Discussion:
Refutation of the first Incompleteness Theorem with Minimal Type Theory (MTT)
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Pete Olcott
2017-06-08 03:30:06 UTC
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<p style="margin-bottom: 0in"><font face="Segoe UI Symbol,
sans-serif"><font style="font-size: 11pt" size="2"><b>G(
~∃Γ ⊂ PM (Γ ⊢ G) )   // </b></font></font><font
face="Segoe UI Symbol, sans-serif"><font style="font-size:
11pt" size="2"><b><span style="background: #ffff00">Incompleteness
Theorem as a named predicate</span></b></font></font><font
face="Segoe UI Symbol, sans-serif"><font style="font-size:
11pt" size="2"><b>
</b></font></font>
</p>
</p>
<p><font face="Segoe UI Symbol">Minimal Type Theory Directed Acyclic
Graph of the above expression<br>
01 ~       (2)        // G is an alias for this node<br>
02 ∃       (3)(6)  <br>
03 ⊂       (4)(5) <br>
04 Γ  <br>
05 PM <br>
06 ⊢       (4)(1)   // cycle indicates error: evaluation
infinite loop<br>
</font></p>
<p><font face="Segoe UI Symbol">Further elaboration provided on link
below: <br>
<a class="moz-txt-link-freetext" href="http://LiarParadox.org/Provability_with_Minimal_Type_Theory.pdf">http://LiarParadox.org/Provability_with_Minimal_Type_Theory.pdf</a>
<br>
</font></p>
<p><b><font face="Segoe UI Symbol">Copyright 2017 Pete Olcott </font></b><b><br>
</b></p>
<br>
<div class="moz-signature">-- <br>
<p class="western" style="margin-bottom: 0in"><b><font
face="Arial, sans-serif"><font style="font-size: 12pt"
size="2">(Γ
</font><font style="font-size: 12pt" size="2">⊨ </font><sub><font
style="font-size: 8pt" size="2">FS</font></sub><font
style="font-size: 12pt" size="2">
A) ≡ (</font><font style="font-size: 12pt" size="2">Γ </font><font
style="font-size: 12pt" size="2">⊢
</font><sub><font style="font-size: 8pt" size="2">FS</font></sub><font
style="font-size: 12pt" size="2">
A)</font></font></b></p>
<p class="western" style="margin-bottom: 0in"><br>
</p>
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DKleinecke
2017-06-08 04:50:23 UTC
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G(
~∃Γ ⊂ PM (Γ ⊢ G) )   // Incompleteness
Theorem as a named predicate
Minimal Type Theory Directed Acyclic
Graph of the above expression
01 ~       (2)        // G is an alias for this node
02 ∃       (3)(6) 
03 ⊂       (4)(5)
04 Γ 
05 PM
06 ⊢       (4)(1)   // cycle indicates error: evaluation
infinite loop
Further elaboration provided on link
http://LiarParadox.org/Provability_with_Minimal_Type_Theory.pdf
Copyright 2017 Pete Olcott
--

⊨ FS
A) ≡ (Γ ⊢
FS
A)
If this has any connection to language it is hidden from me.
Arnaud Fournet
2017-06-08 05:36:29 UTC
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Post by DKleinecke
G(
~∃Γ ⊂ PM (Γ ⊢ G) )   // Incompleteness
Theorem as a named predicate
Minimal Type Theory Directed Acyclic
Graph of the above expression
01 ~       (2)        // G is an alias for this node
02 ∃       (3)(6) 
03 ⊂       (4)(5)
04 Γ 
05 PM
06 ⊢       (4)(1)   // cycle indicates error: evaluation
infinite loop
Further elaboration provided on link
http://LiarParadox.org/Provability_with_Minimal_Type_Theory.pdf
Copyright 2017 Pete Olcott
--

