Is there a sentence that proves itself does not exist? (Laymen's terms for sci.lang)

[peteolcott]

Philosophy of Logic – Reexamining the Formalized Notion of Truth
https://philpapers.org/archive/OLCPOL.pdf

To put this in laymen's terms all of the truth that can be expressed using
words or math symbols is anchored in sentences that are defined to be true:
“A cat is an animal”. Other true sentences are derived from this basic set:
(1) A cat is an animal.
(2) Animals breath.
(3) Therefore cats breath.

The basic truths of English would be called axioms in math.
The derived truths of English would be called theorems in math.
It turns out that all conceptual truth works this same way.

Here is how the mathematical notation works:
(∃G ∈ Sentences(English) (English ⊢ "Cats breath")
In English means:
{There are sentences of English that prove that "Cats breath"}

∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~(F ⊢ G)))
There is at least one language that has a sentence that says the same thing as its own proof does not exist.

When we assume this formalization of the notion of Truth:
∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))
Then the above sentence says:

∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~True(F, G)))
There is at least one language that has a true sentence that says the same thing as it is not true.

[peteolcott]

Philosophy of Logic – Reexamining the Formalized Notion of Truth
https://philpapers.org/archive/OLCPOL.pdf

The above revised paper is a mandatory prerequisite to the following:

Within the (R. B. BRAITHWAITE 1962) correspondence between
formal proof and deductive inference it is impossible to have
any sound deduction that is not also a formal proof to a
theorem consequence.

Within the Haskell Curry definition of formal system the
semantic truth value of axioms can be propagated to theorem
consequences (because valid deduction is truth preserving)
without the need for any alternative system of representation
such as model theory.

These two views taken together provide the basis for this universal Truth predicate:
∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))

Thus showing that truth cannot possibly diverge from provability,
within this (BRAITHWAITE / Curry) analytical framework.

[peteolcott]

Post by peteolcott
Philosophy of Logic – Reexamining the Formalized Notion of Truth
https://philpapers.org/archive/OLCPOL.pdf
Within the (R. B. BRAITHWAITE 1962) correspondence between
formal proof and deductive inference it is impossible to have
any sound deduction that is not also a formal proof to a
theorem consequence.
Within the Haskell Curry definition of formal system the
semantic truth value of axioms can be propagated to theorem
consequences (because valid deduction is truth preserving)
without the need for any alternative system of representation
such as model theory.
∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))
Thus showing that truth cannot possibly diverge from provability,
within this (BRAITHWAITE / Curry) analytical framework.

thus the following sentence would be unsatisfiable:
∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~(F ⊢ G)))

[DKleinecke]

Post by peteolcott
Philosophy of Logic – Reexamining the Formalized Notion of Truth
https://philpapers.org/archive/OLCPOL.pdf
To put this in laymen's terms all of the truth that can be expressed using
(1) A cat is an animal.
(2) Animals breath.
(3) Therefore cats breath.
The basic truths of English would be called axioms in math.
The derived truths of English would be called theorems in math.
It turns out that all conceptual truth works this same way.
(∃G ∈ Sentences(English) (English ⊢ "Cats breath")
{There are sentences of English that prove that "Cats breath"}
∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~(F ⊢ G)))
There is at least one language that has a sentence that says the same thing as its own proof does not exist.

ERROR - meaningless formal statement

In the symbolic system G is a language and not part of any
model (what you mean by G seems to be G is a model of F).

I think that this is not what you intend G to mean. You
have said F is a Formal System - what then are the members
of Language (F)?

Post by peteolcott
∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))
∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~True(F, G)))
There is at least one language that has a true sentence that says the same thing as it is not true.

There is no need for such a theorem because you have defined
True (F, x) as (F ⊢ x)

[peteolcott]

Post by DKleinecke

Post by peteolcott
Philosophy of Logic – Reexamining the Formalized Notion of Truth
https://philpapers.org/archive/OLCPOL.pdf
To put this in laymen's terms all of the truth that can be expressed using
(1) A cat is an animal.
(2) Animals breath.
(3) Therefore cats breath.
The basic truths of English would be called axioms in math.
The derived truths of English would be called theorems in math.
It turns out that all conceptual truth works this same way.
(∃G ∈ Sentences(English) (English ⊢ "Cats breath")
{There are sentences of English that prove that "Cats breath"}
∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~(F ⊢ G)))
There is at least one language that has a sentence that says the same thing as its own proof does not exist.

