R. P. Nettelhorst (you really should check out his writing, even though this is a shameless plug by a friend) pointed out that the Genesis creation account parallels the Mesopotamian creation account, Enuma Elish. This lead me to think that so-called "Creation Science" really destroys the literary genius of the Genesis account. Let me explain.

I've already written about the true meaning of the Genesis creation account. It wasn't written to combat modern 21st century science. It was written to combat polytheism. There are not innumerable gods as in Egypt. There is one God who did all. So already I think "Creation Science" misses the whole point of the Genesis account. (And so I keep using the quotes, because it's not the correct view of creation and it certainly isn't science.)

However, I was missing something that's very cool. I used to cringe when people compared Genesis to Enuma Elish. After all, I believe the Bible is the word of God. How humiliating to see it as just a rip-off of a stupid polytheistic legend. What my friend, R. P., pointed out is that it's not a rip-off, it's a specific counter to the Enuma Elish.

So I looked up Enuma Elish and found this page. The Enuma Elish has six stages of creation. Genesis gives six days of creation. In each stage of the Enuma Elish a new god is created. On each day of Genesis, God creates something new. Here is how they align.

Enuma Stage/Genesis Day	Enuma god (function)	Genesis	Notes
1	Tiamat (primeval chaos & bearer of sky and earth) and Apsu (water)	In the beginning when God created the heavens and the earth, the earth was a formless void and darkness covered the face of the deep (or watery chaos), while a wind from God swept over the face of the waters. (1:1-2)	In the Biblical Hebrew "chaos" is "Tehowim", similar to Tiamat.
2	Lahamu (muddy silt)	And God said, "Let there be a dome in the midst of the waters, and let it separate the waters from the waters." (1:6-7)	God slicing waters with the dome parallels Marduk slicing open Tiamat to make land and sky
3	Kishar (the Earth)	And God said, "Let the waters under the sky be gathered together into one place, and let the dry land appear." And it was so. God called the dry land Earth, and the waters that were gathered together he called Seas. (1:9-10)
4	Anu (sky)	God made the two great lights--the greater light to rule the day and the lesser light to rule the night-- and the stars. (1:16)
5	Ea (speaks and makes all things of the Earth)	So God created the great sea monsters and every living creature that moves, of every kind, with which the waters swarm, and every winged bird of every kind. And God saw that it was good. (1:21)	Ea speaks the things of the Earth into existence. God speaks everything into existence.
6	Marduk (king of gods and creator of man)	Then God said, "Let us make humankind in our image, (1:26) And on the seventh day God finished the work that he had done, and he rested on the seventh day from all the work that he had done. (2:2)	Marduk makes people his slaves so he can rest. God finishes creation with people in his image, rests on seventh day and gives it as a holy day of rest for people.

Again we see the same message. There are not many gods. There is one God. God is telling his people that the story they have heard is all wrong. Genesis isn't a rip-off of Enuma Elish, it's a brilliant literary reversal of the common mythology of the time. "Creation Science", in its haste to refute what it sees as modern scientific evil, perverts the beauty and brilliance of the Genesis account by twisting it into a modern scientific treatise. Very sad indeed.

One more thing. Notice that Enuma Elish teaches that people are slaves of Marduk. Genesis teaches that people are the image of God and get to rest just like he does. What a complete difference. God walks and talks with people in Genesis. He treats us are equals in many ways. He loves us. Genesis is the complete opposite of Enuma Elish, just as the Bible is the complete opposite of so many beliefs that make the world a place of inhuman gods that want to harm us.

There is only one God. His love for us is inconceivable. Why pervert Genesis with any other meaning than that?

Thanksgiving

Most years I have to think about what I'm thankful for. Not this year. Easy: I just got a new job after a long layoff. No-brainer there. We were looking at some tough times starting in January so this was close to our last chance. Very thankful indeed.

However, I'm not just thankful for finding a job. My year of unemployment -- yes a whole year -- has actually been very good for me. First, my former company gave me a good severance package. We were a lot better off than many unemployed, we kept our house, never wanted for food or anything else. Took it easy on the spending, but basically lived life as normal. God has been very good to us.