⊨ FS
A) ≡ (Γ ⊢
FS
A)
If this has any connection to language it is hidden from me.
It very much looks like some ads for a Manual of Engineering for Loonies.
That kind of posts. Spam in short.
A.
Pete Olcott
2017-06-08 13:47:05 UTC
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Post by Arnaud Fournet
Post by DKleinecke
G(
~∃Γ ⊂ PM (Γ ⊢ G) ) // Incompleteness
Theorem as a named predicate
Minimal Type Theory Directed Acyclic
Graph of the above expression
01 ~ (2) // G is an alias for this node
02 ∃ (3)(6)
03 ⊂ (4)(5)
04 Γ
05 PM
06 ⊢ (4)(1) // cycle indicates error: evaluation
infinite loop
Further elaboration provided on link
http://LiarParadox.org/Provability_with_Minimal_Type_Theory.pdf
Copyright 2017 Pete Olcott
--

⊨ FS
A) ≡ (Γ ⊢
FS
A)
If this has any connection to language it is hidden from me.
It very much looks like some ads for a Manual of Engineering for Loonies.
That kind of posts. Spam in short.
A.
https://plato.stanford.edu/entries/compositionality/
The above specifies the math details of sub atomic semantic compositionality.
I am well aware that math details such as this seems to be nothing more than gibberish to most linguists.
The concept of compositionality should be well understood by most linguists.
--
(Γ ⊨ _FS A) ≡ (Γ ⊢ _FS A)
Pete Olcott
2017-06-08 13:34:06 UTC
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Post by DKleinecke
G(
~∃Γ ⊂ PM (Γ ⊢ G) ) // Incompleteness
Theorem as a named predicate
Minimal Type Theory Directed Acyclic
Graph of the above expression
01 ~ (2) // G is an alias for this node
02 ∃ (3)(6)
03 ⊂ (4)(5)
04 Γ
05 PM
06 ⊢ (4)(1) // cycle indicates error: evaluation
infinite loop
Further elaboration provided on link
http://LiarParadox.org/Provability_with_Minimal_Type_Theory.pdf
Copyright 2017 Pete Olcott
--

⊨ FS
A) ≡ (Γ ⊢
FS
A)
If this has any connection to language it is hidden from me.
All linguists have heard of the principle of compositionality. Some erroneously disagree because they think that context is excluded, it is not.

The meaning of an expression is entirely comprised of the meaning of all of its constituent parts, including context. I have derived a formal system of sub atomic semantic compositionality. This system shows exactly how all of the tiniest pieces of meaning precisely fit together to form increasingly larger units of meaning.
--
(Γ ⊨ _FS A) ≡ (Γ ⊢ _FS A)
Peter T. Daniels
2017-06-08 13:43:03 UTC
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Post by Pete Olcott
All linguists have heard of the principle of compositionality. Some erroneously disagree because they think that context is excluded, it is not.
The meaning of an expression is entirely comprised of the meaning of all of its constituent parts, including context.
So now "context" is a "part of an expresion"?
Post by Pete Olcott
I have derived a formal system of sub atomic semantic compositionality. This system shows exactly how all of the tiniest pieces of meaning precisely fit together to form increasingly larger units of meaning.
--
(Γ ⊨ _FS A) ≡ (Γ ⊢ _FS A)
Pete Olcott
2017-06-08 13:50:52 UTC
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Post by Peter T. Daniels
Post by Pete Olcott
All linguists have heard of the principle of compositionality. Some erroneously disagree because they think that context is excluded, it is not.
The meaning of an expression is entirely comprised of the meaning of all of its constituent parts, including context.
So now "context" is a "part of an expresion"?
Context is often a constituent part of the derivation of the semantic meaning of an expression.
Post by Peter T. Daniels
Post by Pete Olcott
I have derived a formal system of sub atomic semantic compositionality. This system shows exactly how all of the tiniest pieces of meaning precisely fit together to form increasingly larger units of meaning.
--
(Γ ⊨ _FS A) ≡ (Γ ⊢ _FS A)
--
(Γ ⊨ _FS A) ≡ (Γ ⊢ _FS A)
Peter T. Daniels
2017-06-08 13:57:05 UTC
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Post by Pete Olcott
Post by Peter T. Daniels
Post by Pete Olcott
All linguists have heard of the principle of compositionality. Some erroneously disagree because they think that context is excluded, it is not.
The meaning of an expression is entirely comprised of the meaning of all of its constituent parts, including context.
So now "context" is a "part of an expresion"?
Context is often a constituent part of the derivation of the semantic meaning of an expression.
But that's not what you said. You need to express yourself more rigorously.

