Tony Thomas was born in England in 1939, and is a retired bureaucrat living in Brisbane, Australia. He has an Australian wife, two adult daughters, a dog and a cat. He holds a degree in economics from the University of Queensland. His interests are catholic, and include: writing fiction, poetry, and political diatribes to the newspapers. Other abiding interests include political and social philosophy, with occasional forays into logic and the foundations of mathematics.  His politics are left wing anarchism but his activities are restricted to the pen rather than the sword.

   God's Shopping List

On the thirteenth night of Christmas 1918, Georg Cantor died and went to heaven. When he arrived at the pearly gates, St. Peter was interrogating an Australian aborigine of the Ningy tribe called Googol-Googol.



Conversing in the Wakka Tongue, Googol-Googol was telling Peter of his baptism by German missionaries and of his faithful service to the church.

Cantor stepped forward, wild eyed but confident, cutting a sorry figure in his
hospital nightgown and untied straitjacket concealed beneath.

Startled, Peter asked, "Name?"

Cantor straightened up as much as the jacket allowed and said, "Georg Ferdinand Ludwig Philipp Cantor."

Peter consulted the Great Book and saw that he was in the presence of one of the world's great men. In the margin, someone had scrawled in red: 'Troublemaker - do not admit'.

"I'm sorry," Peter said, "God is cutting down on people with too many Christian names, they don't fit in the columns properly."

"But, I can see you already have all my names written down," Cantor retorted, with a superior smile.

Embarrassed, Peter mumbled, "Anyway, we're full up at the moment until we complete the extension."

"Ridiculous!" snorted Cantor, "Heaven is infinite so it can't be full up."

Peter, who was no good at maths, went red in the face. "Be very careful what you say, you don't want to end up in Hell, do you? Well, do you Junker?"

Realising that he was talking to a fool, Cantor smiled and said, "Posterity will judge me as a great genius, when the world has had time to understand my ideas. Surely I have a better right to be in heaven than this poor fellow here," he said pointing to Googol-Googol.

"But he was here before you," Peter said.

"The first shall be last and the last shall be first," Cantor quoted quickly.

His hackles rising, Peter looked again at the list of Cantor's achievements. It all seemed like gobbledegook to him, so he decided to consult Jesus before making a final judgement.

"Wait here, I won't be long," he said, heading for Noah's ark, where Jesus was feeding the animals.

On the way, he remembered the debacle of the loaves and fishes, when everyone had to go hungry while the disciples tried to break the food into smaller and smaller pieces. He decided that God would be the better judge as far as mathematics was concerned, so he headed off to the casino.

Donning dark glasses, Peter went into the gaming room where God sat spinning the wheel of fortune.

God knew what Peter wanted right away and told him to give Cantor and Googol-Googol a test to determine who was most fit to enter Heaven.

"Since you're here, I want some things from the shop," God said, meaning from the infinite emporium where all things in Heaven and Earth were stored.

"Give them a list so they can buy what I want and charge it to my credit account," he said, handing Peter his gold card.

Peter knew that he would have to use his own credit card because God's security number was too long for anyone else to remember.

God began to dictate the list, which Peter wrote down on a thin stone tablet screwed to a piece of Cedar wood.

God began: "Sausages", which Peter recorded on the first line of the tablet, carefully avoiding the screw heads, which ran diagonally from left to right.

God paused after saying, "coffee; milk, honey".

Peter knew that God liked his coffee with milk and honey and that he would be very angry if Cantor and Googol-Googol came back from the shop without them.

"Anything else?" Peter asked, immediately regretting it.

God thought for a while and began stuttering "A...A...A..." as if he couldn't quite remember something.

"A what?" Peter wondered, but decided to get on with the test without waiting for God to finish stuttering.

The wheel of fortune came to rest on zero yet again as he left the casino.


Peter returned to the candidates with the list, which looked like this:

There wasn't enough room on the front, so Peter had written the last two items on the back of the tablet:

When Cantor heard about the test he was delighted and seized the tablet eagerly. Googol-Googol looked perplexed, explaining that he could not read and that Cantor was hogging the tablet.

Peter, remembering the items, repeated them several times to Googol-Googol until he had learned them perfectly. A guardian angel was assigned to escort each of the candidates to the infinite emporium.

Googol-Googol, who often went shopping for his masters, soon returned with his angel, but when Peter looked in his bag there were only ten items in it.

"Why have you forgotten the coffee, milk and honey?" Peter asked, feeling disappointed.

"Number belong things belong bag same number belong fingers belong hand,"
Googol-Googol said sagely. "Number belong fingers belong hand
biggest number. Number belong things belong bag same number, so all things in shopping bag."

Peter realised that Googol-Googol had counted the things on his fingers
and had stopped at ten because it was the biggest number he knew. God really should have known better than to set such an unfair test.

