Simulating halt decider axiom

[olcott]

If simulating halt decider H correctly simulates its input D until H
correctly determines that its simulated D would never stop running
unless aborted then H can abort its simulation of D and correctly report
that D specifies a non-halting sequence of configurations.

Once the above is verified to be a tautology then it can be accepted as
an axiom for Simulating halt deciders.

When we apply ordinary software engineering code analysis to H(D,D)
shown below we see that the criteria of the above axiom is met:

simulating halt decider H correctly simulates its
input D until H correctly determines that its
simulated D would never stop running unless aborted

thus it is proven that H correctly reports that D specifies a
non-halting sequence of configurations.

*Every rebuttal to this*
(a) Ignores that the first paragraph above is an axiom, thus failing to
understand that it is a tautology.
(b) Performs the software engineering code analysis incorrectly.

void D(void (*x)())
{
int Halt_Status = H(x, x);
if (Halt_Status)
HERE: goto HERE;
return;
}

int main()
{
H(D,D);
}

_D()
[000019b3] 55 push ebp
[000019b4] 8bec mov ebp,esp
[000019b6] 51 push ecx
[000019b7] 8b4508 mov eax,[ebp+08]
[000019ba] 50 push eax
[000019bb] 8b4d08 mov ecx,[ebp+08]
[000019be] 51 push ecx
[000019bf] e8bff9ffff call 00001383
[000019c4] 83c408 add esp,+08
[000019c7] 8945fc mov [ebp-04],eax
[000019ca] 837dfc00 cmp dword [ebp-04],+00
[000019ce] 7402 jz 000019d2
[000019d0] ebfe jmp 000019d0
[000019d2] 8be5 mov esp,ebp
[000019d4] 5d pop ebp
[000019d5] c3 ret
Size in bytes:(0035) [000019d5]

_main()
[000019e3] 55 push ebp
[000019e4] 8bec mov ebp,esp
[000019e6] 68b3190000 push 000019b3 // push D
[000019eb] 68b3190000 push 000019b3 // push D
[000019f0] e88ef9ffff call 00001383 // call H
[000019f5] 83c408 add esp,+08
[000019f8] 33c0 xor eax,eax
[000019fa] 5d pop ebp
[000019fb] c3 ret
Size in bytes:(0025) [000019fb]

machine stack stack machine assembly
address address data code language
======== ======== ======== ========= =============
[000019e3][00102a39][00000000] 55 push ebp // begin main
[000019e4][00102a39][00000000] 8bec mov ebp,esp
[000019e6][00102a35][000019b3] 68b3190000 push 000019b3 // push D
[000019eb][00102a31][000019b3] 68b3190000 push 000019b3 // push D
[000019f0][00102a2d][000019f5] e88ef9ffff call 00001383 // call H

When H correctly simulates D it finds that D remains stuck in recursive
simulation

H: Begin Simulation Execution Trace Stored at:112ae5
Address_of_H:1383
[000019b3][00112ad1][00112ad5] 55 push ebp // begin D
[000019b4][00112ad1][00112ad5] 8bec mov ebp,esp
[000019b6][00112acd][00102aa1] 51 push ecx
[000019b7][00112acd][00102aa1] 8b4508 mov eax,[ebp+08]
[000019ba][00112ac9][000019b3] 50 push eax // call D
[000019bb][00112ac9][000019b3] 8b4d08 mov ecx,[ebp+08]
[000019be][00112ac5][000019b3] 51 push ecx // push D
[000019bf][00112ac1][000019c4] e8bff9ffff call 00001383 // call H
H: Infinitely Recursive Simulation Detected Simulation Stopped

We can see that the first seven instructions of D simulated by H
precisely match the first seven instructions of the x86 source-code of
D. This conclusively proves that these instructions were simulated
correctly.

Anyone sufficiently technically competent in the x86 programming
language will agree that the above execution trace of D simulated by H
shows that D will never stop running unless H aborts its simulation of D.

H detects that D is calling itself with the exact same arguments that H
was called with and there are no conditional branch instructions from
the beginning of D to its call to H that can possibly escape the
repetition of this recursive simulation.