ERROR - meaningless formal statement
In the symbolic system G is a language and not part of any
model (what you mean by G seems to be G is a model of F).

That is how Panu Raatikainen told me to say it.
We need to distinguish the language of formal system L(F)
and the formal system F (often equated with the set of
its theorems). Distinct systems may share the same language.

We could also say it like this:
∃F ∈ Formal_Systems (∃G ∈ WFF(F) (G ↔ ~(F ⊢ G)))

Post by DKleinecke
I think that this is not what you intend G to mean. You
have said F is a Formal System - what then are the members
of Language (F)?

The members of a language (generically to include formal and
natural languages) are expressions of this language.

Post by DKleinecke

Post by peteolcott
∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))
∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~True(F, G)))
There is at least one language that has a true sentence that says the same thing as it is not true.

There is no need for such a theorem because you have defined
True (F, x) as (F ⊢ x)

The unsatisfiability of this sentence refutes the conclusion of the

Post by DKleinecke

Post by peteolcott
∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))

G can only be true if G can be proven to be unprovable, thus not true.

Thus proving that there is not G that satisfies the above expression
and thus the 1931 Incompleteness Theorem is refuted.

[DKleinecke]

Post by peteolcott

Post by DKleinecke

Post by peteolcott
Philosophy of Logic – Reexamining the Formalized Notion of Truth
https://philpapers.org/archive/OLCPOL.pdf
To put this in laymen's terms all of the truth that can be expressed using
(1) A cat is an animal.
(2) Animals breath.
(3) Therefore cats breath.
The basic truths of English would be called axioms in math.
The derived truths of English would be called theorems in math.
It turns out that all conceptual truth works this same way.
(∃G ∈ Sentences(English) (English ⊢ "Cats breath")
{There are sentences of English that prove that "Cats breath"}
∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~(F ⊢ G)))
There is at least one language that has a sentence that says the same thing as its own proof does not exist.

ERROR - meaningless formal statement
In the symbolic system G is a language and not part of any
model (what you mean by G seems to be G is a model of F).

That is how Panu Raatikainen told me to say it.
We need to distinguish the language of formal system L(F)
and the formal system F (often equated with the set of
its theorems). Distinct systems may share the same language.
∃F ∈ Formal_Systems (∃G ∈ WFF(F) (G ↔ ~(F ⊢ G)))

It still doesn't work. F is not a member of WFF(F). I think
you need to initially restrict yourself some formal system F.
Then, assuming F, assert
∃G ∈ WFF (G ↔ ~(WFF ⊢ G)))

The meanings of ↔, ~ and ⊢ all are parametric on F. I have
assumed that in A ⊢ B A be can a set of expressions than just
one expression. Your latest BNF for MTT does not define ⊢
that way.

[peteolcott]

Post by DKleinecke

Post by peteolcott

Post by DKleinecke

Post by peteolcott
Philosophy of Logic – Reexamining the Formalized Notion of Truth
https://philpapers.org/archive/OLCPOL.pdf
To put this in laymen's terms all of the truth that can be expressed using
(1) A cat is an animal.
(2) Animals breath.
(3) Therefore cats breath.
The basic truths of English would be called axioms in math.
The derived truths of English would be called theorems in math.
It turns out that all conceptual truth works this same way.
(∃G ∈ Sentences(English) (English ⊢ "Cats breath")
{There are sentences of English that prove that "Cats breath"}
∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~(F ⊢ G)))
There is at least one language that has a sentence that says the same thing as its own proof does not exist.

ERROR - meaningless formal statement
In the symbolic system G is a language and not part of any
model (what you mean by G seems to be G is a model of F).