Cantor did not return, so Peter sent Googol-Googol's angel back to find out what was going on. After what seemed like eternity, the two angels returned, dragging the struggling Cantor between them.

"Why did you take so long to carry out such a simple task?" Peter asked the dishevelled figure.

"I couldn't find one of the items on the shelves," Cantor raved.

"Nonsense," said Peter "The emporium has everything. What was the
item?"

Cantor began to stutter "A...A...A..." sounding a bit like God had done.

"Be quiet," snapped Peter, "let's see what you've got in the bag". He was amazed that the great mathematician had forgotten the milk and honey.

"Why did you only get eleven items," Peter asked, worried that it would soon be time for God's coffee break.

"Dumkopf!" shrieked Cantor, "can't you see that BAKED(-)BEANS has only got eleven letters, including the space, so how can you say I've forgotten some of the items?"

Seeing Peter's puzzled look, Cantor began to explain the diagonal argument to him.

Peter didn't understand a word of it, but when he looked at God's list, sure enough, BAKED(-)BEANS did only have eleven letters, including the screw head, so the list was only eleven letters wide.

In a flash of inspiration, Peter realised that the list had to be a square, because God always chose the most perfect shape. At last, he understood God's wisdom in reserving to himself the milk and honey that he loved so much.

Peter welcomed Cantor with open arms and was just about to dismiss Googol-Googol when Jesus appeared out of nowhere.

  "The square is a holy figure," said Jesus, "but the Tetraktys is more holy still. Googol-Googol brought back ten items but Cantor brought back eleven, which is a very ugly prime number."

"It looks like you'll have to go to Halle after all," Peter told Cantor.

"Nein, nein" screamed the mathematician, "I can be of great service to Heaven, I know all about infinity."

"Perhaps he could help you with the inventory of the infinite
emporium," Jesus suggested. "You've made a complete mess of it so far using those stone tablets. Why don't you use the special form that Zeno designed for me?"

"I couldn't find it; which item number is it?"

"The last one of course, just negate the diagonal number," Jesus said walking off over the duck pond.

"Look here," said Peter gruffly, grabbing Cantor by the scruff of his strait jacket,
"You have to help me find this number."

"But I have already told you that you can't list all the items and you certainly can't list the last item because there isn't one," Cantor said with an air of finality.

"Talk to Zeno," echoed Jesu's voice over the water, as he climbed back into the Ark.

Peter picked up the gold phone with the pearl trim and dialled zero.

"Eleatics," snapped a voice at the other end.

"This is Peter, please come to the front desk, I need your assistance right away."

"Impossible," snapped Zeno. "When I have walked half way there, I have to walk a quarter of the way, and then..."

Peter, fed up with hearing this story from Zeno, said, "Shut up and get over here right now," and slammed the phone down.

"We gotta phone like that at the mission," observed Googol-Googol.

Zeno soon appeared.

"So, how did you manage to get here?" Peter snarled sarcastically.

"Achilles gave me a lift," Zeno said, unperturbed.

I need some of that silly graph paper of yours; the type you designed for Jesus," Peter said. "I can't remember the inventory number."

"No problem, said Zeno, enthusiastically. We can draw up some more."

"I'll eat my hat if you can do that," mumbled Cantor under his breath.

"Lucky you're not wearing one, then," Zeno replied, taking out a roll of parchment from under his cloak. "You'll need a ruler and a pen," Zeno said.

Peter pulled out a battered ruler from under the counter with 'Property of the Roman Empire' stamped on it. He took out an angel quill, dipped it into the ink well and looked expectantly at Zeno.

"First, draw a square," Zeno began.

Peter dipped the pen into the sepia again and drew a ragged line across the top of the parchment.

"Now complete the square," Zeno said.

" I bet he can't do that," Cantor muttered.

Zeno continued: "Divide the square into two columns and divide the right hand column in two. Keep doing this until you reach the right hand side of the square."

Cantor groaned.

"This could take forever," Peter said.

"Exactly," Zeno replied. "When you have done that, divide up the square into rows, in the same way. When you have done that, divide each row in the same way that you divided the square. This will give you enough lines to record all the items in the infinite store."

Before Cantor could raise an objection Peter thrust the ruler and pen into his free hand saying, "Here, you carry on now, and don't make any blotches or you'll have to start again."

Peter suddenly remembered that Jesus had said the graph paper was the last item on the list, so all he had to do was go in by the back door of the store and take it off the last shelf. Feeling very pleased with himself, he took pity on Cantor.

"You can stay in Heaven while you are working on the list. If you need any help you can use Googol-Googol to check the items on the shelves."

By this time, quite a long queue had formed at the counter. "Next," said Peter brightly.