[000019f5][00102a39][00000000] 83c408 add esp,+08
[000019f8][00102a39][00000000] 33c0 xor eax,eax
[000019fa][00102a3d][00000018] 5d pop ebp
[000019fb][00102a41][00000000] c3 ret // end main
Number of Instructions Executed(972) == 15 Pages

Simulating Halt Decider Applied to the Halting Theorem
https://www.researchgate.net/publication/364657019_Simulating_Halt_Decider_Applied_to_the_Halting_Theorem



--
Copyright 2022 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

[Skep Dick]

On Saturday, 3 December 2022 at 19:34:59 UTC+2, olcott wrote:
> If simulating halt decider H correctly simulates its input D until H
> correctly determines that its simulated D would never stop running

Olcott, shut up already. Your decider is using proof by induction. You are extrapolating a general conclusion from a small set of observations.

The argument isn't whether your decider is "correct" or "incorrect".

The argument is whether your generalization is "faulty"; or not.

https://en.wikipedia.org/wiki/Faulty_generalization

[olcott]

As you just said in your rebuttal of Richard H is a partial halt
decider. H correctly determines whether or not elements of its domain
specify halting computations.

One of the elements of its domain is the "impossible" input required by
all of the conventional halting problem proofs:

void D(void (*x)())
{
int Halt_Status = H(x, x);
if (Halt_Status)
HERE: goto HERE;
return;
}

int main()
{

Output("Input_Halts = ", H(D, D));

[olcott]

On 12/3/2022 2:11 PM, Skep Dick wrote:

It can be seen on the basis of induction that D correctly simulated by H
would never stop running unless aborted.

void D(void (*x)())
{
int Halt_Status = H(x, x);
if (Halt_Status)
HERE: goto HERE;
return;
}

int main()
{

Output("Input_Halts = ", H(D, D));
}

[Skep Dick]

On Saturday, 3 December 2022 at 22:28:33 UTC+2, olcott wrote:
> On 12/3/2022 2:11 PM, Skep Dick wrote:
> > On Saturday, 3 December 2022 at 19:34:59 UTC+2, olcott wrote:
> >> If simulating halt decider H correctly simulates its input D until H
> >> correctly determines that its simulated D would never stop running
> > Olcott, shut up already. Your decider is using proof by induction. You are extrapolating a general conclusion from a small set of observations.
> >
> > The argument isn't whether your decider is "correct" or "incorrect".
> >
> > The argument is whether your generalization is "faulty"; or not.
> >
> > https://en.wikipedia.org/wiki/Faulty_generalization
> As you just said in your rebuttal of Richard H is a partial halt
> decider. H correctly determines whether or not elements of its domain
> specify halting computations.
>
> One of the elements of its domain is the "impossible" input required by
> all of the conventional halting problem proofs:
> void D(void (*x)())
> {
> int Halt_Status = H(x, x);
> if (Halt_Status)
> HERE: goto HERE;
> return;
> }
>
> int main()
> {
> Output("Input_Halts = ", H(D, D));
> }

You are not even attempting to address my point. Your decider is drawing its conclusion inductively.
Induction is a problem: https://en.wikipedia.org/wiki/Problem_of_induction

Just because you've observed a call (with the same parameters) N times; it doesn't mean that the next call won't be different and break the pattern.

[Skep Dick]

On Saturday, 3 December 2022 at 22:33:20 UTC+2, olcott wrote:
> It can be seen on the basis of induction that D correctly simulated by H
> would never stop running unless aborted.

It might have stopped. You don't know!

https://en.wikipedia.org/wiki/Problem_of_induction

[olcott]

The fact that there is no conditional code that can break the pattern
establishes a recursion invariant that proves non-halting.

http://web.stanford.edu/class/archive/cs/cs161/cs161.1176/Sections/161-section-1.pdf

[Skep Dick]

On Saturday, 3 December 2022 at 22:40:59 UTC+2, olcott wrote:
> The fact that there is no conditional code that can break the pattern
> establishes a recursion invariant that proves non-halting.

There is conditional code. Do you not see the "IF" statement?

[olcott]

Are you ignoring that analytic versus synthetic distinction on purpose
or have you never heard of this?

The POI does not apply to mathematical induction.

[olcott]

The execution trace of D correctly simulated by H proves that it cannot
possibly reach its conditional code. I have already showed this in my OP.