Because I was no longer commuting I became much more plugged into our community. Some of this was just the simple things like driving some streets I didn't usually drive, using the library regularly, shopping, or getting out during the week. But I also volunteered. I already had volunteered with a local high school robot club. With more schedule flexibility I was able to be much more involved. Very enjoyable and rewarding.

I also joined a job club, AVPPNG. They helped me tremendously with moral support and navigating unemployment and job searching. I eventually was elected president and got to practice seldom used leadership and people skills. I made several friends as well as networking contacts.

Our church held a "Biggest Loser" contest. I was motivated to lose 45lbs. In winning, I am now officially the biggest loser at church. The contest motivated me to up my running schedule and start a running group. Over the course of the year I became much more fit and reduced my 5k PR by 4 1/2 minutes (29:55 to 25:17 -- not very fast, but still a great improvement). At one point I could run fast enough and do enough push-ups, sit-ups, mountain climbers and pull-ups to pass the LA County Sheriff's PT 500 physical exam (arthritis problems have since forced me to back down a little). Through our running group I got involved with a soccer team and now play once or twice a week. Our group met a lot at Apollo Park. While running there on my own one day (group dissolved) I met a fellow runner who is the very well-know dog trainer, Kyra Sundance.

Lately I realized that what I love doing is developing software. I took initiative to start learning on my own. I realized that such learning and initiative would help me in my job search. It also was fun and re-energized me. It has made me more aware of how much I do not know and need to keep learning. I am much more excited now about my career and about learning many new things. This learning and enthusiasm, the lessons I also learned from job searching and interviewing, led directly to me landing my new job.

The real negative about being laid off is the stress of not knowing, of seeing the money dwindling, of having to be ready to network and search and respond to employer requests at a moment's notice. In some ways that stress is self-induced. I too easily forget Matthew 6:33-34: "But seek first his kingdom and his righteousness, and all these things will be added to you. Do not worry about tomorrow, for tomorrow will take care of itself. Each day has enough trouble of its own."

Self induced or not, the stress is very real. Waking up at 4am stressed out about not having work is no fun. Also, I don't like being forced into anything, especially not having to reach out (kiss-up?) to people. But job search forces me to do just that. And a funny thing happened. I've become much more comfortable with it. Happy I found a job and don't have to be so uncertain? Ecstatic!! Still, now I know I can survive this.

Bottom line: the process has been very good for me. There are still many problems in life, but I feel better able to handle them. My new job is temporary -- a three month contract. I could be right back out there looking very soon. I'm not thrilled about this, but am OK with it. When I was first laid off a trainer said on average we'd do this again in 3-5 years. That scared the crap out of me. Now, it's OK. I know more about what to do. More importantly, I have a much better vision of what I want to do and how to get there. I'm much more confident in myself and in God working in my life. It's been a very good year.

Please Stop Hating

I am Christian and it hurts when people who call themselves Christian do hateful things. I am referring to the threatened Quran burning by Terry Jones. Christianity is about love, not hate. Enough evil has been done in the name of Christ, a man who said love is the greatest good. Please, Mr. Jones and my fellow Christians, stop violating the message of Jesus.

Jesus called us to love our enemies. How is provocation love? OK, Mr. Jones, I know loving your enemies is hard. How about showing a little consideration for your fellow believers? Burning copies of the Quran in Florida is relatively safe. How about the violent reaction it causes in other parts of the world? Do you really have so little regard for others that you're OK with the persecution, injury and death of Christians around the world that your actions will incite? Do you ever think about or pray for believers in places like Pakistan, Saudia Arabia and Syria?

Mr. Jones, we Christians send people out to tell of God's love in sending his son. Why do you instead want to advertise hate and nullify everything we stand for? If you won't listen to me, perhaps you can hear the words of Martin Luther King: "Hate cannot drive out hate, only love can do that." Please, Mr. Jones, do not follow through on your plan to burn copies of the Quran.

Video Resume

Here is the video resume I posted on Mycareeri.com. Employers can set up free accounts to search for my resume and many others.

______________________________

A Great Day.

Yesterday was a great day.

