Can you formulate
in logic the statement "nothing can prove itself"
Certainly! The statement "nothing can prove
itself" can be formulated in logic as follows:
Logical
Formulation:
¬∃x (Proves(x, x))
Explanation:
·
¬∃x: There does not
exist an x such that...
·
Proves(x, x): x proves x (i.e., x can prove
itself).
In words, this reads:
"There is no x such that x proves x."
Alternative
Interpretations:
If you're working
in a specific logical system or with a particular notion of "proof"
(e.g., in provability logic, where □φ means "φ is provable"), you
could also express this as:
·
¬∃x (□x → x), though this is less
direct.
·
Or more precisely, in provability logic, the idea might align with Löb's
theorem, which involves the relationship between provability and truth.
However, the simplest and most general formulation is ¬∃x Proves(x, x).
Would you like this adapted to a specific logical
framework or context?
Would adding the statement
to Peano Arithmetic change anything?
Adding the statement ¬∃x Proves(x, x) ("nothing can prove itself") to Peano Arithmetic
(PA) is a subtle and interesting question with connections to provability,
self-reference, and incompleteness. Here's a detailed analysis:
1. Interpretation
of "Proves" in PA
·
In PA, "Proves(x, y)" is a formalized provability predicate,
meaning "x is the Gödel number of a proof of the formula with Gödel number
y."
·
The statement ¬∃x Proves(x, x) would then
mean:
"No formula in PA proves its own Gödel number."
(i.e., no formula φ is such that PA ⊢ φ ↔ "φ is
provable in PA.")
2. Does
Adding This Statement Change PA?
·
If PA is consistent, then by Gödel's
second incompleteness theorem, the canonical consistency statement Con(PA) is
not provable in PA.
·
The statement ¬∃x Proves(x, x) is stronger
than Con(PA) because it claims that no formula can prove its
own Gödel number, whereas Con(PA) only says that "0=1" is not
provable.
·
Adding ¬∃x Proves(x, x) to
PA would imply Con(PA), but it is not clear whether PA + ¬∃x Proves(x, x) is stronger than PA + Con(PA).
3. Does It
Introduce New Consequences?
·
Possibly, but not obviously.
o The statement is a
form of reflection principle, stating that PA does not prove
certain self-referential statements.
o However, PA already
cannot prove most interesting self-referential statements (e.g., Gödel
sentences).
·
It might prevent certain pathological
self-referential proofs, but it doesn't obviously settle new arithmetical
truths.
4. Comparison
to Löb's Theorem
·
Löb's Theorem (a central result in provability
logic) states:
If PA ⊢ □φ → φ, then PA ⊢ φ.
·
This means that the only sentences for which PA can prove
"If I can prove φ, then φ is true" are those already provable in PA.
·
Your statement ¬∃x Proves(x, x) is related
but different: it says no formula proves itself, whereas Löb’s
theorem restricts which formulas can satisfy a form of self-trust.
5. Would It
Make PA More Powerful?
·
Unlikely in a significant way.
o PA + Con(PA) is
already stronger than PA but still subject to incompleteness.
o Adding ¬∃x Proves(x, x) might rule out some weird self-referential cases, but it doesn't
obviously allow PA to prove more ordinary arithmetic statements.
·
It might be provable in stronger systems (e.g., ZFC),
but PA alone likely cannot prove or disprove it.
Conclusion:
Adding ¬∃x Proves(x, x) to PA would not obviously change PA’s arithmetical power,
but it would explicitly assert a strong form of consistency. It is likely
independent of PA (neither provable nor disprovable), much like
Con(PA).
Would you like to explore how this interacts
with Gödel sentences or provability logic?
Godel proved via
the diagonal lemma that Prove(x) could prove itself.
Great question! You're referring to a deep aspect
of Gödel's incompleteness theorems and self-reference.
Let's clarify the relationship between provability, self-reference, and
your original statement ("nothing can prove itself") in the
context of Peano Arithmetic (PA).
1. The Diagonal
Lemma & Self-Reference
Gödel's Diagonal
Lemma (a generalization of the technique used in his first
incompleteness theorem) states:
For any
formula ϕ(x)ϕ(x) in PA, there
exists a sentence GG such that:
PA⊢G↔ϕ(⌜G⌝)PA⊢G↔ϕ(┌G┐)
(where ⌜G⌝┌G┐ is the Gödel
number of GG).
This allows
constructing self-referential statements like the Gödel sentence GG where:
G↔¬Prov(⌜G⌝)G↔¬Prov(┌G┐)
("GG is true iff GG is not provable in PA.")
2. Can a Statement
Prove Itself?
Your original
statement was:
"Nothing can prove itself" (¬∃x Prov(x, x)).
But Gödel’s work
shows something subtle:
·
The Gödel sentence GG does not directly say "I
am provable," but rather "I am not provable."