_D()
[000019b3] 55 push ebp
[000019b4] 8bec mov ebp,esp
[000019b6] 51 push ecx
[000019b7] 8b4508 mov eax,[ebp+08]
[000019ba] 50 push eax
[000019bb] 8b4d08 mov ecx,[ebp+08]
[000019be] 51 push ecx

[000019bf] e8bff9ffff call 00001383 // can't get past here

[000019c4] 83c408 add esp,+08
[000019c7] 8945fc mov [ebp-04],eax
[000019ca] 837dfc00 cmp dword [ebp-04],+00
[000019ce] 7402 jz 000019d2
[000019d0] ebfe jmp 000019d0
[000019d2] 8be5 mov esp,ebp
[000019d4] 5d pop ebp
[000019d5] c3 ret
Size in bytes:(0035) [000019d5]

[Skep Dick]

On Saturday, 3 December 2022 at 23:03:26 UTC+2, olcott wrote:
> The POI does not apply to mathematical induction.

It doesn't apply to inductively-defined objects, but it absolutely applies to function analysis!

You have absolutely NO idea whether the function you are studying is inductively defined with some upper bound!

https://en.wikipedia.org/wiki/Wittgenstein_on_Rules_and_Private_Language#The_rule-following_paradox

[Skep Dick]

On Saturday, 3 December 2022 at 23:07:33 UTC+2, olcott wrote:
> The execution trace of D correctly simulated by H proves that it cannot
> possibly reach its conditional code. I have already showed this in my OP.

Huh?!?!

Here is the conditional code which would lead to a halt. It looks very reachable to me.

> _D()
> [000019ce] 7402 jz 000019d2

[Skep Dick]

On Saturday, 3 December 2022 at 23:07:33 UTC+2, olcott wrote:

> _D()

> [000019ce] 7402 jz 000019d2
> [000019d0] ebfe jmp 000019d0
> [000019d2] 8be5 mov esp,ebp

Let me translate that the instruction @ 000019ce into English for you.
it says "Jump OVER the Infinite loop."

[Richard Damon]

On 12/3/22 12:34 PM, olcott wrote:
> If simulating halt decider H correctly simulates its input D until H
> correctly determines that its simulated D would never stop running
> unless aborted then H can abort its simulation of D and correctly report
> that D specifies a non-halting sequence of configurations.

Right, and the DEFINITON of "Correct Simultion" means a simulation that
demonstrates the behavior of the direct exectution of this exact input.

Since D(D) WILL Halt BECAUSE ITS copy of H aborts another simulation of
D(D), this H doesn't NEED to abort its simulation, so the condition is
not satisfied.

>
> Once the above is verified to be a tautology then it can be accepted as
> an axiom for Simulating halt deciders.
>
> When we apply ordinary software engineering code analysis to H(D,D)
> shown below we see that the criteria of the above axiom is met:
>
>    simulating halt decider H correctly simulates its
>    input D until H correctly determines that its
>    simulated D would never stop running unless aborted

Which you either do not apply correctly or is defined incorrectly making
your definition inconsistent with the Computation Theory definition of a
Halting Decider.

Meaning you are not working on the Halting Problem

>
> thus it is proven that H correctly reports that D specifies a
> non-halting sequence of configurations.

Only by the WRONG definiotion.

>
> *Every rebuttal to this*
> (a) Ignores that the first paragraph above is an axiom, thus failing to
>     understand that it is a tautology.

No, it isn't an axiom.

It is a statement which can be proven witht ee CORRECT meaning pof teh
words from teh DEFINTION of a Halt Decider, which shows what the correct
meaning of the words needs to be.

> (b) Performs the software engineering code analysis incorrectly.

What, you think running the input program isn't a correct analysis to
determine if the program halt?

Since it is the DEFINITION, YOU sre the one doing an incorrect analysis.

[olcott]

Because we could possibly be brains in vats connected to computers we
cannot trust that any aspect of what appears to be physical reality is
actual reality. This does not apply to mathematics.

Whether or not we are brains in vats 2+2 = 4 either way.

[olcott]

Only because you did not carefully study my original post. D simulated
by H remains stuck in recursive simulation until aborted thus cannot
possibly reach its own conditional code whether aborted or not.

[Skep Dick]

On Saturday, 3 December 2022 at 23:48:31 UTC+2, olcott wrote:
> On 12/3/2022 3:40 PM, Skep Dick wrote:
> > On Saturday, 3 December 2022 at 23:03:26 UTC+2, olcott wrote:
> >> The POI does not apply to mathematical induction.
> > It doesn't apply to inductively-defined objects, but it absolutely applies to function analysis!
> >
> > You have absolutely NO idea whether the function you are studying is inductively defined with some upper bound!
> >
> > https://en.wikipedia.org/wiki/Wittgenstein_on_Rules_and_Private_Language#The_rule-following_paradox
> Because we could possibly be brains in vats connected to computers we
> cannot trust that any aspect of what appears to be physical reality is
> actual reality. This does not apply to mathematics.

