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␈↓ α∧␈↓␈↓ u1


␈↓ α∧␈↓␈↓ β{␈↓αComments on Dennett's ␈↓↓The Abilities of Men and Machines␈↓
␈↓ α∧␈↓␈↓ ∧n(preliminary version for today's discussion)

␈↓ α∧␈↓␈↓ αTDennett starts out with

␈↓ α∧␈↓"␈α...␈αthe␈αconstraints␈αof␈αlogic␈αexert␈αtheir␈αforce␈αnot␈αon␈αthe␈αthings␈αof␈αthe␈αworld␈αdirectly,␈αbut␈αrather␈αon
␈↓ α∧␈↓what we are to count as defensible descriptions or interpretations of things".

␈↓ α∧␈↓␈↓ αTAs␈αI␈αunderstand␈αthis␈αstatement,␈αI␈αdon't␈α
agree␈αwith␈αit;␈αI␈αthink␈αlogic␈αconstrains␈αwhat␈α
machines
␈↓ α∧␈↓can␈αdo␈αand␈αnot␈αjust␈α
how␈αwe␈αcan␈αdescribe␈αthem.␈α (See␈α
remark␈α4).␈α However,␈αI␈αthink␈αthis␈α
contention
␈↓ α∧␈↓is␈αnot␈α
required␈αto␈αshow␈α
that␈αG␈↓:␈↓odel's␈αtheorem␈α
can't␈αbe␈αused␈α
to␈αshow␈αthat␈α
machines␈αcan␈α
do␈αthings
␈↓ α∧␈↓people can't.

␈↓ α∧␈↓␈↓ αTDennett␈α⊃goes␈α⊃on,␈α⊃"The␈α⊃common␈α⊃skeleton␈α⊂of␈α⊃the␈α⊃anti-mechanistic␈α⊃arguments␈α⊃is␈α⊃this:␈α⊂any
␈↓ α∧␈↓computing␈α
machine␈αcan␈α
be␈αrepresented␈α
as␈α
some␈αTuring␈α
machine,␈αbut␈α
a␈αman␈α
cannot,␈α
for␈αsuppose
␈↓ α∧␈↓Jones␈αover␈α
there␈αwere␈α
a␈αrealization␈α
of␈αsome␈α
Turing␈αmachine␈α
TM␈↓βj␈↓,␈αthen␈α
(by␈αG␈↓:␈↓odel)␈α
there␈αwould␈α
be
␈↓ α∧␈↓something␈α␈↓↓A␈↓␈αthat␈αJones␈α␈↓↓could␈αnot␈αdo␈↓␈α(namely␈αprove␈αTM␈↓βj␈↓'s␈αG␈↓:␈↓odel␈αsentence).␈α But␈αwatch!␈αthis␈αis␈αthe
␈↓ α∧␈↓crucial␈α
empirical␈α
part␈α
of␈α
the␈α∞argument␈α
-␈α
Jones␈α
can␈α
do␈α
␈↓↓A;␈↓␈α∞therefore␈α
Jones␈α
is␈α
not␈α
a␈α∞realization␈α
of
␈↓ α∧␈↓TM␈↓βj␈↓,␈α
and␈α∞since␈α
␈↓↓it␈α
can␈α∞be␈α
seen␈↓␈α
that␈α∞this␈α
will␈α
be␈α∞true␈α
whatever␈α
Turing␈α∞machine␈α
we␈α∞choose,␈α
Jones
␈↓ α∧␈↓transcends, angel-like, the limits of mechanism".

␈↓ α∧␈↓␈↓ αTIn␈α↔the␈α↔citation␈α↔of␈α⊗this␈α↔argument,␈α↔there␈α↔is␈α⊗a␈α↔patchable␈α↔mathematical␈α↔error␈α↔and␈α⊗an
␈↓ α∧␈↓unpatchable omission of one of the hypotheses of G␈↓:␈↓odel's theorem.