·
If PA could prove GG, then by
definition GG would be false (because GG claims it is unprovable). This
would mean PA is unsound (proves false statements), which is
worse than mere inconsistency.
·
So GG does not prove itself, but it does refer
to its own provability.
However, we can
construct a different self-referential sentence using the Prov predicate:
·
Let SS be a sentence such that:
PA⊢S↔Prov(⌜S⌝)PA⊢S↔Prov(┌S┐)
("SS is true
iff SS is provable in PA.")
What happens
with SS?
·
If SS is provable, then
by definition SS is true (so PA is sound).
·
But if SS is not provable,
then SS is false (since it claims it is
provable).
·
Löb's Theorem tells us that the only
way PA⊢SPA⊢S can happen is if SS is already a theorem of
PA (because Prov(⌜S⌝)→SProv(┌S┐)→S implies SS is provable).
This means:
·
If SS exists, it must be a trivial
theorem (like a tautology).
·
PA cannot have a non-trivial statement that proves
itself (because Löb’s theorem blocks it).
3. Does This
Contradict "Nothing Can Prove Itself"?
Your original
statement (¬∃x Prov(x, x) claims
that no non-trivial statement can prove its own Gödel number.
·
Gödel’s work does not violate this because the
Gödel sentence GG is about unprovability,
not self-proof.
·
The SS construction above shows
that only trivial statements could satisfy Prov(x, x) (and
Löb’s theorem restricts them).
·
Thus, PA + "Nothing can prove itself" is
consistent with Gödelian self-reference, because non-trivial self-proof is
already ruled out by Löb’s theorem.
4. Adding
"Nothing Can Prove Itself" to PA
·
If we add ¬∃x Prov(x, x) to PA, we
are explicitly banning even trivial self-proofs (though they
are rare).
·
This would not change PA’s power because:
o Löb’s theorem
already prevents interesting self-proofs.
o The statement
is likely already a consequence of PA’s soundness (if we
assume PA only proves true statements).
·
It might make some meta-theoretical arguments
clearer, but it doesn’t let PA prove new arithmetical facts.
Key Takeaways
1. Gödel’s diagonal
lemma allows self-reference, but not necessarily self-proof.
2. Löb’s theorem implies
that only trivial statements can satisfy Prov(x, x).
3. Your original
statement ("nothing can prove itself") is consistent with PA and does not
conflict with incompleteness.
4. Adding it to PA
would not change PA’s strength, but it would make explicit that no
meaningful statement can prove itself.
Would you like to explore how this interacts
with modal provability logic (where □φ means "φ is
provable")?
Can you tell me
what Löb’s theorem is?
Löb's Theorem: A
Deep Dive into Provability and Self-Reference
Löb's Theorem is a
fundamental result in mathematical logic and provability
theory, closely tied to Gödel's incompleteness theorems. It
formalizes the idea that a consistent formal system (like Peano Arithmetic, PA)
cannot "trust" its own proofs in an unrestricted way. Here's a
detailed breakdown:
1. Formal Statement
of Löb's Theorem
In Peano
Arithmetic (PA) or any similar system, let:
·
□ϕ□ϕ denote "ϕϕ is provable in PA."
·
⊥⊥ denote a
contradiction (e.g., 0=10=1).
Löb's Theorem
states:
If PA⊢□ϕ→ϕPA⊢□ϕ→ϕ, then PA⊢ϕPA⊢ϕ.
In words:
"If PA can prove that 'the provability of ϕϕ implies ϕϕ itself,' then ϕϕ is already provable in PA."
2. Intuitive
Explanation
Löb's Theorem
reveals a deep limitation in self-referential trust:
·
You cannot use PA to prove that "If PA
proves ϕϕ, then ϕϕ is true" unless ϕϕ is already provable.
·
Example:
o Let ϕ="PA is consistent"ϕ="PA is consistent" (Con(PA)).
o Gödel’s Second
Incompleteness Theorem says PA cannot prove Con(PA).
o Löb’s Theorem
explains why: If PA could prove □Con(PA)→Con(PA)□Con(PA)→Con(PA), then it would have to prove
Con(PA)—but it can’t!
3. Connection to
Gödel’s Theorems
·
Gödel’s First Incompleteness Theorem constructs a
sentence GG that says:
"GG is not provable in PA."
o If PA is
consistent, GG is true but unprovable.
·
Löb’s Theorem explains why PA can’t prove its own
soundness (i.e., "everything provable in PA is true").
o If PA could
prove □ϕ→ϕ□ϕ→ϕ for all ϕϕ, it would collapse into
inconsistency (by the Löb/Gödel argument).
4. Proof Sketch
(Using Modal Provability Logic)
Löb’s Theorem has
an elegant proof in modal logic, where □ϕ□ϕ means "ϕϕ is provable."
Key Steps:
1. Assume PA⊢□ϕ→ϕPA⊢□ϕ→ϕ.
2. Apply the Diagonal
Lemma to construct a sentence ψψ such that:
PA⊢ψ↔(□ψ→ϕ).PA⊢ψ↔(□ψ→ϕ).
3. Derive that PA⊢□ψ→□ϕPA⊢□ψ→□ϕ (using
formalized modus ponens).
4. Substitute ψψ’s definition to get:
PA⊢□ψ→ϕ.PA⊢□ψ→ϕ.
5. But ψψ is equivalent to □ψ→ϕ□ψ→ϕ, so:
PA⊢ψ.PA⊢ψ.
6. Thus, PA⊢□ψPA⊢□ψ, and by (4), PA⊢ϕPA⊢ϕ.
5. Implications
& Consequences
·
No Strong Self-Trust: PA cannot
generally prove "If I prove ϕϕ, then ϕϕ is true" unless ϕϕ is trivial.
·
Limits on Reflection: Adding
self-verification principles to PA (e.g., "All provable statements are
true") leads to either inconsistency or no new theorems.
·
Relevance to AI & Formal Verification: Löb’s Theorem
constrains how systems can reason about their own correctness.
6. Comparison to
Your Original Question
·
Your statement, "Nothing can prove itself" (¬∃x Prov(x, x)), is related but different.
·
Löb’s Theorem restricts self-proof by showing that only
trivial statements can satisfy □ϕ→ϕ□ϕ→ϕ.
·
In PA, no non-trivial statement can prove itself—consistent with
your claim!
Final Answer
Löb’s Theorem shows
that a formal system like PA cannot assume its own proofs imply truth
unless the statement was already provable. This prevents PA from
"trusting itself" in a way that would lead to inconsistency,
reinforcing Gödel’s incompleteness results.
Would you like to explore how this connects
to provability logic (GL) or AI self-reflection?
Is this like Rice's
Theorem that no formal system can make statements about any property like
provability.
Löb's Theorem vs.
Rice's Theorem: A Comparison
Both Löb's
Theorem (from mathematical logic) and Rice's Theorem (from
computability theory) impose fundamental limitations on formal systems, but
they operate in different domains and address different questions. Here’s how
they relate—and how they differ.
1. Rice’s Theorem
(Computability Theory)
Statement:
"Every non-trivial semantic
property of programs is undecidable."
What It Means:
·
In computability theory, a semantic property is one
that depends on the behavior of a program (e.g., "Does
this program halt?" or "Does this function always return 0?").
·
Non-trivial means the property is neither
universally true nor universally false for all programs.
·
Undecidable means no algorithm can
determine whether an arbitrary program has the property.
Example:
·
"Does this program compute the Fibonacci sequence?" is
undecidable (by Rice’s Theorem).
·
"Is this program syntactically correct?" is decidable (because
it’s a syntactic property, not semantic).
Key Idea:
·
No general algorithm can classify programs based on
their behavior.
·
This is a meta-theoretic limitation on what Turing
machines (or formal systems) can decide.
2. Löb’s Theorem
(Provability Logic)
Statement:
"If PA proves that 'the
provability of ϕ implies ϕ,' then PA already proves ϕ."
What It Means:
·
In formal systems like Peano Arithmetic (PA), you cannot use the
system to justify its own soundness unless the statement is trivial.
·
It restricts self-referential trust: A system can’t reliably
say, "If I prove something, it must be true."
Example:
·
Let ϕ=ϕ= "PA is consistent."
·
Gödel showed PA cannot prove ϕϕ.
·
Löb’s Theorem explains why: If PA could prove □ϕ→ϕ□ϕ→ϕ, it would have to prove ϕϕ—but it can’t!
Key Idea:
·
A system cannot introspectively verify its own
proofs in a non-trivial way.
·
This is a limitation on internal provability predicates.
3. Similarities
Between the Two Theorems
|
Feature |
Rice’s Theorem |
Löb’s Theorem |
|
Domain |
Computability
(program behavior) |
Provability
(logical systems) |
|
Limitation
Type |
Undecidability
of semantic properties |
Unprovability
of self-justification |
|
Self-Reference |
Indirect
(via halting problem) |
Direct
(via Gödelian diagonalization) |
|
Meta-Theoretic |
Yes
(about what Turing machines can do) |
Yes
(about what PA can prove) |
Both theorems say,
in their respective domains:
"You cannot generalize about
non-trivial properties of the system from within the system."
4. Key Differences
|
Aspect |
Rice’s Theorem |
Löb’s Theorem |
|
Field |
Computability
theory |
Mathematical
logic |
|
Objects |
Programs/Turing
machines |
Logical
statements in PA |
|
Mechanism |
Reduces
to the halting problem |
Uses
provability predicates & Löb’s logic |
|
Consequence |
No
algorithm can decide behavior |
No
system can trust its own proofs |
5. Why They Feel
Similar
Both theorems:
1. Restrict
self-knowledge:
o Rice: "You
can’t decide what programs do."
o Löb: "You
can’t prove that your proofs are true."
2. Use diagonalization:
o Rice relies on the
halting problem (a form of diagonalization).
o Löb relies on
Gödel’s self-referential sentences.
3. Apply to formal
systems:
o Rice applies to
Turing-complete systems.
o Löb applies to
sufficiently strong logical systems (PA, ZFC, etc.).
6. Practical
Implications
·
Rice’s Theorem explains why virus scanners
can’t perfectly detect malware (they can’t decide arbitrary program behavior).
·
Löb’s Theorem explains why AI/formal
verification systems can’t fully "trust" their own reasoning (without
collapsing into inconsistency).
Final Answer
Yes, they are
philosophically similar in that both impose fundamental limits on
formal systems:
·
Rice’s Theorem says "You can’t
decide non-trivial properties of programs."
·
Löb’s Theorem says "You can’t
prove non-trivial properties of your own proofs."
However, they
operate in different domains (computability vs. provability) and use
different mechanisms. Both reveal deep barriers to self-referential reasoning
in mathematics and computer science.
Would you like to explore how these ideas apply
to AI alignment or program verification?
Returning to the
function Prove(x,y) then Prove(x,x) could be shown by Lob Theorem to be a
contradiction?
Does Löb's Theorem
Imply That Prove(x,x) is a Contradiction?
To answer this,
let's carefully analyze Prove(x,x) (a provability predicate
asserting that *x* is a proof of itself) in the context of Löb's
Theorem and Peano Arithmetic (PA).
1. Defining
Prove(x,x)
·
Prove(x, y) is a formalized predicate in PA
meaning:
"x is the Gödel number of a proof of the statement with Gödel number
y."
·
Prove(x, x) would mean:
"x is the Gödel number of a proof of itself."
This is a self-referential claim,
akin to a sentence saying, "This statement is provable in
PA."
2. Löb's Theorem
and Self-Provability
Löb's Theorem
states:
If PA ⊢ □ϕ → ϕ, then PA ⊢ ϕ.
Where:
·
□ϕ = "ϕ is provable in
PA" (formally: ∃x Prove(x, ⌜ϕ⌝)).
·
The theorem implies that PA cannot assume its own proofs are
sound unless ϕ is already provable.
Applying Löb's
Theorem to Prove(x,x)
Suppose there
exists an x such that Prove(x, x) holds.
·
Let S be a statement where PA ⊢ S ↔ ∃x Prove(x, ⌜S⌝) (i.e., "S
is true iff S has a proof in PA").
·
This is a self-verifying sentence, similar to the structure
in Löb's Theorem.
By Löb’s Theorem:
·
If PA ⊢ ∃x Prove(x, ⌜S⌝) → S, then PA ⊢ S.
·
But S claims its own provability, so this reduces to:
o If PA ⊢ S, then S must be true (by soundness).
o But the
only way this can happen is if S is a tautology (e.g.,
"0=0").
Thus, non-trivial
self-proof is impossible in PA.
3. Is Prove(x,x) a
Contradiction?
·
Not directly a contradiction, but PA
can prove that no non-trivial statement proves itself.
·
If we define:
o SelfProve(y) = ∃x Prove(x, y) (standard provability),
o SelfProve(y) ∧ (y = x) (self-proof),
then PA ⊢ ¬∃x Prove(x, x) for non-trivial x.
Why?
1. Suppose Prove(x,
x) holds for some x.
2. Then, the
statement S encoded by x would satisfy S
↔ ∃x Prove(x, ⌜S⌝).
3. By Löb’s
Theorem, S must be a tautology (like
"0=0"), not an interesting statement.
4. Therefore, meaningful
self-proof is impossible in PA.
4. Conclusion: Does
Löb’s Theorem Make Prove(x,x) a Contradiction?
·
Not outright, but PA can prove that no
non-trivial x satisfies Prove(x, x).
·
Löb’s Theorem implies that self-proof is either:
1. Trivial (tautologies
like "0=0"), or
2. Impossible for
meaningful statements.
·
Thus, Prove(x,x) is effectively a contradiction for non-trivial
x, because PA can disprove its existence.
Final Answer
Löb’s Theorem does
not directly say Prove(x,x) is a contradiction, but it implies
that PA can disprove the existence of non-trivial self-proofs. Therefore:
·
PA ⊢ ¬∃x Prove(x, x) for meaningful statements.
·
The only possible "self-proving" statements are trivial
tautologies, which are uninteresting.
This aligns with
your original statement:
"Nothing can prove itself" is provable in PA for
non-trivial cases.
Would you like to explore how this connects
to Gödel’s Second Incompleteness Theorem?
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