«In order to make efﬁciency savings and achieve faster throughput, Pre-Judgement Reception outside the Pearly Gates has been reorganised and it has ...»
Leon Horsten & Philip Welch
Dept. of Philosophy and School of Mathematics,
University of Bristol,
September 17 2012
In order to make efﬁciency savings and achieve faster throughput, Pre-Judgement Reception outside the Pearly Gates has been reorganised and it has been decided that philosophers, scientists, mathematicians, theologians, and literary theorists would be processed
centrally in batches from now on, and indeed only in inﬁnite batches.
Whilst waiting in the “Pre-Judgement Antechamber” (more commonly known locally as the Limbo Room) for the requisite numbers to turn up, the transmigrating thinkers from various worlds discover their new found powers of perception, which include being able to take in and process inﬁnite amounts of information at a glance: the mathematicians amuse themselves playing two person inﬁnite games of perfect information, they nod knowingly at the conﬁrmation of Goldbach’s Conjecture and discover the consistency (or otherwise) of their favourite theories such as Peano Arithmetic or Zermelo-Fraenkel set theory; there are groans of dismay and even some emotional moments when they learn that the ﬁrst zeroes of the Riemann zeta function off the critical line occur at distances of the order of 1010 ; the cosmologists have mixed feelings about the partial conﬁrmation of the Big Bang theory; the one and only absolute reading of Genesis dismays the literary theorists, whilst the biologists are crestfallen at the triumph of Creationism.
When a countably inﬁnite number have been gathered in, they are told that Judgement will proceed: ENTER God (looking entirely like you would expect, as from a Blake watercolour); He makes the following announcement. “Welcome to the Judgement Precinct. You will shortly be proceeding next door to the Hall of Judgement where the Final Reckoning will take place. As you well know, those of Sufﬁcient Virtue will gain admittance within these Gates, but those of you deemed of, how shall We put it? of Insufﬁcient Virtue will have to...”, here He trailed off, looking tired and grave for a moment, ”will have to go to,..., go to the... Other Place.” Then brightening up, he said that, as was widely rumoured, he was merciful. Each would have a second opportunity of Redemption by meditating on His, God’s, Judgement.
There were, he continued, some differing courses of action open to them: the ﬁrst such was that they could use their own, now enhanced powers of judgement (both of morals and of other relevant factors of esteem), and for searching through the souls of others (which they had not noticed whilst all the fun with Goldbach’s Conjecture etc. was going on).
They might then care, the Almighty continued, whilst in the Hall of Judgement, to introspect and also come to some judgement about themselves, and accordingly decide on what should be their own fate. “However...” said God, “It is important for you to understand, that your moral judgement about yourself or others will mostly agree with mine, indeed each individual if they were to judge all those present would only make ﬁnitely many mistakes. However I make no particular promise as to how you might judge yourselves, other than that you need not fear mis-judging the beam in your own eye, compared to the mote of others: I can assure you, you will be as adept, and as fair, at self-judgement as adjudicating other souls.” God gave his solemn promise on this. This they all believed (they were generally in a chastened mood after the debacle over Evolution).
“You see, getting it right, even if it is a self judgement of Insufﬁcient...”, the Almighty coughed, somewhat embarrassedly, and started again: “ I regard correct moral self-judgement, even in the case of Insufﬁcient..., as pertaining to some form of repentance, ﬁnal value-added confession so-to-speak, worthy of redemption.” “Unfortunately, if inﬁnitely many of you make bad judgements and assess your own Judgements incorrectly, then, in the interests of..., then I am afraid you will all... you will all have to go down... the Other Place.” Of course, He continued, they might try exercising their inalienable right to Free Will (at this point the Lord faintly smiled): some other plan or tactic could be deployed. They would even be allowed to choose to appoint one person to represent the whole group, a champion so to speak: if the appointed person arrives at a correct self-judgement, then everyone is saved, if not, all are sent down. What would not be permitted (on pain, or even - they were left in no doubt - inﬁnite pain, being the inevitable outcome of collective and total expulsion into the hands of You Know Who), would be for them to start now informing each other of their moral judgements on each other, or even their own self-judgements, thus pre-empting the One and Final Judgement. Discussion now would be strictly limited to whether to use their own judgements or which action or tactic to take.
Once they had decided on a course of action they would be ushered in to the Judgement Hall, where, when the Reckoning was completed, they would have the opportunity to observe His, God’s, Judgements: each would ﬁnd they had a hat on their heads, either white (indicating their admittance), or else a somewhat orangey red. Each person would then be required to indicate their ﬁnal choice of the quality of his or her Virtue on their personal Self-Assessment Form (VG51) (Please tick Box 1 Sufﬁcient or 2 Insufﬁcient Virtue (Required Boxes - black ink only)). If the whole group would opt to be represented by a single chosen individual, then the procedure would be simpler: only the representative would enter the Judgement Hall, and the others would wait outside.
God then gave them 10 minutes to come to a collective decision.
After God had left the room there was a brief moment of stunned silence, and then uproar. Turmoil, shouting, and even (one would think for such a place) unseemly threats. The normally sober Pre-Judgement Area was transformed into a scene more worthy of Pandemonium. Eventually a Mathematician held the ﬂoor.