␈↓ α∧␈↓␈↓ αTFirst,␈α∞a␈α∞Turing␈α∞machine␈α∞in␈α∞general␈α∞does␈α
not␈α∞have␈α∞a␈α∞G␈↓:␈↓odel␈α∞sentence.␈α∞ G␈↓:␈↓odel␈α∞sentences␈α
are
␈↓ α∧␈↓associated␈αwith␈αtheories␈αof␈αarithmetic␈α(or␈αother␈αsystems␈αcapable␈αof␈αrepresenting␈αelementary␈αsyntax)
␈↓ α∧␈↓-␈αnot␈αwith␈αmachines␈αper␈αse.␈α This␈αcan␈αbe␈αpatched␈αup,␈αbecause␈αwith␈αevery␈αtheory␈α␈↓↓Th␈↓␈αof␈αarithmetic,
␈↓ α∧␈↓a␈α∞Turing␈α
machine␈α∞␈↓↓TM(Th)␈↓␈α
can␈α∞be␈α
effectively␈α∞constructed␈α
that␈α∞enumerates␈α
its␈α∞theorems,␈α∞and␈α
the
␈↓ α∧␈↓G␈↓:␈↓odel␈α
sentence␈α
␈↓↓G␈↓:␈↓↓odel(Th)␈↓␈α
will␈α
not␈α
be␈α∞enumerated␈α
by␈α
that␈α
Turing␈α
machine.␈α
 A␈α∞universal␈α
Turing
␈↓ α∧␈↓machine␈α∂is␈α∂not␈α∂associated␈α∂with␈α∂any␈α⊂particular␈α∂theory␈α∂of␈α∂arithmetic,␈α∂but␈α∂we␈α∂can␈α⊂also␈α∂effectively
␈↓ α∧␈↓construct␈αa␈αprogram␈αfor␈αthe␈αuniversal␈αmachine␈α
that␈αenumerates␈αthe␈αsentences␈αof␈α␈↓↓Th␈↓␈αor␈α
any␈αother
␈↓ α∧␈↓first␈α∞order␈α
theory.␈α∞ (Indeed␈α
we␈α∞could␈α
make␈α∞it␈α
enumerate␈α∞the␈α
sentences␈α∞of␈α
all␈α∞first␈α∞order␈α
theories,
␈↓ α∧␈↓labelling␈α∞each␈α∞theorem␈α∞with␈α∂the␈α∞theory␈α∞it␈α∞that␈α∞generated␈α∂it,␈α∞provided␈α∞we␈α∞didn't␈α∂mind␈α∞including
␈↓ α∧␈↓sentences␈α∂from␈α∞inconsistent␈α∂theories).␈α∞ Again␈α∂this␈α∂program␈α∞will␈α∂not␈α∞print␈α∂␈↓↓G␈↓:␈↓↓odel(Th)␈↓␈α∂However,␈α∞it
␈↓ α∧␈↓may print a sentence whose intuitive content is

␈↓ α∧␈↓1)␈↓ αt ␈↓↓consistent(Th) ⊃ G␈↓:␈↓↓odel(Th)␈↓,

␈↓ α∧␈↓because␈α
that␈α∞may␈α
be␈α
a␈α∞theorem␈α
␈↓↓Th.␈↓␈α
If␈α∞the␈α
G␈↓:␈↓odel␈α
sentence␈α∞is␈α
taken␈α
to␈α∞be␈α
the␈α
statement␈α∞that␈α
the
␈↓ α∧␈↓theory␈α%is␈α%consistent,␈α%then␈α%it␈α%will␈α&print␈α%(1),␈α%because␈α%its␈α%content␈α%will␈α&be␈α%just
␈↓ α∧␈↓␈↓↓consistent(Th) ⊃ consistent(Th)␈↓.␈α↔ However,␈α_no␈α↔human␈α↔␈↓↓knows␈↓␈α_that␈α↔even␈α↔Peano␈α_arithmetic␈α↔is
␈↓ α∧␈↓consistent.