No idea why you are even comparing the two.

Mathematics is invented. Reality is not.

You've confused synthetic truths (Mathematics) with analytic truths (studying reality).

[olcott]

That wild guess of the analytic/synthetic distinction is woefully
incorrect.

*I divided it up a little more precisely*
(a) Analytic truth is verified as true entirely on the basis of the
semantic meaning of expressions of language.

(b) Expressions of language of Empirical truth requires sense data from
the sense organs to verify its truth.

The problem of induction only applies to the latter and not to the former.

[Skep Dick]

On Sunday, 4 December 2022 at 00:14:38 UTC+2, olcott wrote:
> *I divided it up a little more precisely*
> (a) Analytic truth is verified as true entirely on the basis of the
> semantic meaning of expressions of language.

How are you going to verify the semantic meaning when you continuously confuse syntax for semantics?

> (b) Expressions of language of Empirical truth requires sense data from
> the sense organs to verify its truth.

Which is exactly what the BHK interpretation requires also!

https://en.wikipedia.org/wiki/Brouwer%E2%80%93Heyting%E2%80%93Kolmogorov_interpretation

[Mikko]

On 2022-12-03 17:34:55 +0000, olcott said:

> If simulating halt decider H correctly simulates its input D until H
> correctly determines that its simulated D would never stop running
> unless aborted then H can abort its simulation of D and correctly
> report that D specifies a non-halting sequence of configurations.
>
> Once the above is verified to be a tautology then it can be accepted as
> an axiom for Simulating halt deciders.

Once that is verified to be a tautology then it can (and should) be removed
from the set of axioms for Simulating halt deciders (and other axiom sets).
It is regarded as a bad style to use redundant axioms, and tautologies are
theorems of logic, thus redundant as axioms of a theory.

Mikko

[Skep Dick]

On Sunday, 4 December 2022 at 11:31:18 UTC+2, Mikko wrote:
> Once that is verified to be a tautology then it can (and should) be removed
> from the set of axioms for Simulating halt deciders (and other axiom sets).
> It is regarded as a bad style to use redundant axioms, and tautologies are
> theorems of logic, thus redundant as axioms of a theory.

What is the logical implication of your words? What theorems follow form what you say?
Are you suggesting that axioms are NOT tautologies (true in every model) and are thus subject to falsification?

[Skep Dick]

On Sunday, 4 December 2022 at 11:31:18 UTC+2, Mikko wrote:

> Once that is verified to be a tautology then it can (and should) be removed
> from the set of axioms for Simulating halt deciders (and other axiom sets).
> It is regarded as a bad style to use redundant axioms, and tautologies are
> theorems of logic, thus redundant as axioms of a theory.

Perhaps an example would bode well...

x=x is assumed to be a tautology. That's all fine and well, but is it an axiom or a theorem?

[olcott]

All axioms are tautologies. The only way to correctly determine that a
brand new axiom has been discovered is to verify that it is a tautology.

[Mr Flibble]

It is an axiom and axioms require no proof as they are true by definition.

/Flibble

[Richard Damon]

On 12/4/22 10:20 AM, olcott wrote:
> On 12/4/2022 3:31 AM, Mikko wrote:
>> On 2022-12-03 17:34:55 +0000, olcott said:
>>
>>> If simulating halt decider H correctly simulates its input D until H
>>> correctly determines that its simulated D would never stop running
>>> unless aborted then H can abort its simulation of D and correctly
>>> report that D specifies a non-halting sequence of configurations.
>>>
>>> Once the above is verified to be a tautology then it can be accepted
>>> as an axiom for Simulating halt deciders.
>>
>> Once that is verified to be a tautology then it can (and should) be
>> removed
>> from the set of axioms for Simulating halt deciders (and other axiom
>> sets).
>> It is regarded as a bad style to use redundant axioms, and tautologies
>> are
>> theorems of logic, thus redundant as axioms of a theory.
>>
>> Mikko
>>
>
> All axioms are tautologies. The only way to correctly determine that a
> brand new axiom has been discovered is to verify that it is a tautology.
>

No, a tautology is a statment that by its structure MUST be true for all
cases.

An axiom is a statement of accepted self-evident truth. They are not
"Discovered", except in the creation of a new branch of a field, as they
form the basis of that branch/field.

For example, Euclid's 5th Axiom: Given a line, and a point not on that
line, only one line can be drawn through that point that never touches
that line.