The Mathematician’s Argument (The Shade of Hermann Kahn)
The Mathematician spoke thus: “We now have powers to absorb and retain inﬁnite amounts of information, I propose we invoke the Axiom of Choice. We can adopt the following strategy: we give ourselves numbers 0, 1, 2,..., k,... (since I perceive we are denumerably many). Each person remembers his or her number. Now consider all possible listings of judgements of Sufﬁcient and Insufﬁcient Virtue such as the list J: j0 = I(nsufﬁcient), j1 = S(ufﬁcient), j2 = I, j3 = S,..., jn = S,..., with, e.g., j3 being the outcome on person number 3. There are inﬁnitely many such lists of I’s and S’s, but one of them is God’s Judgement list. Let us say two possible judgement listings are equivalent if from some point k on, the jk in the two listings they identical. This is an equivalence relation. Let us now pick, and so collectively agree on, for each equivalence class E of equivalent listings under this relation, a single representative list from this class, call it JE say.” Seeing a look of puzzlement on some of the faces of the deconstructionists he leaned over towards the foremost and said, “(This is a kind of proxy for the class E - it does not matter which JE from E it is, as long as we all agree on it.) Then at the One and Final Judgement, we use the hat colours we can see and our numbering of ourselves to make the list of God’s Judgements JG. The point is that standing in our enumerated order and looking forwards we shall all see in any case the same JG (apart from God’s judgement on ourselves and those behind us with earlier numbers);
hence we all know in which equivalence class it lies, let us call it EG ; and we all have agreed on our response list for this class: JEG. When we have to commit to a self-judgement, person numbered k will write down the value of the k’th element of the listing JEG we have all previously agreed on. The list of our responses may not be exactly God’s listing JG but, and this is the point, it only differs at most on ﬁnitely many people at the start of the sequence.
Hence only ﬁnitely many of us will be incorrect at most, and only these will be punished.
We should agree on this plan which results in a possible sacriﬁce of, at worst, ﬁnitely few.” Suddenly seeing Pascal in the crowd, he called out “See that man over there? He is a mathematician like me: he will agree with me!” There was a murmuring and a shifting of feet, whilst the implications of this strategy were absorbed. Some even started backing away from the Mathematician in the fear of being allotted a low number.
The man pointed out by the Mathematician elbowed his way to the front, shouting all the while: “No! No! not at all! Please, please, listen, This is quite wrong” (but in French).
The uproar slowly calmed, and a whispering broke out amongst the crowd.
The Shade of Pascal’s Argument (for it is indeed he):
(Pascal pauses for quiet.) “The previous speaker’s argument is specious and faulty. Who believes in this so-called Axiom of Choice? It is a chimæra. Suppose it would be true. Then the argument just given can be used to justify almost anything. For example, suppose God says to one of us ‘I have chosen a function F : R → R’ (rather than a judgement sequence JG ). Suppose God allows us to choose instead of Sufﬁcient or Insufﬁcient, one real number x say. He then shows us the complete set of values F (z) for every z = x. We do not know the value at x, but we are each then allowed to ‘guess’ the value F (x) to gain our salvation. Of course we cannot do it! The probability of ﬁnding the correct F (x) is that of choosing the right real number out of all the uncountably many real numbers that exist (as Herr Cantor has shown). But the Mathematician here would have you believe, that we could pick by this so-called ’Axiom’ of Choice, a representative function of each equivalence class of functions from R to R which only differ on ﬁnitely many x; he would then say that our representative function can only differ from God’s function at ﬁnitely many places. Hence the conclusion is that given God’s choice of function F, the probability would even be certainty that any one of us would be correct! We could only be wrong ﬁnitely often! This is nonsense, and you should not listen to it.” “The situation is laughably simple. God has promised us, and which of you” (he now glowered at the biologists), “which of you can now doubt His Word? He has promised us that our moral judgements about others will coincide with his when we enter the Hall of Judgement; we shall all have the evidence, even an inﬁnite amount of evidence that He is wise, just, and more to the point, agrees with us. It is true that we have not been vouchsafed that we are correct in our own self-worth. Truly, in order to enjoy His Gift of full Redemption, God in his Mercy has allowed us one more shot at it. Listen to your own moral reason.” (Having gained their full attention, Pascal pauses once more for breath; all the while he is beaming at his audience. He is, after all, feeling lucky.) “We should observe the following: Although God says that we may be wrong about our self-judgements, as we could make ﬁnitely many errors in our judgements, what is the probability that we are wrong about ourselves? We shall even have an abundance of evidence in the Hall of Judgement that we are good, nay even almost perfect, judges of peoples’ souls?
Given that each and every one of us will have been given an inﬁnite amount of evidence that our faculties of moral judgement are in accord with that of the Supreme Being, the probability for each one of us that he is wrong about his self-judgement is surely one of Mr Newton’s inﬁnitesimals? We should choose one worthy representative, perhaps Cardinal Richelieu here, and let our lot depend on the correctness of his own moral self-judgement.
Listen to your own God-given Faculty of Judgement and we shall all be saved.” The uproar returned, agitated with especial vigour by the Calvinists.
What should they do? Is Pascal correct?————–
With a shout someone calls out: they have performed the impossible, and demonstrated
that the Banach-Tarski paradox is no paradox at all, but that the Axiom of Choice is true: