Church’s Thesis and Hume’s Problem: Justification as Performance
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Transcript of Church’s Thesis and Hume’s Problem: Justification as Performance
Church’s Thesis Church’s Thesis and Hume’s and Hume’s
Problem:Problem:Justification as Justification as PerformancePerformance
Kevin T. KellyDepartment of Philosophy
Carnegie Mellon [email protected]
Further ReadingFurther Reading(With C. Glymour) “Why You'll Never Know Whether Roger Penrose is a Computer”, Behavioral and Brain Sciences, 13: 1990.•The Logic of Reliable Inquiry, Oxford: Oxford University Press, 1996.•(with O. Schulte) “Church's Thesis and Hume's Problem,” in Logic and Scientific Methods, M. L. Dalla Chiara, et al., eds. Dordrecht: Kluwer, 1997.•(with O. Schulte and C. Juhl) “Learning Theory and the Philosophy of Science”, Philosophy of Science 64: 1997.“The Logic of Success”, British Journal for the Philosophy of Science, 51:2000.“Uncomputability: The Problem of Induction Internalized,” Theoretical Computer Science 317: 2004.Computability and Learnibility, Textbook manuscript, 2005.
Fateful First CutFateful First Cut
Relations of IdeasRelations of Ideas Matters of factMatters of fact
AnalyticAnalytic SyntheticSynthetic
CertainCertain UncertainUncertain
A PrioriA Priori A PosterioriA Posteriori
Philosophy of MathPhilosophy of Math Philosophy of Philosophy of ScienceScience
Proofs and Proofs and algorithmsalgorithms ConfirmationConfirmation
Truth-findingTruth-finding ““Rationality”Rationality”
ComputabilityComputability ProbabilityProbability
Painful ProgressPainful Progress
Relations of IdeasRelations of Ideas Matters of factMatters of fact
AnalyticAnalytic SyntheticSynthetic
CertainCertain UncertainUncertain
A PrioriA Priori A PosterioriA Posteriori
Philosophy of MathPhilosophy of Math Philosophy of Philosophy of ScienceScience
Proofs and Proofs and algorithmsalgorithms ConfirmationConfirmation
Truth-findingTruth-finding ““Rationality”Rationality”
ComputabilityComputability ProbabilityProbability
Slight RearrangementSlight Rearrangement
Philosophy of MathPhilosophy of Math Philosophy of Philosophy of ScienceScience
CertainCertain UncertainUncertain
A PrioriA Priori A PosterioriA Posteriori
Proofs and Proofs and algorithmsalgorithms ConfirmationConfirmation
Truth-findingTruth-finding ““Rationality”Rationality”
ComputabilityComputability ProbabilityProbability
Philosophy of Science 101Philosophy of Science 101
Formal QuestionsFormal Questions Scientific Scientific QuestionsQuestions
UnverifiableUnverifiable
““The sun will neverThe sun will never
stop rising”stop rising”
VerifiableVerifiable
““The given number The given number
is even”is even”
Wrong Twice!Wrong Twice!
Formal QuestionsFormal Questions Scientific Scientific QuestionsQuestions
UnverifiableUnverifiable
““The given The given computation will computation will
never halt”never halt”
UnverifiableUnverifiable
““The sun will neverThe sun will never
stop rising”stop rising”
VerifiableVerifiable
““The given number The given number
is even”is even”
VerifiableVerifiable
““The next emerald The next emerald
is green”is green”
Relations of IdeasRelations of Ideas Matters of factMatters of fact
Classical ExcuseClassical Excuse
Rational Animal
Rational ?
Reason Observation
UnboundedexperienceCompleted analysis
??Always black?
Relations of IdeasRelations of Ideas Matters of factMatters of fact
But Maybe…But Maybe…
Rational ?
Reason Observation
Unboundedexperience
Never white?Always black?
Unboundedanalysis
????
Synthetic A PrioriSynthetic A Priori A PosterioriA Posteriori
KantKant
Intuition Observation
Unboundedexperience
3+2 = 5 ?
3 4 5
Never white?Always black?
temporal intuition
23
??
Synthetic A PrioriSynthetic A Priori A PosterioriA Posteriori
Oops!Oops!
Intuition Observation
Unboundedexperience
Never cross? Never white?Always black?
spatial intuition
????
Formal ProblemsFormal Problems Empirical Empirical ProblemsProblems
After TuringAfter Turing
Never halts?
Computer Observation
Unboundedreality
Unboundedcomputation
Input
n
Never white?Always black??? ??
Bad First CutBad First Cut
Formal QuestionsFormal Questions Scientific Scientific QuestionsQuestions
UnverifiableUnverifiable
““The given The given computation will computation will
never halt”never halt”
UnverifiableUnverifiable
““The sun will neverThe sun will never
stop rising”stop rising”
VerifiableVerifiable
““The given number The given number
is even”is even”
VerifiableVerifiable
““The next emerald The next emerald
is green”is green”
Good First CutGood First Cut
Formal QuestionsFormal Questions Scientific Scientific QuestionsQuestions
UnverifiableUnverifiable
““The given The given computation will computation will
never halt”never halt”
UnverifiableUnverifiable
““The sun will neverThe sun will never
stop rising”stop rising”
VerifiableVerifiable
““The given number The given number
is even”is even”
VerifiableVerifiable
““The next emerald The next emerald
is green”is green”
Computational EpistemologyComputational Epistemology
For formal For formal andand empirical empirical questions:questions:
Justification =Justification = truth-finding truth-finding performanceperformance
Find the truth in theFind the truth in the best possible best possible sensesenseAnd then as And then as efficiently as efficiently as possiblepossible. .
Verification as Verification as “Support”“Support”
1.0
.5Deductive verification
Inductive verification
Paradigms EvolveParadigms Evolve
•Objective support by evidence.
Paradigms Evolve…Paradigms Evolve…
•Objective support by evidence.
•Coherent personal opinion.•“Rational” update.
Paradigms Evolve…Paradigms Evolve…
•Objective support by evidence.
•Coherent personal opinion.•“Rational” update.
But Leading Metaphors MatterBut Leading Metaphors Matter
•Myopic •Bedrock = “rationality” intuitions•Success = being “rational”•Truth-finding performance deferred or ignored•Problem complexity ignored•Little resemblance to computability
Verification as Verification as HaltingHalting
Conclusion
Data
Verification as Verification as HaltingHalting
Conclusion
Data
Yes!
Verification as Verification as HaltingHalting
Conclusion
Data
Verification as Verification as HaltingHalting
Conclusion
Data
Verification as Verification as HaltingHalting
Conclusion
Data
Bang!
Verification as Verification as HaltingHalting
Conclusion
Data
Can’t take it back.
VerifiabilityVerifiability
Conclusion true
Yes!
Conclusion false
Laws are Not VerifiableLaws are Not Verifiable
Always green?
Laws are Not VerifiableLaws are Not Verifiable
Always green?
Laws are Not VerifiableLaws are Not Verifiable
Always green?
Laws are Not VerifiableLaws are Not Verifiable
Always green?
Laws are Not VerifiableLaws are Not Verifiable
Always green?
You must halt with “yes” if it’s true!
Laws are Not VerifiableLaws are Not Verifiable
Always green?
You must halt with “yes” if it’s true!
Laws are Not VerifiableLaws are Not Verifiable
Yes!
Always green?
Laws are Not VerifiableLaws are Not Verifiable
Always green?
Laws are Not VerifiableLaws are Not Verifiable
Always green?
A Purely Formal ProblemA Purely Formal Problem
Never shoots?
Analogy?Analogy?
Never shoots?
I ain’t a’ shootin’, yeller belly!
Apparently NotApparently Not
Game’s up, J.R.! Yer program’s showin’!
Never shoots? 666
Apparently NotApparently Not
Now shaddup! I’m calculatin’a-preeorey-like what yer gonna do.
Never shoots? 666
Never shoots?
Apparently NotApparently Not
Never shoots? 2145623455666
?!!!
Never shoots?
But Then Again…But Then Again…
666Never shoots?
I got it! Two kin play that opry oary game!
2145623455
Never shoots?
But Then Again…But Then Again…
666Never shoots? 2145623455
I won’t shoot ‘til Isee him shootin this here opry oarysim-yooo-laytion!
2145623455
666
Never shoots?
But Then Again…But Then Again…
666Never shoots? 2145623455
2145623455
666
I ain’t a’ shootin’.
Never shoots?
But Then Again…But Then Again…
666Never shoots? 2145623455
666
I still ain’t a’ shootin’.
2145623455
Never shoots?
But Then Again…But Then Again…
666Never shoots? 2145623455
666
You check that there opry oary all you want…
2145623455
Never shoots?