"All generalizations are false, including this one."

The above sentence is true and valid. If it doesn't fit into your formalism,
then your formalism is inadequate.
Post by Pete Olcott
Post by Peter T. Daniels
Post by Pete Olcott
I have derived a formal system of sub atomic semantic compositionality. This system shows exactly how all of the tiniest pieces of meaning precisely fit together to form increasingly larger units of meaning.
Pete Olcott
2017-06-08 14:52:59 UTC
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Post by Peter T. Daniels
Post by Pete Olcott
Post by Peter T. Daniels
Post by Pete Olcott
All linguists have heard of the principle of compositionality. Some erroneously disagree because they think that context is excluded, it is not.
The meaning of an expression is entirely comprised of the meaning of all of its constituent parts, including context.
So now "context" is a "part of an expresion"?
Context is often a constituent part of the derivation of the semantic meaning of an expression.
But that's not what you said. You need to express yourself more rigorously.
"All generalizations are false, including this one."
The above sentence is true and valid. If it doesn't fit into your formalism,
then your formalism is inadequate.
The primary purpose of my formalization was to show the error of expressions such as that.

Your example is really a variation of: "This sentence is false".
Another variation is: "This sentence is true".
I show the infinitely recursive structure of: "This sentence is not true".

here: http://LiarParadox.org/
Post by Peter T. Daniels
Post by Pete Olcott
Post by Peter T. Daniels
Post by Pete Olcott
I have derived a formal system of sub atomic semantic compositionality. This system shows exactly how all of the tiniest pieces of meaning precisely fit together to form increasingly larger units of meaning.
--
(Γ ⊨ _FS A) ≡ (Γ ⊢ _FS A)
Peter T. Daniels
2017-06-08 16:58:33 UTC
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Post by Pete Olcott
Post by Peter T. Daniels
Post by Pete Olcott
Post by Peter T. Daniels
Post by Pete Olcott
All linguists have heard of the principle of compositionality. Some erroneously disagree because they think that context is excluded, it is not.
The meaning of an expression is entirely comprised of the meaning of all of its constituent parts, including context.
So now "context" is a "part of an expresion"?
Context is often a constituent part of the derivation of the semantic meaning of an expression.
But that's not what you said. You need to express yourself more rigorously.
"All generalizations are false, including this one."
The above sentence is true and valid. If it doesn't fit into your formalism,
then your formalism is inadequate.
The primary purpose of my formalization was to show the error of expressions such as that.
Your example is really a variation of: "This sentence is false".
Another variation is: "This sentence is true".
I show the infinitely recursive structure of: "This sentence is not true".
Except that no one in real life ever has reason to say "This sentence is true,"
or to invoke Russell's Paradox (who shaves the barber?), whereas my sentence
is used and understood perfectly every day.

Logical analysis is not suited to human language.
Post by Pete Olcott
here: http://LiarParadox.org/
DKleinecke
2017-06-08 17:41:03 UTC
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Post by Peter T. Daniels
Post by Pete Olcott
Post by Peter T. Daniels
Post by Pete Olcott
Post by Peter T. Daniels
Post by Pete Olcott
All linguists have heard of the principle of compositionality. Some erroneously disagree because they think that context is excluded, it is not.
The meaning of an expression is entirely comprised of the meaning of all of its constituent parts, including context.
So now "context" is a "part of an expresion"?
Context is often a constituent part of the derivation of the semantic meaning of an expression.
But that's not what you said. You need to express yourself more rigorously.
"All generalizations are false, including this one."
The above sentence is true and valid. If it doesn't fit into your formalism,
then your formalism is inadequate.
The primary purpose of my formalization was to show the error of expressions such as that.
Your example is really a variation of: "This sentence is false".
Another variation is: "This sentence is true".
I show the infinitely recursive structure of: "This sentence is not true".
Except that no one in real life ever has reason to say "This sentence is true,"
or to invoke Russell's Paradox (who shaves the barber?), whereas my sentence
is used and understood perfectly every day.
Logical analysis is not suited to human language.
Post by Pete Olcott
here: http://LiarParadox.org/
And, of course, languages are not compositional in the
strict sense. Context is not a rigorously defined notion
and semantics is still very poorly understood.

All we really know about the human mind is that it does not
work like a computer.
Pete Olcott
2017-06-08 19:56:07 UTC
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<div class="moz-cite-prefix">On 6/8/2017 11:58 AM, Peter T. Daniels
wrote:<br>
</div>
<blockquote
cite="mid:9811e314-672e-4c07-b143-***@googlegroups.com"
type="cite">
<pre wrap="">On Thursday, June 8, 2017 at 10:53:06 AM UTC-4, Pete Olcott wrote:
</pre>
<blockquote type="cite">
<pre wrap="">On 6/8/2017 8:57 AM, Peter T. Daniels wrote:
</pre>
<blockquote type="cite">
<pre wrap="">On Thursday, June 8, 2017 at 9:50:59 AM UTC-4, Pete Olcott wrote:
</pre>
<blockquote type="cite">
<pre wrap="">On 6/8/2017 8:43 AM, Peter T. Daniels wrote:
</pre>
<blockquote type="cite">
<pre wrap="">On Thursday, June 8, 2017 at 9:34:13 AM UTC-4, Pete Olcott wrote:
</pre>
</blockquote>
</blockquote>
</blockquote>
</blockquote>
<pre wrap="">
</pre>
<blockquote type="cite">
<blockquote type="cite">
<blockquote type="cite">
<blockquote type="cite">
<blockquote type="cite">
<pre wrap="">All linguists have heard of the principle of compositionality. Some erroneously disagree because they think that context is excluded, it is not.
The meaning of an expression is entirely comprised of the meaning of all of its constituent parts, including context.
</pre>
</blockquote>
<pre wrap="">So now "context" is a "part of an expresion"?
</pre>
</blockquote>
<pre wrap="">Context is often a constituent part of the derivation of the semantic meaning of an expression.
</pre>
</blockquote>
<pre wrap="">But that's not what you said. You need to express yourself more rigorously.
"All generalizations are false, including this one."
The above sentence is true and valid. If it doesn't fit into your formalism,
then your formalism is inadequate.
</pre>
</blockquote>
<pre wrap="">
The primary purpose of my formalization was to show the error of expressions such as that.

Your example is really a variation of: "This sentence is false".
Another variation is: "This sentence is true".
I show the infinitely recursive structure of: "This sentence is not true".
</pre>
</blockquote>
<pre wrap="">
Except that no one in real life ever has reason to say "This sentence is true,"
or to invoke Russell's Paradox (who shaves the barber?), whereas my sentence
is used and understood perfectly every day.

Logical analysis is not suited to human language.</pre>
</blockquote>
<br>
That is certainly what most would presume. <br>
As soon as I have completed the process of exhaustively elaborating
the mathematics of meaning (previously the mathematics of meaning of
words) one can verify the falsity of this presumption. <br>
<br>
<blockquote
cite="mid:9811e314-672e-4c07-b143-***@googlegroups.com"
type="cite">
<pre wrap="">

</pre>
<blockquote type="cite">
<pre wrap="">here: <a class="moz-txt-link-freetext" href="http://LiarParadox.org/">http://LiarParadox.org/</a>
</pre>
</blockquote>
</blockquote>
<br>
<p><br>
</p>
<div class="moz-signature">-- <br>
<p class="western" style="margin-bottom: 0in"><b><font
face="Arial, sans-serif"><font style="font-size: 12pt"
size="2">(Γ
</font><font style="font-size: 12pt" size="2">⊨ </font><sub><font
style="font-size: 8pt" size="2">FS</font></sub><font
style="font-size: 12pt" size="2">
A) ≡ (</font><font style="font-size: 12pt" size="2">Γ </font><font
style="font-size: 12pt" size="2">⊢
</font><sub><font style="font-size: 8pt" size="2">FS</font></sub><font
style="font-size: 12pt" size="2">
A)</font></font></b></p>
<p class="western" style="margin-bottom: 0in"><br>
</p>
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Pete Olcott
2017-06-08 19:51:22 UTC
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Post by Pete Olcott
G(
~∃Γ ⊂ PM (Γ ⊢ G) ) // Incompleteness
Theorem as a named predicate
Minimal Type Theory Directed Acyclic
Graph of the above expression
01 ~ (2) // G is an alias for this node
02 ∃ (3)(6)
03 ⊂ (4)(5)
04 Γ
05 PM
06 ⊢ (4)(1) // cycle indicates error: evaluation
infinite loop
Further elaboration provided on link
http://LiarParadox.org/Provability_with_Minimal_Type_Theory.pdf
Copyright 2017 Pete Olcott
--

⊨ FS
A) ≡ (Γ ⊢
FS
A)
This, and the link, remain incomprehensible. I do not
expect you to be able to explain what they mean.
--
(Γ ⊨ _FS A) ≡ (Γ ⊢ _FS A)
The set theory operators specify axiomatic set theory of NBG. If you don't know what that is we cannot proceed until you do.
National Bank of Greece? I don't know what NBG stands for and so
cannot not examine their set theory.
https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory
--
(Γ ⊨ _FS A) ≡ (Γ ⊢ _FS A)
DKleinecke
2017-06-08 21:52:56 UTC
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Post by Pete Olcott
Post by Pete Olcott
G(
~∃Γ ⊂ PM (Γ ⊢ G) ) // Incompleteness
Theorem as a named predicate
Minimal Type Theory Directed Acyclic
Graph of the above expression
01 ~ (2) // G is an alias for this node
02 ∃ (3)(6)
03 ⊂ (4)(5)
04 Γ
05 PM
06 ⊢ (4)(1) // cycle indicates error: evaluation
infinite loop
Further elaboration provided on link
http://LiarParadox.org/Provability_with_Minimal_Type_Theory.pdf
Copyright 2017 Pete Olcott
--

⊨ FS
A) ≡ (Γ ⊢
FS
A)
This, and the link, remain incomprehensible. I do not
expect you to be able to explain what they mean.
--
(Γ ⊨ _FS A) ≡ (Γ ⊢ _FS A)
The set theory operators specify axiomatic set theory of NBG. If you don't know what that is we cannot proceed until you do.
National Bank of Greece? I don't know what NBG stands for and so
cannot not examine their set theory.
https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory
--
(Γ ⊨ _FS A) ≡ (Γ ⊢ _FS A)
Ah so. A piece of jargon I missed.

I was once very familiar with this stuff. I took multiple classes
with both Kelley and Morse (of Morse-Kelley set theory) and heard
most of these ideas ab initio but bowed out before the final
presentations.

Now explain how your ideas fit NBG - as that set theory is usually
understood,
Pete Olcott
2017-06-09 12:58:59 UTC
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Post by DKleinecke
Post by Pete Olcott
G(
~∃Γ ⊂ PM (Γ ⊢ G) ) // Incompleteness
Theorem as a named predicate
https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory
--
(Γ ⊨ _FS A) ≡ (Γ ⊢ _FS A)
Ah so. A piece of jargon I missed.
I was once very familiar with this stuff. I took multiple classes
with both Kelley and Morse (of Morse-Kelley set theory) and heard
most of these ideas ab initio but bowed out before the final
presentations.
Now explain how your ideas fit NBG - as that set theory is usually
understood,
After further pondering, I will not quite be using the proper classes of NBG.

I am forming an inheritance type hierarchy, thus will have some types that are subtypes of other types.

Elements of Type ⊂
Element of Type ∈

I will eventually specify the axioms of whatever theory that I am using, in MTT everything is explicitly specified as an axiom: rules of inference, the meaning of all logical operators.

One thing expressly excluded is that no set, class, or category can be a member of itself.

I figured out a few months ago that a set containing itself as a member is analogous to a tin-can entirely containing itself within itself, thus incoherent.
--
(Γ ⊨ _FS A) ≡ (Γ ⊢ _FS A)
Peter Percival
2017-06-09 14:20:46 UTC
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Post by Pete Olcott
[...]
I will eventually specify the axioms of whatever theory that I am
using, in MTT everything is explicitly specified as an axiom: rules
of inference
How do you specify a rule of inference as an axiom? Do you even know
what a rule of inference is?
Post by Pete Olcott
, the meaning of all logical operators.
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
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