Suddenly, there was a flash of lightning from the direction of the casino.

"Where's my coffee!" God roared.

Prima facie, set theory is an unpromising subject for Satire. Its importance in the hierarchy of intellectual disciplines gives it the status of a modern theology, so satire is not without precedent. Mediaeval philosophers speculated about the nature of the cosmos, but had to keep a wary eye on the Catholic Church, lest their ideas stray beyond the bounds of approved doctrine. The penalty for going too far could lead to rack and stake, or life imprisonment, as Galileo found out. So religious satire was largely out of bounds, at least until the Enlightenment.

The grand church of modern science, like its religious precursors, also rests on sacred foundations, albeit mathematical rather than theological ones. The exacting discipline of mathematics, in turn, sits on the logical masonry of set theory.

At the beginning of the 20th Century, Bertrand Russell and Alfred North Whitehead embarked on a major work of logical theory. The resulting volumes of Principia Mathematica, published between 1910 and 1913, claimed to have reduced the foundations of mathematics to purely logical principles.

An important aspect of this grand work was the use of axiomatic systems, where a few axioms, manipulated by means of fixed rules, established the propositional and predicate calculi necessary to prove that two and two make four.

Now, every village idiot who can add up on his fingers begins to drool at the prospect of pouring scorn upon his betters, who think it necessary to toil for eleven years to reach this level of arithmetical prowess. Unfortunately, he is unlikely to come across them standing round the village pump. If he did, he might cease to tug his forelock, in recognition of their fall from common sense.

The motive behind Principia Mathematica was to construct a foundation for mathematics that would be entirely free of errors. If mathematicians stuck to the formal rules of axiomatic systems in deriving their theorems then they could be sure that no contradictions could arise.

This hope was dented if not dashed by the work of Kurt Godel, who proved in 1929 that the kind of axiomatic systems aimed at by Russell and Whitehead were either insufficiently powerful to cover the whole of mathematics or would be inconsistent if their scope was too wide.

While Russell and Whitehead were finalising Principia Mathematica, the German mathematician Zermelo was hard at work applying axiomatic systems to the ideas of his predecessor, Georg Cantor, the inventor of sets.

Cantor's great contribution to mathematics was the idea that sets were a more fundamental category than numbers, and could be used as the logical basis for defining those intractable little beasts. This innovation allowed him not only to explain the natural numbers but also their more complex derivatives, the rational fractions and the real numbers. Without this reduction of numbers to sets, it is doubtful if the great synthesis of modern mathematics could have been achieved.

After an enormous amount of toil by logicians and mathematicians, we have long since reached the point where set theory is well established on school curricula, and woe betide those parents who frown when faced with the equations of Mr Boole or the diagrams of Mr Venn.

It is unlikely that religious fundamentalists will begin howling for Herr Cantor's blood as they so often do for Mr Darwin's, and demand the removal of set theory from the school curriculum. If only they realised the theological implications of his theory of infinity they might be less sanguine.

If miracles could be explained, they would become mere conjuring tricks or, worse still, theorems derived from the axioms of set theory. Of course, the Scholastics used a logical principle called consequentia mirabilis (Latin names provide the imprimatur of infallibility). In words, this theorem of logic states:

If any proposition is false but implies its own truth then it must be true.

The truth of this principle can be established with a quick truth table (thank you Mr Wittgenstein) or by a reductio test, based on another principle called modus definiens, AKA reductio ad absurdam. Failing that, one must call upon one's understanding of everyday language to appreciate its miraculous properties.

But herein lies the difficulty, for would be philosophers. Thou shalt not rely upon thy native tongue (metalanguage) when talking about logic in general or set theory in particular. Only a special language will do to appraise the ideas of the master race, er mathematicians.

The particular problem for the satirist is to contrast what might be rudely called 'herd mentality' with the high doctrine he wishes to pillory. This is easier said than done, faced with a well-educated populace whose brains are stuffed with calculus and whose piggy eyes are focussed on the bottom line. The target audience is reduced to those gentle philosophers who have so far avoided the frozen peaks of logic and mathematics, preferring to graze in the lush valleys of the liberal arts.

The establishment of set theory did not come without some sacrifice of common intuitive ideas. For example, if I put two apples with three oranges in a bowl then I end up with five pieces of fruit in the bowl.

Now imagine putting all the even numbers greater than zero in a mental bowl. Lets call this number X. Then put all the odd numbers greater than zero in the bowl and call this number Y. We end up with all the natural numbers in the bowl, which, according to ordinary arithmetic would be X even things + Y odd things.

According to set theory, the addition of the odd numbers made no difference to the bowl total, because there are as many natural numbers as there are odd or even ones. To put it crudely, this is one of the central but counter intuitive conclusions of set theory. More formally, the addition of infinite, discrete sets does not affect the cardinality of the combined set. A simple example is that there are as many positive and negative integers as there are natural numbers.

