Tortoise: Not at all. In fact, Achilles, I am looking forward to treating you
to a fine birthday dinner, provided merely that I am convinced that
knowledge of all those yes-answers at once (as supplied by Answer
Schema w) allows me to proceed directly and without any further
detours to the conclusion that today is your birthday. That's the case,
isn't it?
Achilles: Yes, of course it is.
Tortoise: Good. And now I have yes-answer w + 1. Armed with it, I can
proceed to accept the hypothesis that today is your birthday, if it is
valid to do so. Would you be so kind as to counsel me on that matter,
Achilles?
Achilles: What is this? I thought I had seen through your infinite plot.
Now doesn't yes-answer w + 1 satisfy you? All right. I'll give you not
only yes-answer w + 2, but also yes-answers w + 3, w + 4, and so on.
Tortoise: How generous of you, Achilles. And here it is your birthday,
when I should be giving YOU presents instead of the reverse. Or
rather, I SUSPECT it is your birthday. I guess I can conclude that it IS
your birthday, now, armed with the new Answer Schema, which I will
caH "Answer Schema 2w". But tell me, Achilles: Does Answer Schema
2w REALLY allow me to make that enormous leap, or am I missing
something?
Achilles: You won't trick me any more, Mr. T. I've seen the way to end this
silly game. I hereby shall present you with an Answer Schema to end
all Answer Schemas! That is, I present you simultaneously with An-
swer Schemas w, 2w, 3w, 4w, 5w, etc. With thi~ Meta-Answer-Schema, I
have JUMPED OUT of the whole system, kit and caboodle, transcended
this silly game you thought you had me trapped in-and now we are
DONE!
Tortoise: Good grief! I feel honored, Achilles, to be the recipient of such a
powerful Answer Schema. I feel that seldom has anything so gigantic
been devised by the mind of man, and I am awestruck by its power.
Would you mind if I give a name to your gift?
Achilles: Not at all.
Tortoise: Then I shaH call it "Answer Schema w^2 ". And we can shortly
proceed to other matters-as soon as you tell me whether the
possession of Answer Schema w^2 allows me to deduce that today is
your birthday.Achilles: Oh, woe is me! Can't I ever reach the end of this tantalizing trail?
What comes next?
Tortoise: Well, after Answer Schema w^2 there's answer w^2 + 1. And
then answer w^2 + 2. And so forth. But you can wrap those all
together into a packet, being Answer Schema w^2 + w. And then
there are quite a few other answer-packets, such as w^2 + 2w, and
w^2 + 3w ... Eventually, you come to Answer Schema 2w^2 , and
after a while, Answer Schemas 3w^2 and 4w^2 • Beyond them there
Birthday Cantatatata ... 463