This statement is taken as a true statement in Euclidian Geometry.

That statement is NOT a tautology in general, as in non-Euclidian
Geometry it does not hold.

[Mr Flibble]

That isn't Euclid's 5th Axiom at all: you spout more crap than that
deviant, Olcott who you cannot resist engaging with. You are suspect and
unreliable.

/Flibble

[Richard Damon]

There are many forms that the statement is presented in, and all are
called Euclid's 5th Axiom.

Euclid didn't call them axioms, but postulates, as the theory of logic
was still evolving, so that wasn't the term he used, but is what is used
in modern studies of the theory.

What do YOU think Eucid's 5th Axiom is? and does it actually express
something different?

[olcott]

On 12/4/2022 1:22 PM, Richard Damon wrote:
> On 12/4/22 10:20 AM, olcott wrote:
>> On 12/4/2022 3:31 AM, Mikko wrote:
>>> On 2022-12-03 17:34:55 +0000, olcott said:
>>>
>>>> If simulating halt decider H correctly simulates its input D until H
>>>> correctly determines that its simulated D would never stop running
>>>> unless aborted then H can abort its simulation of D and correctly
>>>> report that D specifies a non-halting sequence of configurations.
>>>>
>>>> Once the above is verified to be a tautology then it can be accepted
>>>> as an axiom for Simulating halt deciders.
>>>
>>> Once that is verified to be a tautology then it can (and should) be
>>> removed
>>> from the set of axioms for Simulating halt deciders (and other axiom
>>> sets).
>>> It is regarded as a bad style to use redundant axioms, and
>>> tautologies are
>>> theorems of logic, thus redundant as axioms of a theory.
>>>
>>> Mikko
>>>
>>
>> All axioms are tautologies. The only way to correctly determine that a
>> brand new axiom has been discovered is to verify that it is a tautology.
>>
>
> No, a tautology is a statment that by its structure MUST be true for all
> cases.
>

A tautology is any expression of language that is verified as completely
true entirely on the basis of its meaning.

Any expression of language that is a tautology can be used as if it was
an axiom in that in both cases it is known to be true thus forming a
sound basis for inference.

> An axiom is a statement of accepted self-evident truth. They are not
> "Discovered", except in the creation of a new branch of a field, as they
> form the basis of that branch/field.
>

Simulating halt deciders are the new field.
That the above paragraph is a tautology provides a stronger basis for
its truth than if it was merely assumed to be true as an axiom.

> For example, Euclid's 5th Axiom: Given a line, and a point not on that
> line, only one line can be drawn through that point that never touches
> that line.
>
> This statement is taken as a true statement in Euclidian Geometry.
>

So you are saying that it is false in other cases?

> That statement is NOT a tautology in general, as in non-Euclidian
> Geometry it does not hold.

[Richard Damon]

What your source of that definition?

From: https://en.wikipedia.org/wiki/Tautology_(logic)

In mathematical logic, a tautology (from Greek: ταυτολογία) is a formula
or assertion that is true in every possible interpretation. An example
is "x=y or x≠y". Similarly, "either the ball is green, or the ball is
not green" is always true, regardless of the colour of the ball.

So,

> Any expression of language that is a tautology can be used as if it was
> an axiom in that in both cases it is known to be true thus forming a
> sound basis for inference.

Nope. Source for that statement?

>
>> An axiom is a statement of accepted self-evident truth. They are not
>> "Discovered", except in the creation of a new branch of a field, as
>> they form the basis of that branch/field.
>>
>
> Simulating halt deciders are the new field.

But must conform to all the requirmeent of Halt Deciders, or that aren't
Halt Decider.

> That the above paragraph is a tautology provides a stronger basis for
> its truth than if it was merely assumed to be true as an axiom.

But it isn't one (at least with your meaning of the words), so you whole
theory colapses.

>
>> For example, Euclid's 5th Axiom: Given a line, and a point not on that
>> line, only one line can be drawn through that point that never touches
>> that line.
>>
>> This statement is taken as a true statement in Euclidian Geometry.
>>
>
> So you are saying that it is false in other cases?

Yes.

You never heard of the Non-Euclidean Geometries?

For example, in Spherical Geometry, it is IMPOSSIBLE to place a
(straight) line through a point not on another line, that never touches
that other line.

[Richard Damon]

On 12/4/22 3:14 PM, olcott wrote:

> A tautology is any expression of language that is verified as completely
> true entirely on the basis of its meaning.
>

If so, then it is PROVABLE by the definitions of the words.

Of course to do this, you need to put yourself on record for the meaning
of the words, and need to be using them with the same defintion
everywhere, and a definition compatible with the things you are using in
your proof.

Since your definition of "A Correct Simulation" is going to fall flat
there, you have your problem.

To make the simulation actually show the behavior of the program, it
will need to include complete, but to show your H is "correct", it needs
to allow partial.

Thus, your "tautology" fails to be a true statement, so it CAN'T be a
Tautology.

YOU FAIL.

[André G. Isaak]

I'd drop the 'self-evident' part. That might have been the view of the
Greeks, but I don't think people today would consider axioms to be
self-evident. Your example of the Fifth Postulate is a case in point:
while it might have seemed self-evident to Euclid, the existence of
non-Euclidian geometries suggest otherwise.

Axioms are simply statements which are taken as given in a particular
system.

André

> For example, Euclid's 5th Axiom: Given a line, and a point not on that
> line, only one line can be drawn through that point that never touches
> that line.
>
> This statement is taken as a true statement in Euclidian Geometry.
>
> That statement is NOT a tautology in general, as in non-Euclidian
> Geometry it does not hold.

--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.

[Skep Dick]

On Sunday, 4 December 2022 at 17:20:30 UTC+2, olcott wrote:
> All axioms are tautologies. The only way to correctly determine that a
> brand new axiom has been discovered is to verify that it is a tautology.

That's incoherent. A tautology is a statement which is "true in all models".

But the trivial case to manufacture a model in which the axiom is not true is to define the negation of the axiom as true.

x=x is not a tautology because not(x=x) can be defined as an axiom.

Trivial example: https://ibb.co/9yV0nBy

[Skep Dick]

On Monday, 5 December 2022 at 04:18:36 UTC+2, richar...@gmail.com wrote:
> On 12/4/22 3:14 PM, olcott wrote:
> > A tautology is any expression of language that is verified as completely
> > true entirely on the basis of its meaning.
> >
> If so, then it is PROVABLE by the definitions of the words.

Uh. Prove that x=x "from the definition of the words".

> Of course to do this, you need to put yourself on record for the meaning
> of the words, and need to be using them with the same defintion
> everywhere, and a definition compatible with the things you are using in
> your proof.

To prove that something is a tautology requires you to prove it true in ALL models.

I have no idea how you are going to enumerate ALL models of any statement.

> Since your definition of "A Correct Simulation" is going to fall flat
> there, you have your problem.

You mean exactly like all of your definitions are going to fall flat?

That's not a problem with any particular definition. That's a problem with definability.

> To make the simulation actually show the behavior of the program, it
> will need to include complete, but to show your H is "correct", it needs
> to allow partial.
>
> Thus, your "tautology" fails to be a true statement, so it CAN'T be a
> Tautology.
>
> YOU FAIL.

But that's a problem with ALL Mathematics in general - not Olcott's decider in particular.

What's your point?

[Skep Dick]

On Monday, 5 December 2022 at 02:58:44 UTC+2, richar...@gmail.com wrote:
> From: https://en.wikipedia.org/wiki/Tautology_(logic)
>
> In mathematical logic, a tautology (from Greek: ταυτολογία) is a formula
> or assertion that is true in every possible interpretation. An example
> is "x=y or x≠y". Similarly, "either the ball is green, or the ball is
> not green" is always true, regardless of the colour of the ball.

Shame. Those are terrible examples of tautologies.

They are tautologies of classical logic (where excluded middle holds).
They are not tautologies of Intuitionistic logic.

So your "tautology" is not true in every model.

🤣🤣🤣🤣

[Richard Damon]

Yes, I agree their examples make assumptions about the system they are
being expressed in.

After all, 'x = x', isn't a tautology in all systems.

The key point is Olcott seem to like to treat Wikipedia as a reliable
source, and they are giving a significantly different definition, it
shows that he needs to find a justification for the one he wants to use.

Especially since he draws an incorrect conclusion from his definition.

[Richard Damon]

Except that adding the axiom that negates the tautology doesn't
invalidate the logic that proved the tautology, it only makes the new
system with the added axiom inconsistent.

Remember, ALL statements are made in the context of a logic system that
implicitly imports all the definitions of that system. That system is
often just implicitly defined by the context of the conversation.

Yes, a Tautology will often only be a Tautology in the system it is
being talked about in.

[Richard Damon]

On 12/5/22 2:05 AM, Skep Dick wrote:
> On Monday, 5 December 2022 at 04:18:36 UTC+2, richar...@gmail.com wrote:
>> On 12/4/22 3:14 PM, olcott wrote:
>>> A tautology is any expression of language that is verified as completely
>>> true entirely on the basis of its meaning.
>>>
>> If so, then it is PROVABLE by the definitions of the words.
> Uh. Prove that x=x "from the definition of the words".

Given that the statement must be coming from a system that is presumed
to be somewhat "normal", since there are system that define that some
members of their domain do not compare equal to themselvess even if they
are the same as them selves (Like the "Not-a-number" value of IEEE
floating point math.)

A typical prove for a normal system like this could go something like,
nornal system being a reasonable assumption since the convention is that
abnormal sustems need to be declaired before being used:

A variable like "x" when it appears multiple times in a given set of
expressions will refer to the same member of the domain for all occurrences.

Since the statement uses the "=" sign, which by default implies
equality, and the default rules of equality state that equality is
normally a reflexive operation (something is equal to itself)

Since the two "x"s by the first definintion refer to the same item, and
the "=" refers to equality, x = x is saying that some object is equal to
itself, which is a property of the presumed equality rule,

This statment is proven to be true in a normal domain.

>
>> Of course to do this, you need to put yourself on record for the meaning
>> of the words, and need to be using them with the same defintion
>> everywhere, and a definition compatible with the things you are using in
>> your proof.
> To prove that something is a tautology requires you to prove it true in ALL models.
>
> I have no idea how you are going to enumerate ALL models of any statement.

I guess that just shows your model of thinking can't handle infinities.

>
>> Since your definition of "A Correct Simulation" is going to fall flat
>> there, you have your problem.
> You mean exactly like all of your definitions are going to fall flat?
>
> That's not a problem with any particular definition. That's a problem with definability.

Nope. since you define things in a context, which ultimately gets you
back to those first principles that were just accepted as defined.

Yes, if you reject the concept of First Principles, you end up not being
able to know anything.

>
>> To make the simulation actually show the behavior of the program, it
>> will need to include complete, but to show your H is "correct", it needs
>> to allow partial.
>>
>> Thus, your "tautology" fails to be a true statement, so it CAN'T be a
>> Tautology.
>>
>> YOU FAIL.
> But that's a problem with ALL Mathematics in general - not Olcott's decider in particular.
>
> What's your point?
>

Can you prove THAT statement?

Or are you just showing the limitations of the systems you have stuck
yourself into.

[Mikko]

On 2022-12-04 15:20:27 +0000, olcott said:

> On 12/4/2022 3:31 AM, Mikko wrote:
>> On 2022-12-03 17:34:55 +0000, olcott said:
>>
>>> If simulating halt decider H correctly simulates its input D until H
>>> correctly determines that its simulated D would never stop running
>>> unless aborted then H can abort its simulation of D and correctly
>>> report that D specifies a non-halting sequence of configurations.
>>>
>>> Once the above is verified to be a tautology then it can be accepted as
>>> an axiom for Simulating halt deciders.
>>
>> Once that is verified to be a tautology then it can (and should) be removed
>> from the set of axioms for Simulating halt deciders (and other axiom sets).
>> It is regarded as a bad style to use redundant axioms, and tautologies are
>> theorems of logic, thus redundant as axioms of a theory.
>>
>> Mikko
>>
>
> All axioms are tautologies. The only way to correctly determine that a
> brand new axiom has been discovered is to verify that it is a tautology.

Not necessarily, as Richard Damon already pointed out. Of course, if you
wish, you can call the rest "postulates" instead of axioms. What exactly
is the difference between an axiom and a postulate is a matter of opinion,
but the most common opinion is that there is no difference, or at least
no important difference, and the words "axiom" and "postulate" are
synonymes.

Mikko

[Mikko]

Actually, Euclid did not consider his fifth postulate, as he called it,
self-evident. Other philosophers had proposed that it is possible, given
a straight line and a point outside it, draw from the point two asymptotic
straight lines so that they intersect each other but not the given line.
Euclid thought otherwise but failed to prove, so he had to make it
a postulate. In his constructions and proofs Eiclid avoided the use of
the fifth postulate as much as possible.

The difference between postulates and axioms, as Euclid used the words,
is that the postulates are specific to geometry but irrelevant to e.g.
accounting and axioms like "when eequals are added to equals the sums
are equal" are relevant to both and other purposes.

