**Summary**

- Humans are used to things being limited.
- That makes infinity hard to think about.
- God has unlimited love, and can give an unlimited amount of love and attention to each of us.
- I illustrate this with a numerical analogy.

Posted 30 September 2012. Upated 10 December 2015.

Earlier today, I gave my sixth "Thought for the week" on BBC Radio Manchester. I shared some simple mathematics which has helped me to think about God's ability to give each of us unlimited amounts of love and attention.

We are used to things being limited. If I have a bigger garden, there is less land in the country for everyone else.

Since almost everything we have is limited, it is easy to think that everything is limited. That would mean that if God gives you more love, He has less love left over for me.

However Judaism, Christianity and Islam all teach that God has an unlimited amount of love. He can shower unlimited amounts of love on each of us, without ever running out of love. His love is infinite, not limited.

Thinking about infinity is hard, so I want to share something which helps me. We all know that if you start counting one, two, three, the whole numbers go on forever; they are infinite.

Try to share the whole numbers amongst your friends and aim to give each friend an infinite amount of numbers.

Itâ€™s not easy. You might give your first friend the odd numbers, and your next friend the even numbers. Each of them is happy, but you have now run out of numbers to give away.

However, it can be done fairly easily.

Give your first friend the number two, and all the numbers you can make by multiplying twos together, in other words four, eight, sixteen, thirty two, sixty four and so on. That is an infinite amount of numbers.

Give your second friend the number three, and all the numbers you make by multiplying threes together, nine, twenty seven, eighty one and so on.

Your next friend receives five, twenty five, one hundred and twenty five and so on, just multiplying the number five by itself.

The next person gets all the numbers you can make by multiplying sevens together and the next person numbers made by multiplying elevens together.

In this way, you can give each person you meet an infinite amount of numbers, without ever running out. You could give an infinite amount of numbers to every person on the planet; indeed to every person who will ever be borne.

This thought process has helped me to understand how God can lavish unlimited amounts of love and attention on me, without limiting what He gives to others.

The "Thought for the week" slot operates to a strict time limit and the talk needs to be about 360 words. (The above is 369 words.) Accordingly I avoided using the words "prime number" since that either assumes mathematical knowledge or requires many words to explain what a prime number is.

The above method of giving away an infinite amount of numbers to each person, (without giving the same number to more than one person) only works easily if the base number in each case is prime. Accordingly the i^{th} person is given the set of numbers p_{i}, p_{i}^{2}, p_{i}^{3} p_{i}^{4}... p_{i}^{∞}or more concisely the i^{th} person is given the set of numbers p_{i}^{j} where j=1 to ∞. Here ∞ is the symbol for infinity.

If you give away composite numbers, (numbers which are not prime numbers), you have to work much harder to ensure that none of the numbers you are trying to give away has not already been given to some else earlier.

This of course assumes that there are an infinite number of prime numbers p_{i} where i=1 to ∞. The Greeks proved that a couple of thousand years ago in the following way.

Assume that there is only a finite quantity of prime numbers. If so they can be listed as p

_{i}where i=1 to N where N is the last prime number.Consider the number p

_{1 }x p_{2}x p_{3}... x p_{N}+ 1. That number cannot be divided by any of the prime numbers in our list p_{i}where i=1 to N since there is a 1 left over when you divide by any p_{i}.Accordingly there are only two possibilities. Either:

- The number p
_{1 }x p_{2}x p_{3}... x p_{N}+ 1 is prime.- The number p
_{1 }x p_{2}x p_{3}... x p_{N}+ 1 is not prime, which means that it has at least one prime divisor. If so, the prime divisor cannot be in our list p_{i}where i=1 to N since none of the primes in that list can divide p_{1 }x p_{2}x p_{3}... x p_{N}+ 1 since there is a 1 left over when you divide p_{1 }x p_{2}x p_{3}... x p_{N}+ 1 by any p_{i}Either of the two possibilities above contradicts our starting assumption that we had a complete list of prime numbers p

_{i}where i=1 to N since each possibility leads to at least one more prime number. Accordingly by contradiction we have proved that there are an unlimited number of prime numbers.

Finally, there is more than one kind of infinity. I was 16 when I came across Georg Cantor's Theory of Transfinite Numbers, and found it intellectually mind blowing. What I read at that age has never left me, even though I never had occasion to go back to it during my mathematics degree.

A reader's comment on another page made me realise that the purpose of this page can be misunderstood.

For the avoidance of doubt, this page is not intended as a proof of the existence of God. Philosophers have attempted to prove the existence of God for centuries, if not millenia, and have failed to do so. I believe that the existence of God cannot be proved; it is a matter of faith during our lifetimes, although if God does exist, as I believe, the afterlife will demonstrate that.

The purpose of the page is to illustrate that if you accept that God has infinite power, then he can allocate an infinite amount of love and attention to each of us. As Jesus says in Matthew 10:29:

"Aren't two sparrows sold for a penny? Yet not one of them falls to the ground without your Father's consent."

Holman Christian Standard Bible

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