It started in the morning. I got up and got out running. The weather was cool and damp but no rain. I felt good and ran faster than I had been. I got back to my home just as the rain started. Cool! But even better, while running, God changed my attitude about searching for a job. I worry a lot about getting a new job. I worry about the future, get depressed and then mope around and am too discouraged to do much. God affirmed that he will take care of me and my family. Even if we have no money and are on the street, he'll take care of us. I don't expect that -- he's always treated us way better than that. But whatever, the guarantee is that he'll take care of us. I need to let him handle the results and I need to focus on doing what I should every day.

Two quotes came to mind. The first is "Pray like it's up to God and work like it's up to me." Easy to say, hard to remember and do. The second is Matthew 6:25-34 where Jesus shows how God takes care of the birds and flowers and says we are a lot more important than them. In particular, Matthew 6:33-34 "Seek first his kingdom and his righteousness and all these things will be given to you as well. Do not worry about tomorrow, for tomorrow will worry about itself. Each day has enough trouble of its own."

This all put me in a great mood. And I stayed in a great mood despite a couple of things. First, during my run I felt a twinge on the side of my right leg, just below the knee. When I got home it was mildly sore and I had to be careful to not bend too much. Discouraging, especially since I always talk about avoiding injury. But also ironic. Just Saturday in our beginning running group, Ruth mention she felt something on the side of her calf. I'd never heard of it before. Now I empathize.

Second, a little after I started up my computer I saw the Blue Screen of Death. Fortunately I got the computer restarted and was able to back up all my docs. And it has been OK since.

So, I got a little frustrated, but my mood alteration held and I left for my Wednesday job club (AVPPNG, a network club for job seekers) in high spirits. Our meeting was good. Part of the meeting was a presentation by an EDD person, Margie, about the WIA program, a program to help people get training they need. Many in our group had already applied and were asking what was going on with their applications. Margie explained many times how there were so many applications and so few workers, so many steps involved and why it took so long. Then she dropped a great quote, "Next week with the government is two or three months." In other words, what we'd normally expect in a week usually takes the government two or three months.

After the meeting four of us went out to lunch. We had a fun and encouraging time. Plus we had lots of stimulating conversation, including some good spiritually oriented discussions. One of the encouraging things was that the other three had good chances of landing jobs soon. I commented that I was the odd man out but they re-assured me that I was also the newbie (they'd all been at it a lot longer than me) and good things would come my way too.

After all that, I went to the Robolopes robot club at Antelope Valley High. This year's competition is building robots that play a form of soccer. I help with the software. It was a good time as usual, although we are currently at a roadblock.

When I got home, I got dinner and checked my email. About two weeks ago I had sent an email asking how my application for a flight test engineer at the Edwards Flight Test Center was doing. This is an application that I put in about four months ago (Margie's quote about government weeks became relevant in a new context). My email question and big concern was whether it had passed the human resources screening. It's not exactly my field and I was concerned (OK, worried!) that the HR people would just reject it outright. Yesterday, I got an answer. It had a lot of government speak that I didn't understand well, but it seemed to be saying I passed and my application was sent to the hiring managers.

In the evening we went to our usual Wednesday prayer meeting. Instead of being at church it was in the pastor's home. This made it warmer and cozier than normal on a cold rainy night. Plus lots of people showed up and we had a very good time -- lots of encouragement and uplifting interaction.

Of particular interest to me was that Sherman was there. Sherm is my contact at Edwards and the reason I applied for the job at the FTC. He helped me interpret my email (I'd printed it out hoping he'd be there). He confirmed that I'd "made the cert" and that my resume was now referred out to the hiring managers. YEAH! (Plus I'm not such an odd man out anymore!)

One slightly discouraging thing is that I'm one of 224 who made the cert. However, Sherm reassured me that usually half are no longer available. Plus that many applicants means there are a lot of jobs available -- even a hundred openings is not that big for a place as huge as Edwards. I sent him the email and he is going to ask his bosses what's happening. Plus Sherm had already introduced me to his big boss many months ago. He is one of the hiring managers and spoke to me about how they needed lots of experienced engineers and how they'd get me talking to lots of different people to see where I fit in. Now that I've made the cert I feel good about presenting all my skills and am excited about being able to talk with people out there. It's still a long way from having a job, but it's a huge, exciting step forward.

Well, that's my great day. Hallelujah and Praise God!

Computer Problems Solved

This is too good to pass up. And it's true! Now you all know how to solve your computer problems on your own.