That is how Panu Raatikainen told me to say it.
We need to distinguish the language of formal system L(F)
and the formal system F (often equated with the set of
its theorems). Distinct systems may share the same language.
∃F ∈ Formal_Systems (∃G ∈ WFF(F) (G ↔ ~(F ⊢ G)))

It still doesn't work. F is not a member of WFF(F). I think
you need to initially restrict yourself some formal system F.
Then, assuming F, assert
∃G ∈ WFF (G ↔ ~(WFF ⊢ G)))

The author of the SEP article specified the first one.
Many people of sci.logic accepted the second one.

Post by DKleinecke
The meanings of ↔, ~ and ⊢ all are parametric on F.

Not at all. Not in the least tiny bit.

Post by DKleinecke
I have
assumed that in A ⊢ B A be can a set of expressions than just
one expression.

I was told (by you and others) that F ⊢ G is correct.
The author of the SEP article specified it as correct.

Post by DKleinecke
Your latest BNF for MTT does not define ⊢
that way.

⊢ specifies the binary predicate Provable(thing A, thing B).
Provable(Snake, Ice_Cream_Cone) evaluates to false.

[DKleinecke]

Post by peteolcott

Post by DKleinecke

Post by peteolcott

Post by DKleinecke

Post by peteolcott
Philosophy of Logic – Reexamining the Formalized Notion of Truth
https://philpapers.org/archive/OLCPOL.pdf
To put this in laymen's terms all of the truth that can be expressed using
(1) A cat is an animal.
(2) Animals breath.
(3) Therefore cats breath.
The basic truths of English would be called axioms in math.
The derived truths of English would be called theorems in math.
It turns out that all conceptual truth works this same way.
(∃G ∈ Sentences(English) (English ⊢ "Cats breath")
{There are sentences of English that prove that "Cats breath"}
∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~(F ⊢ G)))
There is at least one language that has a sentence that says the same thing as its own proof does not exist.

ERROR - meaningless formal statement
In the symbolic system G is a language and not part of any
model (what you mean by G seems to be G is a model of F).

That is how Panu Raatikainen told me to say it.
We need to distinguish the language of formal system L(F)
and the formal system F (often equated with the set of
its theorems). Distinct systems may share the same language.
∃F ∈ Formal_Systems (∃G ∈ WFF(F) (G ↔ ~(F ⊢ G)))

It still doesn't work. F is not a member of WFF(F). I think
you need to initially restrict yourself some formal system F.
Then, assuming F, assert
∃G ∈ WFF (G ↔ ~(WFF ⊢ G)))

The author of the SEP article specified the first one.
Many people of sci.logic accepted the second one.

Post by DKleinecke
The meanings of ↔, ~ and ⊢ all are parametric on F.

Not at all. Not in the least tiny bit.

That probably depends on your definition of formal
system.

Mine is:
There is a set of entities,a non-overlapping set
of operators and pair of things OPDN and CLOSE
which are neither entities nor operators.
Some operators are unary and the rest are binary.
Some unary operators are prefixs and rest are
postfixes.
An expression is a string of things - entities,
operators, OPEN and CLOSE.
A term is an entity or an expression enclosed in
in a OPEN-CLOSE pair.
A factor is a term optionally preceded by prefixes
and optionally followed by postfixes.
An expression is a string of factors separated by
binary operators.

I was sloppy about how strings work but that is all
conventional detail.

What's your definition of a formal system?

[peteolcott]

Post by DKleinecke

Post by peteolcott

Post by DKleinecke

Post by peteolcott

Post by DKleinecke

Post by peteolcott
Philosophy of Logic – Reexamining the Formalized Notion of Truth
https://philpapers.org/archive/OLCPOL.pdf
To put this in laymen's terms all of the truth that can be expressed using
(1) A cat is an animal.
(2) Animals breath.
(3) Therefore cats breath.
The basic truths of English would be called axioms in math.
The derived truths of English would be called theorems in math.
It turns out that all conceptual truth works this same way.
(∃G ∈ Sentences(English) (English ⊢ "Cats breath")
{There are sentences of English that prove that "Cats breath"}
∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~(F ⊢ G)))
There is at least one language that has a sentence that says the same thing as its own proof does not exist.