␈↓ α∧␈↓␈↓ αTBesides␈α∀that,␈α∪a␈α∀computer␈α∪program␈α∀can␈α∀have␈α∪other␈α∀relations␈α∪to␈α∀theories.␈α∀ Suppose␈α∪for
␈↓ α∧␈↓example,␈α∞it␈α∞will␈α∞generate␈α∞on␈α∞command␈α∞the␈α∞theorems␈α∞of␈α∞any␈α∞theory␈α∞you␈α∞describe␈α∞to␈α∞it␈α∞but␈α∞has␈α∞a
␈↓ α∧␈↓preferred␈α∞theory␈α∞of␈α∂arithmetic␈α∞that␈α∞it␈α∞uses␈α∂when␈α∞asked␈α∞to␈α∞try␈α∂to␈α∞prove␈α∞a␈α∞sentence␈α∂of␈α∞arithmetic
␈↓ α∧␈↓without additional stipulation.

␈↓ α∧␈↓␈↓ αTJones␈αcan␈αconstruct␈α
␈↓↓G␈↓:␈↓↓odel(Th)␈↓,␈αbut␈αhis␈αconfidence␈α
that␈αit␈αis␈αtrue␈α
is␈αbased␈αon␈α
his␈αconfidence
␈↓ α∧␈↓␈↓ u2


␈↓ α∧␈↓in␈α∞the␈α∞consistency␈α∞of␈α∞␈↓↓Th,␈↓␈α∞or␈α∞even␈α∞the␈α∞␈↓	w␈↓-consistency␈α
of␈α∞␈↓↓Th,␈↓␈α∞and␈α∞he␈α∞has␈α∞no␈α∞more␈α∞reason␈α∞to␈α
be
␈↓ α∧␈↓confident␈αof␈α
that␈αthan␈α␈↓↓Th␈↓␈α
has␈α-␈α
admitting␈αthe␈αabuse␈α
of␈αlanguage␈α
involved␈αin␈αascribing␈α
confidence
␈↓ α∧␈↓to␈α∂␈↓↓Th.␈↓␈α∂Of␈α∂course,␈α∂we␈α∞are␈α∂inclined␈α∂to␈α∂believe␈α∂in␈α∞the␈α∂consistency␈α∂of␈α∂the␈α∂usual␈α∂Peano␈α∞arithmetic.
␈↓ α∧␈↓Someone␈αwho␈αis␈αwilling␈αto␈αadd␈αto␈αPeano␈αarithmetic,␈αan␈αarithmetic␈αtranslation␈αof␈αits␈αconsistency␈αis
␈↓ α∧␈↓using a theory that we may call ␈↓↓Peano'.␈↓

␈↓ α∧␈↓␈↓ αTWhile␈αDennett␈αlater␈αmentions␈αthat␈α␈↓↓Th␈↓␈αmust␈αbe␈αa␈αconsistent␈αsystem,␈αhe␈αleaves␈αthat␈αcondition
␈↓ α∧␈↓out␈αof␈α
his␈αdescription␈αof␈α
the␈αargument␈αfor␈α
the␈αsuperiority␈αof␈α
humans.␈α However,␈αsince␈α
the␈αG␈↓:␈↓odel
␈↓ α∧␈↓sentence␈α
is␈α
typically␈α
taken␈α
as␈α
a␈α
translation␈α
into␈αthe␈α
language␈α
of␈α
the␈α
theory␈α
of␈α
the␈α
statement␈αthat␈α
the
␈↓ α∧␈↓theory is consistent, this omission is crucial.

␈↓ α∧␈↓␈↓ αTWe␈α⊃might␈α⊃elaborate␈α⊂our␈α⊃Turing␈α⊃machine␈α⊂to␈α⊃a␈α⊃computer␈α⊂program␈α⊃that␈α⊃could␈α⊃conduct␈α⊂a
␈↓ α∧␈↓dialog with Jones, and this dialog might run as follows:

␈↓ α∧␈↓Jones: Tell me your theory of arithmetic.

␈↓ α∧␈↓Machine:␈α∞I␈α∞can␈α∞consider␈α
any␈α∞theory␈α∞you␈α∞like,␈α
but␈α∞the␈α∞one␈α∞I␈α
use␈α∞when␈α∞people␈α∞ask␈α∞me␈α
arithmetic
␈↓ α∧␈↓questions is the following.  (Here the machine prints a set of first order axioms).

␈↓ α∧␈↓Jones: Nyaa, nyaa.  Here's a true sentence you'll never prove.

␈↓ α∧␈↓Machine:␈α
It␈α
seems␈α
to␈α
be␈α
a␈αG␈↓:␈↓odel␈α
sentence␈α
for␈α
the␈α
theory␈α
I␈αjust␈α
told␈α
you.␈α
 Why␈α
do␈α
you␈α
think␈αit's
␈↓ α∧␈↓true?

␈↓ α∧␈↓Jones: If your arithmetic is consistent, it's true.

␈↓ α∧␈↓Machine:␈αQuite␈α
so,␈αand␈αif␈α
it's␈αtrue␈αmy␈α
arithmetic␈αis␈αconsistent.␈α
 I␈αwas␈αhoping␈α
you␈αwould␈α
give␈αme
␈↓ α∧␈↓some additional reason for believing my arithmetic is consistent.

␈↓ α∧␈↓Jones: I'll think about it and let you know.

␈↓ α∧␈↓Machine:␈αNow␈α
to␈αturn␈α
the␈αtables,␈α
tell␈αme␈αyour␈α
theory␈αof␈α
arithmetic,␈αand␈α
I'll␈αtell␈α
you␈αa␈αtrue␈α
sentence
␈↓ α∧␈↓of␈α∩arithmetic␈α⊃you'll␈α∩never␈α∩prove.␈α⊃ (We␈α∩machines␈α∩are␈α⊃obliged␈α∩by␈α∩law␈α⊃to␈α∩assume␈α∩that␈α⊃theories
␈↓ α∧␈↓propounded␈α
by␈α
humans␈α
are␈α
consistent).␈α
 I'm␈α
very␈α∞good␈α
at␈α
G␈↓:␈↓odel's␈α
theorem␈α
you␈α
know;␈α
I␈α∞prove␈α
it
␈↓ α∧␈↓every morning for breakfast.

␈↓ α∧␈↓Jones:␈α
You␈α
mean␈α
you␈α
prove␈α
it␈α
before␈α
breakfast?␈α
 Anyway,␈α
I'm␈α
not␈α
committed␈α
to␈α
any␈αspecific␈α
theory
␈↓ α∧␈↓of arithmetic.

␈↓ α∧␈↓Machine: I meant what I said.  We machines don't eat; we prove theorems for breakfast.

␈↓ α∧␈↓Remarks:

␈↓ α∧␈↓␈↓ αT1.␈α⊃It␈α⊃is␈α⊃important␈α⊃to␈α⊃distinguish␈α⊃G␈↓:␈↓odel's␈α⊂theorem␈α⊃from␈α⊃the␈α⊃G␈↓:␈↓odel␈α⊃sentence␈α⊃of␈α⊃a␈α⊂theory.
␈↓ α∧␈↓G␈↓:␈↓odel's␈α∞theorem␈α∂is␈α∞a␈α∂single␈α∞theorem␈α∂of␈α∞metamathematics␈α∂and␈α∞can␈α∂be␈α∞known␈α∂by␈α∞a␈α∂machine.␈α∞ In
␈↓ α∧␈↓principle␈αit␈αcan␈αbe␈αproved␈αby␈αa␈αcomputer␈αprogram,␈αbut␈αpresent␈αprograms␈αare␈αnot␈αsmart␈αenough␈α
to
␈↓ α∧␈↓do␈α
it␈αin␈α
any␈αhonest␈α
sense.␈α
 This␈αis␈α
important,␈αbecause␈α
it␈α
is␈αimportant␈α
to␈αestablish␈α
that␈α
the␈αG␈↓:␈↓odel
␈↓ α∧␈↓game can be played by the machine as well as by a person.
␈↓ α∧␈↓␈↓ u3


␈↓ α∧␈↓␈↓ αT2.␈α∀Even␈α∀more␈α∀metamathematics␈α∀than␈α∀G␈↓:␈↓odel's␈α∀theorem␈α∀seems␈α∀relevant␈α∀to␈α∪philosophical
␈↓ α∧␈↓consideration of this problem.

␈↓ α∧␈↓␈↓ αTFeferman's␈α
"Transfinite␈α
Progressions␈α
of␈α
Theories"␈α
is␈α
of␈α
interest.␈α
 Turing␈α
(I␈α
don't␈α
have␈α
the
␈↓ α∧␈↓reference␈α∪at␈α∪the␈α∀moment)␈α∪pointed␈α∪out␈α∪that␈α∀any␈α∪theory␈α∪of␈α∪arithmetic␈α∀can␈α∪have␈α∪added␈α∀to␈α∪it
␈↓ α∧␈↓"principles␈αof␈α
self-confidence".␈α For␈αexample,␈α
if␈α␈↓↓T␈↓␈αis␈α
a␈αtheory␈αcapable␈α
of␈αrepresenting␈α
syntax,␈αe.g.
␈↓ α∧␈↓Peano␈α
arithmetic,␈α
you␈α
can␈α
get␈α
a␈α
new␈α
theory␈α
␈↓↓T'␈↓␈α
by␈α
adding␈α
a␈α
sentence␈α
whose␈α
content␈α
is␈α
that␈α
␈↓↓T␈↓␈αis
␈↓ α∧␈↓consistent.␈α Repeating␈αthe␈αprocess␈α
gives␈αyou␈αthe␈αsequence␈α␈↓↓T,␈↓␈α
␈↓↓T',␈↓␈α␈↓↓T'',␈↓␈αetc.␈αall␈α
different.␈α Turing
␈↓ α∧␈↓studied␈α∪such␈α∪sequences␈α∩of␈α∪theories.␈α∪ Feferman␈α∩proposed␈α∪a␈α∪stronger␈α∪self-confidence␈α∩expressed
␈↓ α∧␈↓roughly␈α∞by␈α∞␈↓↓∀n.provable((P(n)) ⊃ ∀n.P(n)␈↓.␈α∞ It␈α∞says␈α∞that␈α∂if␈α∞you␈α∞have␈α∞a␈α∞sequence␈α∞of␈α∂sentences,␈α∞and
␈↓ α∧␈↓you␈α_can␈α_conclude␈α_that␈α_all␈α_the␈α_sentences␈α_are␈α_provable,␈α_you␈α_may␈α_conclude␈α_the␈α_universal
␈↓ α∧␈↓quantification␈α
of␈α
the␈α
senntences.␈α
 My␈α
memory␈α
of␈α
this␈α
isn't␈α
precise,␈α
but␈α
I␈α
think␈α
it␈α
amounts␈α
to␈α
the
␈↓ α∧␈↓␈↓	w␈↓-consistency of ␈↓↓Th.␈↓

␈↓ α∧␈↓␈↓ αTSome␈αfuss␈αwith␈αquotation␈αor␈αG␈↓:␈↓odel␈αnumbers␈αis␈αrequired␈αto␈αstate␈αthis␈αprecisely.␈α The␈αprocess
␈↓ α∧␈↓of␈αiterating␈αthe␈α
theories␈αcan␈αbe␈α
carried␈αinto␈αthe␈αtransfinite,␈α
i.e.␈αyou␈αcan␈α
construct␈αa␈αtheory␈α␈↓↓T␈↓#
␈↓	␈↓#
w␈↓↓␈↓#
␈↓#␈↓␈α
that
␈↓ α∧␈↓includes␈α∞the␈α∂whole␈α∞sequence␈α∂␈↓↓T,␈↓␈α∞␈↓↓T',␈↓␈α∂etc.␈α∞ Indeed␈α∂for␈α∞any␈α∂recursive␈α∞ordinal,␈α∂the␈α∞process␈α∂can␈α∞be
␈↓ α∧␈↓iterated␈α
that␈α
far,␈α
but␈αat␈α
some␈α
kinds␈α
of␈αlimit␈α
ordinals,␈α
you␈α
have␈αsome␈α
notational␈α
choices␈α
to␈αmake,
␈↓ α∧␈↓and␈αthere␈αis␈αno␈αsingle␈αway␈αthat␈αyou␈αcan␈αmake␈αall␈αthese␈αchoices␈αonce␈αand␈αfor␈αall␈αin␈α
advance.␈α For
␈↓ α∧␈↓this␈α
reason,␈α
the␈αprocess␈α
of␈α
iteration␈α
through␈αthe␈α
constructive␈α
ordinals␈α
cannot␈αbe␈α
all␈α
defined␈αat␈α
once
␈↓ α∧␈↓and␈αtherefore␈αyou␈αcan't␈αtake␈αit␈αto␈αthe␈αlimit.␈α Feferman␈αfurther␈αshows␈αthat␈αthere␈αare␈αwhat␈α
he␈αcalls
␈↓ α∧␈↓"progressions␈α∂of␈α∂theories"␈α∂such␈α∂that␈α∂the␈α∂limit␈α∂of␈α∂the␈α∂progression␈α∂is␈α∂the␈α∂set␈α∂of␈α∂true␈α⊂sentences␈α∂of
␈↓ α∧␈↓arithmetic.␈α∞ Thus␈α∞one␈α∞can␈α∞say␈α∞that␈α∞all␈α
Peano␈α∞arithmetic␈α∞lacks␈α∞is␈α∞self-confidence,␈α∞but␈α∞it␈α∞lacks␈α
an
␈↓ α∧␈↓infinite␈αamount␈αof␈αself-confidence,␈αand␈αno␈αfinite␈αamount␈αof␈αreassurance␈αwill␈αcomplete␈αit.␈α No-one
␈↓ α∧␈↓has really studied this from a philosophical point of view as far as I know.

␈↓ α∧␈↓␈↓ αTThe␈α∂philosophically␈α∂interesting␈α∂part␈α∂of␈α∂this␈α∂is␈α∂what␈α∂Feferman␈α∂tells␈α∂us␈α∂about␈α∂how␈α∂and␈α∞to
␈↓ α∧␈↓what␈α⊂extent␈α⊂we␈α⊂can␈α⊂transcend␈α⊂our␈α⊂limitations.␈α⊂ The␈α⊂self-confidence␈α⊂principles␈α⊂can't␈α⊂be␈α∂proved
␈↓ α∧␈↓within␈α
the␈α
theories,␈α
but␈α
one␈α
is␈α
inclined␈αto␈α
believe␈α
them.␈α
 The␈α
unconstructability␈α
of␈α
a␈αsingle␈α
definite
␈↓ α∧␈↓progression␈α
should␈αalso␈α
have␈αinteresting␈α
philosophical␈α
consequences.␈α If␈α
we␈αjust␈α
tell␈α
our␈αmachine
␈↓ α∧␈↓about␈αG␈↓:␈↓odel's␈αtheorem,␈αthen␈αthere␈αmay␈αbe␈αa␈αsense␈αin␈αwhich␈αFeferman␈αknows␈αmore.␈α Of␈α
course,␈αa
␈↓ α∧␈↓smart␈α∞enough␈α∞machine␈α∞might␈α∞␈↓↓outfeff␈↓␈α∞him.␈α∞ Suppose␈α∞we␈α∞want␈α∞to␈α∞express␈α∞as␈α∞much␈α∂confidence␈α∞as
␈↓ α∧␈↓possible␈αin␈αPeano␈αarithmetic␈α
and␈αits␈αextensions.␈α This␈αseems␈α
to␈αdepend␈αon␈αwhat␈αrecursive␈α
ordinals
␈↓ α∧␈↓we␈αcan␈αname,␈αhow␈αsure␈αwe␈αare␈αthat␈αthey␈αare␈αwell-orderings,␈αand␈αthe␈αdetails␈αof␈αthe␈αconstruction␈αof
␈↓ α∧␈↓the␈α
iterates␈α∞of␈α
Peano␈α∞arithmetic.␈α
 If␈α∞a␈α
man␈α∞and␈α
a␈α
machine␈α∞or␈α
two␈α∞men␈α
or␈α∞two␈α
machines␈α∞hold␈α
a
␈↓ α∧␈↓contest,␈α∂it␈α⊂would␈α∂be␈α⊂unfair␈α∂to␈α∂allow␈α⊂one␈α∂to␈α⊂say,␈α∂"Let's␈α∂see␈α⊂who␈α∂can␈α⊂name␈α∂the␈α⊂largest␈α∂recursive
␈↓ α∧␈↓ordinal.  You first".

␈↓ α∧␈↓␈↓ αT3.␈α∂On␈α∂page␈α⊂262,␈α∂we␈α∂have␈α∂the␈α⊂improbable␈α∂assumption␈α∂that␈α∂the␈α⊂activities␈α∂of␈α∂an␈α⊂infant␈α∂or
␈↓ α∧␈↓moron␈αadmit␈αinterpretation␈αas␈αproving␈αthe␈αtheorems␈αof␈αarithmetic.␈α For␈αthe␈αcryptographic␈αreasons
␈↓ α∧␈↓I␈α∪have␈α∩previously␈α∪mentioned␈α∪in␈α∩the␈α∪seminar,␈α∩I␈α∪would␈α∪argue␈α∩that␈α∪the␈α∩probability␈α∪of␈α∪this␈α∩is
␈↓ α∧␈↓incredibly␈αlow.␈α
 However,␈αthis␈α
error␈αdoesn't␈α
really␈αcontribute␈α
to␈αwhat␈α
I␈αbelieve␈α
is␈αthe␈α
main␈αerror␈α
of
␈↓ α∧␈↓the paper.

␈↓ α∧␈↓␈↓ αT4.␈α∩I␈α⊃am␈α∩not␈α∩sure␈α⊃whether␈α∩the␈α∩following␈α⊃would␈α∩be␈α∩considered␈α⊃a␈α∩counterexample␈α∩to␈α⊃the
␈↓ α∧␈↓contention␈α⊂that␈α⊂the␈α⊂constraints␈α⊂of␈α⊂logic␈α⊂are␈α⊂a␈α⊂restriction␈α⊂on␈α⊂our␈α⊂interpretations␈α⊃or␈α⊂descriptions
␈↓ α∧␈↓rather␈αthan␈αon␈αthe␈αworld␈αitself.␈α Suppose␈αa␈αmaterial␈αobject␈αwere␈αequipped␈αwith␈αan␈αoracle␈αfor␈αtrue
␈↓ α∧␈↓sentences␈αof␈αarithmetic␈αor␈αeven␈αmerely␈αan␈α
oracle␈αfor␈αwhether␈αa␈αTuring␈αmachine␈αwould␈α
stop.␈α We
␈↓ α∧␈↓suppose␈αthat␈α
the␈αoracle␈αanswers␈α
in␈αa␈αtime␈α
proportional␈αto␈αthe␈α
length␈αof␈αthe␈α
sentence␈αor␈αthe␈α
Turing
␈↓ α∧␈↓␈↓ u4


␈↓ α∧␈↓machine␈α∩description␈α∩asked␈α∩about.␈α∩ No␈α∩cryptographic␈α∩system␈α∩whose␈α∩key␈α∩was␈α∩shorter␈α∩than␈α⊃the
␈↓ α∧␈↓message␈αwould␈αbe␈α
secure␈αagainst␈αsuch␈αa␈α
machine,␈αbecause␈αone␈αcould␈α
play␈α20␈αquestions␈α
(or␈αrather
␈↓ α∧␈↓10,000␈α∞questions)␈α∞asking␈α∞first␈α∞whether␈α∞there␈α∞is␈α∞a␈α∞briefly␈α∞describable␈α∞cryptographic␈α∞system␈α∞and␈α
a
␈↓ α∧␈↓possible␈αplaintext␈αsatisfying␈αa␈αfew␈αknown␈αconstraints␈αof␈αEnglish.␈α Most␈αlikely␈αwe␈αwould␈αonly␈αhave
␈↓ α∧␈↓to␈α
ask␈α
whether␈αthere␈α
was␈α
a␈αsequence␈α
consisting␈α
almost␈αentirely␈α
of␈α
English␈αwords␈α
that␈α
could␈α
be␈αa
␈↓ α∧␈↓plaintext.␈α Receiving␈αan␈αaffirmative␈αanswer,␈αwe␈αthen␈αask␈αwhether␈αthere␈αis␈αsuch␈αa␈αplaintext␈αwhose
␈↓ α∧␈↓first␈αbit␈αis␈αzero␈α(in␈αan␈αencoding␈αwe␈αspecify).␈α We␈αthen␈αask␈αwhether␈αthe␈αsuccessive␈αbits␈αcan␈αbe␈αzero
␈↓ α∧␈↓or␈α∪one,␈α∪thus␈α∩decrypting␈α∪the␈α∪message.␈α∩ Notice␈α∪that␈α∪this␈α∩decryption␈α∪system␈α∪doesn't␈α∪depend␈α∩on
␈↓ α∧␈↓ascribing␈α
real␈αcontent␈α
to␈α
the␈αmessage;␈α
merely␈αthat␈α
it␈α
be␈αinterpretable␈α
as␈α
almost␈αa␈α
sequence␈αof␈α
words
␈↓ α∧␈↓from the ␈↓↓Oxford English Dictionary␈↓.

␈↓ α∧␈↓␈↓ αTWith␈α
somewhat␈α∞more␈α
attention␈α∞to␈α
detail,␈α
we␈α∞could␈α
also␈α∞ask␈α
it␈α
whether␈α∞there␈α
is␈α∞a␈α
scientific
␈↓ α∧␈↓theory␈α
in␈α
a␈αcertain␈α
formal␈α
language,␈α
briefly␈αdescribed␈α
please,␈α
that␈α
would␈αaccount␈α
for␈α
a␈αcollection␈α
of
␈↓ α∧␈↓observations without assuming that more than ten percent of them were errors.

␈↓ α∧␈↓␈↓ αT5.␈α
The␈αpostscript␈α
seems␈αincoherent,␈α
because␈α
it␈αseems␈α
to␈αdepend␈α
on␈α
the␈αidea␈α
that␈αeach␈α
Turing
␈↓ α∧␈↓machine,␈αincluding␈α
universal␈αmachines,␈αhas␈α
a␈αG␈↓:␈↓odel␈αsentence.␈α
 Maybe␈αthis␈αcould␈α
be␈αfixed␈αby␈α
some
␈↓ α∧␈↓reinterpretation, but I don't see how.