On 12/9/2023 7:22 PM, Jim Burns wrote:
On 12/8/2023 12:29 AM, olcott wrote:
On 12/7/2023 6:11 PM, olcott wrote:
On 12/7/2023 10:20 AM, olcott wrote:
On 12/6/2023 9:56 PM, olcott wrote:
On 12/6/2023 4:35 PM, Jim Burns wrote:
[...]
...14
Every epistemological antinomy can likewise
be used for a similar undecidability proof...
(Gödel 1931:43-44)
Thus Gödel really screwed up.
Epistemological antinomies
The epistemological antinomy
| This sentence is false
|
is not in Gödel's proof.
| This sentence is false
|
is the blueprint, which guides
Gödel placement of (metaphorically) actual
bricks and mortar.
You live in a building of some kind, I'd bet.
What odds would you give on whether
that building's blueprints are incorporated
into its construction?
If you ripped plaster off walls,
would you find particular sheets paper
holding up waterlines?
In note 14, Gödel is mentioning that
other blueprints can guide the placement of
(metaphorically) actual bricks and mortar
for other proofs.
Nor are those other blueprints incorporated
into those other proofs.
Epistemological antinomies
are neither
...here nor there.
Since no epistemological antinomy can ever be used for
any proof at all Gödel proved that it didn't have a clue
about the subject matter of his paper.
https://liarparadox.org/Tarski_247_248.pdf
Tarski said that he used Gödel as a basis
and in the above link shows that he anchored
his whole proof in the actual Liar Paradox.
*Here is his actual proof*
https://liarparadox.org/Tarski_275_276.pdf
On Saturday, December 9, 2023
at 5:23:02 PM UTC-8, Jim Burns wrote:
On 12/8/2023 12:29 AM, olcott wrote:
Epistemological antinomies
are neither
...here nor there.
If, say,
you don't have a language with
quantifiers and truth values,
but only a language of terms that
as words
can't refer to themselves,
is there any way to arrive at
the Liar paradox at all?
If, say,
you don't have a language with
quantifiers and truth values,
but only a language of terms that
as words
can't refer to themselves,
is there any way to arrive at
the Liar paradox at all?
How about the diagonal method| a.k.a. anti-diagonalization?
in such a setting,
Goedel's does, involve finding
the wrong-ish term, which
you wouldn't necessarily bar just because
it augments what you said to provide
a counterexample.
Jim or Mr. or Dr. Burns as the case may be,
On 12/10/2023 9:27 AM, Jim Burns wrote:
On 12/9/2023 10:23 PM, Ross Finlayson wrote:
On Saturday, December 9, 2023
at 5:23:02 PM UTC-8, Jim Burns wrote:
On 12/8/2023 12:29 AM, olcott wrote:
Epistemological antinomies
are neither
...here nor there.
If, say,
you don't have a language with
quantifiers and truth values,
but only a language of terms that
as words
can't refer to themselves,
is there any way to arrive at
the Liar paradox at all?
Gödel doesn't arrive at the Liar paradox.
Gödel uses the Liar paradox as a map.
Epimenides addresses
| Each truth isn't this.
Gödel addresses
| Each F.conclusion isn't this.
QuinedNoProofꟳ("QuinedNoProofꟳ(u)"):
| Each F.conclusion isn't
| QuinedNoProofꟳ("QuinedNoProofꟳ(u)")
QuinedNoProofꟳ("H(u)"):
| Each F.conclusion isn't H("H(u)").
QuinedNoProofꟳ("H(u)"):
| ∃y: Quined("H(u)",y) ∧ NoProofꟳ(y)
Quined("H(u)",y):
| Subst("H(u)","H(u)",y)
Subst("H(u)","K(u)",y)
| y = "H("K(u)")"
If, say,
you don't have a language with
quantifiers and truth values,
but only a language of terms that
as words
can't refer to themselves,
is there any way to arrive at
the Liar paradox at all?
Some formal languages cannot express
Subst(x,x,y) and Proves(z,y)
We can't use Gödel's scheme to
prove they're incomplete.
Some theories can be proven complete.
Some of which are surprisingly expressive.
Like Presburger arithmetic: induction and
addition, but not multiplication.
A language without truth values?
The only thing I can imagine would
be a language which can't _say_
"true" or "false"
It doesn't matter what we say.
Call truth values "grue" and "bleen".
A language without quantifiers sounds like
it can't say much.
A language without the Liar because
it says so little sounds like
it would surprise no one.
Do you mean no _explicit_ quantifiers?
This is implicitly quantified
| Little lambs eat ivy.
How about the diagonal method| a.k.a. anti-diagonalization?
Without quantifiers? How?
in such a setting,
Goedel's does, involve finding
the wrong-ish term, which
you wouldn't necessarily bar just because
it augments what you said to provide
a counterexample.
Gödel's
QuinedNoProofꟳ("QuinedNoProofꟳ(u)"):
is already a counter-example to
the claim that all sentences in F
have proofs or disproofs.
You're asking about a counter-counter-example.
A counter-counter-example is not a thing.
Jim or Mr. or Dr. Burns as the case may be,
Good heavens, no Dr!
"I am only an egg."
Some questions I can answer, even so.
You can tell the ones I can't answer
by the way I don't answer them.
*Tarski anchors his proof in the Liar Paradox* https://liarparadox.org/Tarski_247_248.pdf
*Tarski's actual proof*
https://liarparadox.org/Tarski_275_276.pdf
Here is the Tarski Undefinability Theorem proof
(1) x ∉ Provable if and only if p // assumption
(2) x ∈ True if and only if p // assumption
(3) x ∉ Provable if and only if x ∈ True. // derived from (1) and (2)
(4) either x ∉ True or x̄ ∉ True; // axiom: ~True(x) ∨ ~True(~x)
(5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x) (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: Provable(~x) → True(~x)
(7) x ∈ True
(8) x ∉ Provable
(9) x̄ ∉ Provable
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