This brings us to the miracle of the loaves and fishes. We need only consider one of the loaves, since five Cantorian loaves would not provide any more bread than one. Assuming that the loaves distributed by Jesus had been touched by the divine, it seems reasonable to suppose that they might behave like infinite sets. When such a loaf is broken in half, there is as much bread in each half as there was in the whole loaf. The feeding of the five thousand then becomes a simple matter of efficient distribution. The mathematically minded might wish to calculate how many times Jesus would have to divide five loaves successively in half to feed the multitude and how many pieces would be left over.

The fish are a bit more complex, because of their asymmetry. If I break an infinite fish in half, then I have a tail at one end and a head at the other, a bit like the case of the odd and even division of the natural numbers. But it is more like chopping the natural numbers into a lower part and an upper part, leaving the odd and even numbers in their natural order. The extreme tail end is clear enough, since it contains familiar numbers, but one wonders a bit about the numbers in the head.

The doctrine has it that the upper parts would taste as good as the lower parts because all the natural numbers are finite, even though they are infinitely numerous. The 'cardinal' number, which is assigned to the totality of the set of natural numbers, though, is not to be found within the set. If you hanker after the ambrosia of infinity, you must proceed beyond the cardinality of the natural numbers to higher infinities.

Seemingly more numerous infinite sets, such as the rational fractions, also share the miraculous attribute of being the same size as the set of natural numbers. Even triples, quadruples and, indeed, numbers of any dimensional power fail to escape the bounds of the Cantorian cardinal aleph-null.

The piece de resistance in the conjurors rigmarole comes when the cardinality of the set of real numbers is considered. This is where the modus operandi of the diagonal argument is deployed. Thanks to our modern system of decimal notation, irrational numbers like the square root of two or pi can be expressed as infinite strings of digits after the decimal point. A feature of such numbers is that no matter how far the string extends, it will always include a non-zero digit.

By a clever trick, it had proved possible to arrange the rational fractions, comprising all possible pairs of natural numbers, in a matrix and then show that the whole set could be transformed into a linear list. If the real numbers could be arranged using the same principle, then their cardinal would turn out to be the same as that of the natural numbers.

Cantor showed that this was not possible. Any arbitrary arrangement of non-terminating decimal strings in the form of a square matrix contains a diagonal line of digits, running from left to right. If each of these digits is altered systematically, say by adding one then the resulting string of digits could never be found in the rows of the matrix. So no matter how long the string of digits was, there would always be a unique string that was not included in the list.

Example:

Diagonal string: .14159265… Derived string: .25260376…

The inescapable conclusion was that the real numbers were uncountable, or non-denumerable in set theory speak. More importantly, the set of real numbers has a cardinal that is incomparably greater than that of the natural numbers, aleph-one. And here endeth the lesson, so to speak, until some enterprising mathematician proves that he has got a bigger one than Cantor's real whopper.

Having amazed themselves to find that a single cardinal was capable of measuring the size of an infinite number of different sets, they then re-amazed themselves by discovering a set that was the exception to their rule. "I don't believe it," Cantor is alleged to have cried on discovering the bathwater never changed height when he sat down for his daily water therapy. Imagine his surprise when one day it did. "Eureka," by Georg I've got it!"

Anyway, that's all water under the electric shock machine; set theory is here to stay and there is no point worrying about whether it makes sense to the man in the street, or on the couch more likely. Still, if one were to take a vacation to infinity land there might be a few nasty surprises.

Firstly, one's credit card limit would be finite, and therefore zero as far as infinite prices were concerned. Even if it were raised to aleph-null, it would be exhausted by the first transaction, since all the prices would equal the limit. This is contentious though. Mathematicians allow you to add to aleph-null but not to subtract from it, so perhaps everything would be free, after all. There would be no inflation, simply because aleph-null cannot be increased.

On a more abstract plane, considering the cosmos of Thales is instructive. He thought that everything was made of water but was almost certainly unaware that water is composed of fixed proportions of hydrogen and oxygen. Now if his universe were of infinite size, there would have to be as many hydrogen atoms as there are oxygen atoms, to satisfy Cantorian mapping. This either proves that the cosmos is not composed of water or that water is not composed of H2O. The third possibility that bijection has been wrongly applied to infinite sets need not be considered.

Anyway, no reductio ad absurdum, satirical or otherwise, can prevail against the fortress of set theory, because it is founded on an infallible axiomatic system (pace Godel) tried and tested by the finest minds that a century of academic research has attracted to the halls of mathematics. If you can't beat us, join us, goes up the cry from the battlements as a rain of logical barbs rains down upon the would be invader.

I might have to do that, what about you?