ERROR - meaningless formal statement
In the symbolic system G is a language and not part of any
model (what you mean by G seems to be G is a model of F).

That is how Panu Raatikainen told me to say it.
We need to distinguish the language of formal system L(F)
and the formal system F (often equated with the set of
its theorems). Distinct systems may share the same language.
∃F ∈ Formal_Systems (∃G ∈ WFF(F) (G ↔ ~(F ⊢ G)))

It still doesn't work. F is not a member of WFF(F). I think
you need to initially restrict yourself some formal system F.
Then, assuming F, assert
∃G ∈ WFF (G ↔ ~(WFF ⊢ G)))

The author of the SEP article specified the first one.
Many people of sci.logic accepted the second one.

Post by DKleinecke
The meanings of ↔, ~ and ⊢ all are parametric on F.

Not at all. Not in the least tiny bit.

That probably depends on your definition of formal
system.
There is a set of entities,a non-overlapping set
of operators and pair of things OPDN and CLOSE
which are neither entities nor operators.
Some operators are unary and the rest are binary.
Some unary operators are prefixs and rest are
postfixes.
An expression is a string of things - entities,
operators, OPEN and CLOSE.
A term is an entity or an expression enclosed in
in a OPEN-CLOSE pair.
A factor is a term optionally preceded by prefixes
and optionally followed by postfixes.
An expression is a string of factors separated by
binary operators.
I was sloppy about how strings work but that is all
conventional detail.
What's your definition of a formal system?

Peter Percival pointed me to this one:
https://en.wikipedia.org/wiki/Theory_(mathematical_logic)

This (and your comment) gave me a huge boost in that I can now
finally anchor a half of my universal truth predicate in the
credible source of Haskell Curry. (I also bought his book).

I figured out on my own that I could anchor the other half
in R. B. Braithwaite. Summing it all up as this:

Axioms are True:
The elementary statements which belong to T are called the elementary
theorems of T and said to be true.
(Haskell Curry, Foundations of Mathematical Logic, 2010).

Formal proofs express deductive Inference:
The chain of symbolic manipulations in the calculus corresponds to
and represents the chain of deductions in the deductive system.
(R. B. BRAITHWAITE 1962).

∴ Formal proofs to theorem consequences represent sound deductions to true conclusions.
Formalized as: ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))

[peteolcott]

Post by DKleinecke

Post by peteolcott

Post by DKleinecke

Post by peteolcott
Philosophy of Logic – Reexamining the Formalized Notion of Truth
https://philpapers.org/archive/OLCPOL.pdf
To put this in laymen's terms all of the truth that can be expressed using
(1) A cat is an animal.
(2) Animals breath.
(3) Therefore cats breath.
The basic truths of English would be called axioms in math.
The derived truths of English would be called theorems in math.
It turns out that all conceptual truth works this same way.
(∃G ∈ Sentences(English) (English ⊢ "Cats breath")
{There are sentences of English that prove that "Cats breath"}
∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~(F ⊢ G)))
There is at least one language that has a sentence that says the same thing as its own proof does not exist.

ERROR - meaningless formal statement
In the symbolic system G is a language and not part of any
model (what you mean by G seems to be G is a model of F).

That is how Panu Raatikainen told me to say it.
We need to distinguish the language of formal system L(F)
and the formal system F (often equated with the set of
its theorems). Distinct systems may share the same language.
∃F ∈ Formal_Systems (∃G ∈ WFF(F) (G ↔ ~(F ⊢ G)))

It still doesn't work. F is not a member of WFF(F). I think
you need to initially restrict yourself some formal system F.
Then, assuming F, assert
∃G ∈ WFF (G ↔ ~(WFF ⊢ G)))
The meanings of ↔, ~ and ⊢ all are parametric on F.

Think of a generic ontological engineering system that has specific
algorithms associated with all of the logical operators such that
they always operate with their conventional symbolic logic semantics.

A formal system that only has a single axiom: A (the Fonz system?)
would still be able to express WFF within this system. So the
formal language comes with an alphabet (UTF-8) rules-of-inference
specified by the logical operators. Formation rules (MTT syntax).

All the we need to make this into a formal system is to add axioms.

Post by DKleinecke
I have
assumed that in A ⊢ B A be can a set of expressions than just
one expression. Your latest BNF for MTT does not define ⊢
that way.

[DKleinecke]

Post by peteolcott

Post by DKleinecke

Post by peteolcott

Post by DKleinecke

Post by peteolcott
Philosophy of Logic – Reexamining the Formalized Notion of Truth
https://philpapers.org/archive/OLCPOL.pdf
To put this in laymen's terms all of the truth that can be expressed using
(1) A cat is an animal.
(2) Animals breath.
(3) Therefore cats breath.
The basic truths of English would be called axioms in math.
The derived truths of English would be called theorems in math.
It turns out that all conceptual truth works this same way.
(∃G ∈ Sentences(English) (English ⊢ "Cats breath")
{There are sentences of English that prove that "Cats breath"}
∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~(F ⊢ G)))
There is at least one language that has a sentence that says the same thing as its own proof does not exist.

ERROR - meaningless formal statement
In the symbolic system G is a language and not part of any
model (what you mean by G seems to be G is a model of F).

That is how Panu Raatikainen told me to say it.
We need to distinguish the language of formal system L(F)
and the formal system F (often equated with the set of
its theorems). Distinct systems may share the same language.
∃F ∈ Formal_Systems (∃G ∈ WFF(F) (G ↔ ~(F ⊢ G)))

It still doesn't work. F is not a member of WFF(F). I think
you need to initially restrict yourself some formal system F.
Then, assuming F, assert
∃G ∈ WFF (G ↔ ~(WFF ⊢ G)))
The meanings of ↔, ~ and ⊢ all are parametric on F.

Think of a generic ontological engineering system that has specific
algorithms associated with all of the logical operators such that
they always operate with their conventional symbolic logic semantics.
A formal system that only has a single axiom: A (the Fonz system?)
would still be able to express WFF within this system. So the
formal language comes with an alphabet (UTF-8) rules-of-inference
specified by the logical operators. Formation rules (MTT syntax).
All the we need to make this into a formal system is to add axioms.

You also need at least one rule of inference.

[peteolcott]

Post by DKleinecke

Post by peteolcott

Post by DKleinecke

Post by peteolcott

Post by DKleinecke

Post by peteolcott
Philosophy of Logic – Reexamining the Formalized Notion of Truth
https://philpapers.org/archive/OLCPOL.pdf
To put this in laymen's terms all of the truth that can be expressed using
(1) A cat is an animal.
(2) Animals breath.
(3) Therefore cats breath.
The basic truths of English would be called axioms in math.
The derived truths of English would be called theorems in math.
It turns out that all conceptual truth works this same way.
(∃G ∈ Sentences(English) (English ⊢ "Cats breath")
{There are sentences of English that prove that "Cats breath"}
∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~(F ⊢ G)))
There is at least one language that has a sentence that says the same thing as its own proof does not exist.

ERROR - meaningless formal statement
In the symbolic system G is a language and not part of any
model (what you mean by G seems to be G is a model of F).

That is how Panu Raatikainen told me to say it.
We need to distinguish the language of formal system L(F)
and the formal system F (often equated with the set of
its theorems). Distinct systems may share the same language.
∃F ∈ Formal_Systems (∃G ∈ WFF(F) (G ↔ ~(F ⊢ G)))

It still doesn't work. F is not a member of WFF(F). I think
you need to initially restrict yourself some formal system F.
Then, assuming F, assert
∃G ∈ WFF (G ↔ ~(WFF ⊢ G)))
The meanings of ↔, ~ and ⊢ all are parametric on F.

Think of a generic ontological engineering system that has specific
algorithms associated with all of the logical operators such that
they always operate with their conventional symbolic logic semantics.
A formal system that only has a single axiom: A (the Fonz system?)
would still be able to express WFF within this system. So the
formal language comes with an alphabet (UTF-8) rules-of-inference
specified by the logical operators. Formation rules (MTT syntax).
All the we need to make this into a formal system is to add axioms.

You also need at least one rule of inference.

As I already said the logical operators defined directly in the
language already bring with them the semantics of rules-of-inference.

If you only have the single axiom: A
You can specify the inference: A ∨ A → A

Likewise the placement of non-logical symbols in the grammar
indicate whether they are predicates or functions. Predicates
evaluate to Boolean. Functions evaluate to non-Boolean.

So all that is left to add to the formal language to make
a formal system is axioms.

[Franz Gnaedinger]

Post by peteolcott
"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein

(Allgod, mathematical and natural logic, liar paradox, conclusion;
ambiguity or better flexibility, Einstein and Goedel)

Meanwhile Peter Olcott officially announced that he is God. Being God,
and the only one, he can dismiss the proven theorems of Goedel and Turing
and suffocate sci.logic and sci.lang and other fora by starting ever more
parallel threads.

I see the main problem in academe that has no clear idea of logic, or rather
a one-sided one, regarding mathematical logic as logic per se. Mathematical
logic is the logic of building and maintaining based on the formula a = a
while there is a wider logic formulated by Goethe: All is equal, all unequal ...,
known to artists of all times.

Goedel is very hard to understand for laymen when you consider mathematical
logic the only real logic, but most easy when you have a look at both sides.
Goedel proved that mathematics can't really and completely be separated from
general logic and the principle of equal unequal, it can only be secured
from case to case, for example divisions by zero are forbidden. Why? these
divisions yield infinite, which is equal unequal in itself. If you restrict logic
to mathematical logic you encounter paradoxa, which are quite natural
in the real world.

The problem is academe. Universities are not really universal. Allgod tries
to solve the problem by cramming the realm of wider logic (that includes
for example language) into mathematical logic, hoping he will thus regain
totality and completeness - absolute and complete and total being his
mantra words - and does it for the price of his career and sanity.

If only he would stop cross-posting to sci.lang! (he promised to do so,
but he doesn't stand by his word).

The logic of equal unequal blossoms in language. Allgod can't have that,
He tries to force language into the logic of a = a with his "mathematics
of the meaning of words" that led him nowhere. He castrates language
in the name of mathematical logic, and mathematical logic by dismissing
proven theorems.

I feel entitled to defend sci.lang from those who suffocate it with ever
more parallel threads.

(on the liar paradox)

A Cretan says all Cretans are liars. He is right. Psychologists found that
we are lying many times a day, and in different ways. We humans are liars,
Cretans are humans, ergo they are liars. QED. The famous liar paradox arises
when the natural logic of equal unequal is reduced to the mathematical logic
of a = a. A liar is a liar, alaways lying, only ever lying. But such a person
does not exist in the real world, on the contrary, a professional liar cares
to tell the truth as often as ever possible in order to gain the confidence
of a potential victim.

(conclusion, rigid denial vs Goedel's rigor)

Allgod tries to reduce natural logic to mathematical logic, however,
he can't escape the equal unequal, it haunts him in the form of
proven = proven = not proven. He replaces Goedel's rigor with
his naive but rigid denial. Every advice to come down from his trip
and write a modest but useful program that may then be extended
was in vain. He is a satellite that flies too low, destined to burn out
in the atmosphere. All warning failed.

(ambiguity or better flexibility)

The end of the world is near! This may be the warning of a preacher:
prepare yourself. Or it may be the exclamation of a housewife
addressing her hubby: How many times did I tell you to screw
back on the lid of the toothpaste tube after brushing your teeth?
I told you so for years. Now, finally, I see that you oblige. The end
of the world is near!

The same exclamation can have very different meanings. In the first
case it expresses a secret hope: if the apocalypse arrives in our
lifetime we shall go to heaven directly, without having to die,
so prepare yourself. And in the second case it formulates a reproach
in an exaggerated manner that makes it funny: you are so incredibly
obstinate that only a shockwave can move you, must be the approaching
end of the world ...

Ambiguity or better flexibility is the genius of natural language
that escapes from the cage of any formal system.

(Einstein and Goedel)

Einstein and Goedel were good friends who held each other's work
in high esteem. What would Einstein say about a naive Goedel denier?