On why 3D printers cannot print 3D printers
and related theological considerations
P.K. Chomsky.
1. Some introductory results
Lemma
1: 3D printers are Universal Turing Machines.
Consider a 3D printer1 which has the capacity
not only to print, but also to carve cubes of plastic out of
a tape, and via a comfortable webcam, read the strings of
code in input. Now, consider that space is innite, and that
3D printers use space as their working memory: they store
data in these cubes (and voids) of matter, and they "move"
on along the tape. Clearly, this entails that 3D printers have
innite memory. So, 3D printers are Universal Turing Machines.
Proof.
Fact
1: then, 3D printers are powerful enough to express
arithmetic.
Theorem (3D non-recursion):
as a consequence of Fact
1
and of
Gödel's rst incompleteness theorem, 3D printers cannot print other
3D printers.
Observation 1:
Man can reproduce itself.
Proof. We will prove Observation
us rst state our working premises:
2
1 by reductio
ad absurdum. Let
(1) the sentence I am here is
a clear example of a necessarily always true sentence (even though,
some say, it is never necessarily true); and (2) every man is son of his
parents. Now the proof: if man could not reproduce itself, then, by (2),
there would be no son, and no parent for further sons and daughters.
So, there would be no man. Now, clearly one may see that I am here
would necessarily be always false, for there would be nobody there, and
1
Theoretical machine whose existence was rst suggested to me by a 1934 letter
by von Neumann. I was 3, at the time.
2Mainly,
this procedure is known as intercourse, or the naughty things. See
Church (1920) and Priest (1954) for opposite views on the topic.
2. 3D PRINTERS AS THE ULTIMATE EMOLLIENT FOR BAD METAPHYSICS1
no I to utter the sentence. We are thereby compelled to conclude that
man can reproduce itself.
Corollary 1:
Fact
2,
As a consequence of the 3D non-recursion theorem and
man is not a Universal Turing Machine.
2. 3D printers as the ultimate emollient for bad metaphysics
Suppose that God exists.
Lemma (Ta):
If God exists, then there must be a 3D printer capable
of printing it.
Proof. It is well-known that God is omnipotent. So, he must be
able to create any number of 3D printers, some of which able to print
God itself. If he couldn't, then he would be not omnipotent. Therefore,
he can. But what if he could, but for some reason he never would create
such a
cælestiale machina. It has been proved3 that if God exists, then
it exists necessarily. Similarly, by replacing systematically `crucixion'
for `sentence', and `3D printer' for `Meinong' in Kripke's
Naming and
Necessity, one can obtain in a very straightforward way a proof of the
fact that if there can be a printer capable of printing God, then it must
exist necessarily.
Not only:
Kripke has also showed that such a printer is necessarily
4
printing. So, there is a slowly growing amount of God(s), consideration
which triggers the next lemma:
Corollary (Tb):
There can be
Corollary (Tbo):
∞
God(s).
And since God is such that there is nothing greater
than it, this must be an utterly overpopulated Universe.
This is a fundamental step. From Peter Abelard to William Heytesbury, there has been a whole array of convincing proofs that God is
`such that there is nothing greater than it'.
But if there can be
God(s), then this considerations lead immediately to paradox.
5
∞
If we
want to maintain that the set of deities is wellfounded , we must drop
the
Ta lemma.
Theorem (¬ Ta):
Conclusion: there can be no 3D printer capable of
printing God.
Theorem (omnipotence):
3D printers (if big enough) can print ev-
erything.
Proof. Trivial.
3By my grandma (1871), unpublished.
4Printing a God is Pspace-complete : cfr.
5There is a good reason for doing so:
Krunenbach (1998).
Nietzsche argued the same in his
philosophical-theological play in honour of Paul Rée `Siphilis or siphilisn't '.
2
Theorem (extended omnipotence):6
If God exists, then it is a 3D
printer.)
Proof. By the
¬ Ta theorem, we have to conclude that 3D printers
are such that they can't print God.
So, there is at least something
they can't print, consideration which conicts with the omnipotence
theorem.
We have showed that 3D printers being able to print God
leads to absurdity; and as well, one may think, would happen if we
supposed that God were able to print another God. What if God then
could print a 3D printer? This is not at all problematic. A 3D printer,
so it seems, is no God. Therefore, being God able to print the universal
Turing machine, it must be able to print everything that an universal
Turing machine could print by itself.
We are therefore obliged to conclude that if God exists, then it is a 3D
printer.
But then, by the omnipotence theorem, plus the
¬ Ta theorem, we have
that God can print itself. If it could not, it would not be omnipotent.
But this leads to paradox:
Proof.
•
•
Let
φ(x) = 1
i
x
can print itself.
δ(p) = 1 i φ(δ(x)) = 0.
Intuitively, this means that δ(x) = 1 i δ(x) cannot print itself.
• Then, let Ω = {x | φ(x) = 1}.
Consider any set X ⊆ Ω. Then the following i), ii) hold:
i) δ(X) ∈
/ X . Proof. Suppose that δ(X) ∈ X . By denition of δ , this
means that X is incapable of printing itself. But we have assumed that
δ(X) ∈ X , and X ⊆ Ω. So, δ(X) ∈ Ω. Therefore, by denition of Ω,
we have that φ(δ(X)) = 1: something which can't print itself (namely,
δ(X)) can print itself. But this is clearly absurd. We must conclude
that δ(X) ∈
/ X.
ii) δ(X) ∈ Ω. Proof. Suppose that δ(X) ∈
/ Ω. This means that
φ(δ(X)) = 0. I.e., δ(X) cannot print itself. But by the meaning of δ ,
then, we have that δ(δ(X)) = 1. But then, φ(δ(δ(X))) = 1, and so
δ(δ(X)) ∈ Ω. But there may be something (God) which can't print
itself but can print a 3D printer. ⊥. Therefore: δ(X) ∈ Ω.
If we then take X ≡ Ω, we reach the familiar paradox that δ(Ω) ∈ Ω
and δ(Ω) ∈
/ Ω.
∴ God can't exist.
6Also
Now dene a function
δ
such that
known in the literature as the Necessary Selfprintability Theorem, or
Chomsky's conclusion.
2. 3D PRINTERS AS THE ULTIMATE EMOLLIENT FOR BAD METAPHYSICS3
P.K. Chomsky, MsC Logic, ILLC
Amsterdam, 1974