But Then Again…But Then Again…
666Never shoots? 2145623455
666
An’ all you’ll see is I ain’t a-shootin’
2145623455
Never shoots?
But Then Again…But Then Again…
666Never shoots? 2145623455
666
2145623455
Aw, shucks, J.R. I reckon you ain’t never gonna shoot.
Aw, shucks, J.R. I reckonYou ain’t never gonna shoot.
Never shoots?
But Then Again…But Then Again…
666Never shoots?
666
Ban
g!
23455
23455
Never shoots?
But Then Again…But Then Again…
666Never shoots?
666
23455
23455
Never shoots?
But Then Again…But Then Again…
666Never shoots?
666
23455
See ya inthe grand ol’ opry oary, sucker!
23455
Never shoots?
But Then Again…But Then Again…
666Never shoots?
23455
666
Ban
g!Take that!
Never shoots?
But Then Again…But Then Again…
Never shoots?
234556
66
Never shoots?
But Then Again…But Then Again…
Never shoots?
234556
66
ROBOSHERIFF
FooledA priori
Rice-Shapiro TheoremRice-Shapiro Theorem
QDocQ45863
Whatever a computable cognitive scientist could verify about an arbitrary computer’s input-output behavior by formally analyzing its program…
Rice-Shapiro TheoremRice-Shapiro Theorem
QDocQ45863
…could also have been verified by a behaviorist computer who empirically studies only the arbitrary computer’s input-output behavior!
Uncomputable InductionUncomputable Induction
Will see a non-halting machine
Yes!
H N
Verifiable by “ideal agent”.
Uncomputable InductionUncomputable Induction
Will see a non-halting machine
?
Q
H N
Not verifiable by computable agent.
SimilaritySimilarity
Halting ProblemHalting Problem Universal LawUniversal Law
UnverifiableUnverifiable UnverifiableUnverifiable
Demon can fool Demon can fool computablecomputable agent agent
Demon can fool Demon can fool
idealideal agent agent
Answer runs Answer runs beyond beyond formalformal
experienceexperience
Answer runs Answer runs beyond beyond empiricalempirical
experienceexperience
A DifferenceA Difference
Halting ProblemHalting Problem Universal LawUniversal Law
Input endsInput ends Input never endsInput never ends
Can handle some Can handle some examples a priori…examples a priori…
““Internal” problem Internal” problem of induction of induction
eventually catches eventually catches up!up!
Problem of Problem of inductioninduction
Give and TakeGive and Take
Formal inputFormal input Empirical inputEmpirical input
Jaw BreakerJaw Breaker NoodleNoodle
Fits in mouth but Fits in mouth but may never meltmay never melt
May never fit in May never fit in mouthmouth
May never swallowMay never swallow May never swallowMay never swallow. . .
Another Difference?Another Difference?
Formal ScienceFormal Science Empirical ScienceEmpirical Science
IncompletenessIncompleteness Problem of Problem of inductioninduction
Not NecessarilyNot Necessarily
Formal ScienceFormal Science Empirical ScienceEmpirical Science
IncompletenessIncompleteness Problem of Problem of inductioninduction
Add more powerful Add more powerful axiomaxiom Conjecture a theoryConjecture a theory
Who knows if new Who knows if new axiom is consistent axiom is consistent
with old?with old?
Who knows if Who knows if theory is consistent theory is consistent
with data?with data?
Assume it is, keep Assume it is, keep checking, and hopechecking, and hope
Assume it is, keep Assume it is, keep checking, and hopechecking, and hope
ThesisThesisThe problem of induction and the problem of uncomputability are essentially similar.
So the two should be understood similarly.
From the ground upward.
•Unverifiability•“Confirmation”•“Support”•“Coherence”•“Rational” update, etc.
•Uncomputability. •Suck it up•Logic vs. engineering•Try to approximate, etc.
Philosophy: A House DividedPhilosophy: A House Divided
?!
Unified Approach: Gun ControlUnified Approach: Gun Control•Find the truth in the best feasible sense.•So drop the halting requirement if infeasible.
Q
Start with Start with ProblemsProblems, Not , Not MethodsMethods
B
Partition Q over set K of inputs
A C D
Infinite inputFinite input
ConvergenceConvergence
Convergencein limit(in limit)
Convergencewith halting
? ? ? ?A
? ? ? ? ? B C C . . .?A C
No more revisions
revisions
SuccessSuccess
Converge to right answer
BA C D
Don’t converge to wrong answer
Special CasesSpecial Cases
BA BA
BA
Verify A Refute A
Decide A
Special CasesSpecial Cases
= 0 = 1 = 2 . . .
Compute partial
0 1 2 . . .
Theory selection
Dom(
Solutions and OptimalitySolutions and Optimality
•Solution: method that succeeds on each input.•Optimal solution: solves in best possible sense•Solvable problem: has solution.•Problem complexity: best sense of solvability
Halting Bounds RevisionsHalting Bounds Revisions
BA
Verifiability convergence with 1 revision ending with “no”.
Refutability convergence with 1 revision ending with “yes”.
Decidability convergence with 0 revisions.
n-Decision =n rev
Generalization to Generalization to nn RevisionsRevisions
n-Refutation =n+1 rev ending with -A
n-Verification =n+1 rev ending with A
BA
0-ref0-ver
0-dec
Lim-refLim-ver
Empirical ComplexityEmpirical Complexity
Lim-dec
. . .
1-ref1-ver
1-dec
. . .
clopen
or and
Un
derd
ete
rmin
ati
on open closed
0-ref0-ver
0-dec
Lim-refLim-ver
Formal Formal ComplexityComplexity
Lim-dec
. . .
1-ref1-ver
1-dec
. . .
decidable
or and
r.e. Co-r.e.Un
com
pu
tab
ilit
y
0-ref0-ver
0-dec
Lim-refLim-ver
Formal + EmpiricalFormal + Empirical ComplexityComplexity
Lim-dec
. . .
1-ref1-ver
1-dec
. . .
decidable
or and
r.e. Co-r.e.Un
com
pu
tab
ilit
y +
Un
derd
ete
rmin
ati
on
. . .
Ship-shape EpistemologyShip-shape Epistemology
Ideal empirical
Effectiveformal
Effective empirical
. . .
. . .. . .
. . .
Ship-shape EpistemologyShip-shape Epistemology
Topologicalinvariants
Recursiveinvariants
Recursiveinvariants
. . .
. . .. . .
. . .
Ship-shape EpistemologyShip-shape Epistemology
LowerBound
Lower Bound
Combination
. . .
. . .. . .
. . .
Messy, Odd DistinctionMessy, Odd Distinction
EmpiricalBorel
Formalarithmetical
MixedeffectiveBorel
. . .
. . .. . .
. . .
Neat, Natural DistinctionsNeat, Natural Distinctions
EmpiricalBorel
Formalarithmetical
MixedeffectiveBorel
. . .
. . .. . .
Example: Kantian AntinomyExample: Kantian Antinomy
Matter
Finitelydivisible
Infinitelydivisible
Upper Complexity BoundUpper Complexity Bound
I say inf whenyou let me cut.
Finitelydivisible
Infinitelydivisible
Lim-ref Lim-ver
. . .
. . .
fin fin
inf fin
Lower Complexity BoundLower Complexity Bound
Finitelydivisible
Infinitelydivisible
Lim-ref-Lim-ver
Lim-ver-Lim-ref
I let you cut when you say fin.
. . .
. . .
inf fin
inf inf
Purely Formal AnaloguePurely Formal Analogue
666 2145623455
Finitedomain
Infinitedomain
Lim-ref-Lim-ver
Lim-ver-Lim-ref
Purely Formal AnaloguePurely Formal Analogue
666 2145623455
I halt on another input each time he says fin in my opry oary simulation of him ganderin’ at my program.
Finitedomain
Infinitedomain
Lim-ref-Lim-ver
Lim-ver-Lim-ref
. . .
AnalogiesAnalogies
Ideal empirical
Effectiveformal
Effective empirical
. . .
. . .. . .
Finite divisibility
Finitedomain
Finitedivisibility
Mixed Example: PenroseMixed Example: Penrose
UncomputableComputable
Human subjectCognitive scientist
.
???
Empirical IronyEmpirical Irony
UncomputableComputable
Human subjectCognitive scientist
You can verify human computabilityin the limit…
UCUCCCCCCCC…
Empirical IronyEmpirical Irony
UncomputableComputable
Human subjectCognitive scientist
But only if you aren’t computable!
UCUCUCCUCCU…
ComputabilityComputability
Lim-refLim-ver
Lim-dec
Ideal
ComputabilityComputability
Lim-refLim-ver
Lim-dec
Computable
We can verify our computability in the limit… only if we are not computable!
Gold/PutnamGold/PutnamEven assuming computability,you can converge to the true computable behavior only if you are not computable!
0 32 243 . . .
Total computable functions
Uncomputable PredictionsUncomputable Predictions
-TT
(with Oliver Schulte)
•There exists T such that •T makes a unique prediction at each stage;•The predictions are very uncomputable…
Uncomputable PredictionsUncomputable Predictions
-TT
(with Oliver Schulte)
•There exists T such that •T makes a unique prediction at each stage;•The predictions are very uncomputable…•But T is refutable by a computable method!
Moral 1: Rational Moral 1: Rational HobgoblinsHobgoblins
So “a foolish consistency” precludes truth-finding!
Ideal rationality
No consistent
computable
convergent method.
Exists
inconsistent
computable
refuting method.
Exists
consistent
computable
non-convergent
method.
Moral 2: Coup de GraceMoral 2: Coup de Grace•The very idea of insulating deduction from empirical data restricts the truth-finding power of effective science.
TheoremProver
Empirical data n Yes
Q
?
Empirical data
Moral 2: Coup de GraceMoral 2: Coup de Grace•The very idea of insulating deduction from empirical data restricts the truth-finding power of effective science.
TheoremProver
n
n
Yes
Q
•Justification = truth-finding performance.
•Performance depends on problem complexity.
•Formal and empirical complexity are similar.
•Computational epistemology is unified
•Standard epistemology is divided.
•Insistence on division weakens truth-finding performance of effective science.
ConclusionsConclusionsQ
Closing ImageClosing Image
Standard epistemology
Computational epistemology
Q
Q
Closing ImageClosing Image
Standard epistemology
Computational epistemology
Q
Q
I’m rational
Q
Closing ImageClosing Image
Standard epistemology
Computational epistemology
Q
I’m rational
THE ENDTHE END
actual agent
pure structure
actual world How does inquiry proceed?Have we found a correct answer?Will we?Are we guaranteed to?How might we improve?Is guaranteed success achievable?Is guaranteed success achievable in a completely specified, hypothetical problem?Which possible methods achieve solutions?Which maxims preclude solutions?What is necessary for guaranteed success to be achievable?
actual problem
Pure Form of NaturalismPure Form of Naturalism
Every Problem in Its PlaceEvery Problem in Its Place
lim verlim ref
lim dec
cert vercert ref
cert dec
grad vergrad ref
Infinity (Kant’s Antinomy)Quantized conservationUniformitarian geology
Parallel postulateProbability, given existence
Probability existenceConservation laws
FinitenessComputabilityGlobal warmingChaos (M. Harrell)Duhem’s problem
Phenomenal laws Experimental effects
Puzzle-solving adequacy
more“normal”.
more “revolutionary”.
1919thth c. Geology c. Geology
Uniformitarianism (steady-state world)
Catastrophism (progressively cooling world):
Adv
ance
men
tA
dvan
cem
ent
Stonesfield mammals 1814
creation
Stonesfield mammals 1814
1919thth c. Geology c. Geology
Early mammals
new schedule
Early reptiles
new schedule
U
Cschedule
Uniformitarianismschedule new violation.
lim ref lim ver Catastrophism schedule future fossils, the schedule stands
6. The Reliability Regress 6. The Reliability Regress Reliability = success over a broad range of possibilities.But but how can we tell we are in the range of reliability?
actual input stream
“method succeeds”
?
Answer: Answer: Check with Another Check with Another
Method!Method!Same inputs to all
11
H
1 succeeds
22
2 succeeds
33
3 succeeds
Popperian Regress Popperian Regress SolvedSolved
H
Methods never take back a rejection.
Later methods are as reliable as earlier.
Refutes H as reliably as any method in the regress
11
H
1 succeeds
22
2 succeeds
33
3 succeeds
…
Verification Regress Verification Regress SolvedSolved
H
Methods never take back an acceptance.
Later methods are as reliable as earlier.
…
11
H
1 succeeds
22
2 succeeds
33
3 succeeds
…Refutes H in the limit as reliably as any method in the regress.
Observed freq too close to p
“too close” set closer
“too close” set closer
p
= p
Observed freq too close to p
Background: a probability exists.Background: a probability exists. Hypothesis: its value is exactly Hypothesis: its value is exactly pp..
Frequentist ProbabilityFrequentist Probability
lim ref
Refuted with auxiliaries
New auxiliariesNew auxiliaries
H
H
Refuted with auxiliaries
Background: there exist correct Background: there exist correct auxiliaries with respect to which auxiliaries with respect to which isolated isolated HH is refutable. is refutable.
Duhem’s ProblemDuhem’s Problem
lim ver
Ref Ver
……neither [the world’s] finitude nor its infinitude can neither [the world’s] finitude nor its infinitude can be contained in experience, because [the be contained in experience, because [the following are impossible]:following are impossible]:
1.1. experience … of an experience … of an infinite space…infinite space…, ,
2.2. or again, of the or again, of the boundaryboundary of the world by an of the world by an empty space …empty space …
ProlegomenaProlegomena, Section 52c., Section 52c.
Antinomy of Pure Antinomy of Pure ReasonReason
Experience
??
moreInfinite
Finite wait for morewait for more
more
Antinomy of Pure Antinomy of Pure ReasonReason
Infinityposition
Intuition of more space.
lim ref lim ver Boundedness position intuitions, the position is not extended.
Parallel PostulateParallel Postulate
?
new pairnew pair
pp
pp
intersectintersect
lim ref
“unsolvable puzzle”
RearticulateRearticulatead
ad
“unsolvable puzzle”
Puzzle-solving adequacyPuzzle-solving adequacy: some : some articulation resolves by its standards articulation resolves by its standards all puzzles that arise all puzzles that arise withinwithin it. it.
Paradigm ChoiceParadigm Choice
lim ver
Ref Ver
ParadigmsParadigms
ad
ad
A1
A2A3
A1 solves 1st puzzle
A1 never solves 3rd
puzzle
A1 solves 2nd puzzle
grad ref““Internal Puzzle-solving adequacy”Internal Puzzle-solving adequacy”:: articulation such thatarticulation such that puzzle that arises within the paradigmpuzzle that arises within the paradigm Stage by which diligent effort would resolve Stage by which diligent effort would resolve
the puzzle by the paradigm’s standards the puzzle by the paradigm’s standards without rearticulation.without rearticulation.
Kuhnian MoralKuhnian Moral
Gradual refutability of each alternative does not suffice for convergence to an effective paradigm!
gaplim ref lim ver
ConservationConservation
C
C
R1
R2
R3
R1 unbalanced
Grad ver““Q is conserved in each reaction”Q is conserved in each reaction”:: reactionreaction stage at which the stage at which the QQ in = the in = the QQ out and out and Later stage, no more Later stage, no more QQ is observed in or is observed in or
out out
R1 unbalanced
R2 unbalanced
Cmismatch
new programnew programC
mismatch
lim ver
Ref Ver
CognitivismCognitivism
Computability program such that input the program produces the right output.
““Empirical Irony of Cog Empirical Irony of Cog Sci”Sci”
C
C
Prog 1
Prog 2Prog 3
Prog 1 outputs 1st
datum
Prog 2 outputs 2nd
datum Prog 3 never outputs 3rd
datum
Computability program such that input stage by which the right output is produced.
computer
ideal
grad ref
Transcendental DeductionsTranscendental Deductions
•What kind of problem structure is necessary for a given sort of success?
Certain Verifiability and Certain Verifiability and RefutabilityRefutability
Certain verifiability:Certain verifiability: H H is a union of “cones” of serious possibilities.is a union of “cones” of serious possibilities.Certain refutability:Certain refutability: HH is an intersection of cones of serious possibilities. is an intersection of cones of serious possibilities.
“blue will be seen.”
“green forever”
. . .
cone = all possible
extensions
Sextus Empiricus!Sextus Empiricus!
Non-verifiability: Non-verifiability:
Some input stream in Some input stream in HH never enters a fan included in never enters a fan included in HH..
. . .
H
HHH
Limiting VerifiabilityLimiting Verifiability
Limiting verifiability:Limiting verifiability:
HH is a countable is a countable unionunion of of intersectionsintersections of fans. of fans.
“The inputs will eventually stabilize to blue.”
Transcendental SystemTranscendental System
lim verlim ref
lim dec
cert vercert ref
cert dec
grad ref grad ver
Computable Case Similar!Computable Case Similar!
lim verlim ref
lim dec
cert vercert ref
cert dec
grad refgrad ver
Turing-decidable relations