Mikko

[olcott]

On 12/5/2022 12:57 AM, Skep Dick wrote:
> On Sunday, 4 December 2022 at 17:20:30 UTC+2, olcott wrote:
>> All axioms are tautologies. The only way to correctly determine that a
>> brand new axiom has been discovered is to verify that it is a tautology.
> That's incoherent. A tautology is a statement which is "true in all models".
>

I have broadened the definition so that it equally applies to natural
language.

> But the trivial case to manufacture a model in which the axiom is not true is to define the negation of the axiom as true.
>

The axiom/tautology must be verified as true entirely on the basis of
its meaning.

> x=x is not a tautology because not(x=x) can be defined as an axiom.
>
> Trivial example: https://ibb.co/9yV0nBy
>

That is mere nonsense.
When we define that a herd of cows is a pile of bricks we are wrong.
When we ignore the law of non-contradiction we are wrong.

[olcott]

With axioms and tautologies both are necessarily true thus impossibly
false. We can know that a herd of cows is not a pile of bricks on the
basis of the meaning of the words. These meanings are expressed as
stipulated relations between finite strings.

[olcott]

I am unifying (thus broadening) these concepts across formal systems and
natural language to derive the most basic foundation of the notion of
analytical truth.

[Skep Dick]

On Monday, 5 December 2022 at 13:01:46 UTC+2, richar...@gmail.com wrote:
> After all, 'x = x', isn't a tautology in all systems.

If 'x=x' isn't a tautology then saying "in all systems" is redundant.

You know. By definition.

[Jeff Barnett]

I think you have it right. When I took geometry about 65+ years ago, my
teacher state that there where two classes of assumptions that were used
in geometry: the first were properties and relations of "geometric
objects" such as there is exactly one line that contains two given
points; the second was basically the properties of numbers as yours
above "equals added to equals are equals". I hadn't thought about that
distinction for a while.
--
Jeff Barnett

[Skep Dick]

On Monday, 5 December 2022 at 15:16:36 UTC+2, richar...@gmail.com wrote:
> Except that adding the axiom that negates the tautology doesn't
> invalidate the logic that proved the tautology, it only makes the new
> system with the added axiom inconsistent.

You are confused about the whole thing. Tautology means true in ALL possible interpretations.

If a statement is true when interpreted in system A; but false when interpreted in system B then it's not a tautology! Because there is an interpretation in which it's NOT true.

It has nothing to do with the consistency of the system as Bolyai, Lobachevsky, and Riemann all have shown that going contrary to Euclid's parallel postulate doesn't yield any contradictions.

Euclud assumed one line could be drawn through a point parallel to a given line
Bolyai asumed many lines could be drawn through a point parallel to a given line.
Riemann assumed that no line could be drawn through a point parallel to a given line.

No contradiction - just three different geometries.

> Remember, ALL statements are made in the context of a logic system that
> implicitly imports all the definitions of that system. That system is
> often just implicitly defined by the context of the conversation.

That's an oxymoron. Context is always implicit, not explicit.

[Richard Damon]

So, you ADMIT that you aren't working in the ACTUAL COMPUTATION THEORY
of the Halting Problem, so you are working on that Problem.

Note, you can't change something by "broadning" it, and remain working
with it if you change how things work in the original field.

You H is just WRONG about P(P), and saying that is just because you have
"broadened" the terms is just a way to try to disguise LYING,

As an example of CORRECT broadening, expanding the number system from
Naturals to Interger to Rational to Real, etc. In each case the
operations on the smaller set, when done on the members of the smaller
set, give the same results in the bigger set as they do when done in the
smaller set.

This doesn't happen in your "Broadening" of the Halting Problem.

[Richard Damon]

Nope, because a tautology is a statement of logic, which inplies a field
of operation with which to take its context.

The statement when taken out of context, needs to be put into a context
to have meaning.

The author of that example wasn't thinking about contexts where equality
isn't defined to one self in all cases, and it is a fairly odd case.

When put into a context as imagined by the author, it is a tautolgy.
There exists contexts where it isn't. It just shows that we do need to
be away of the context we are talking about, and avoid pulling
statements out of the area where the context is understood.


YOU have shown that you don't understand that statements can't have
actual meaning without context, which is why you can't know anything.

[Richard Damon]

On 12/5/22 2:17 PM, Skep Dick wrote:
> On Monday, 5 December 2022 at 15:16:36 UTC+2, richar...@gmail.com wrote:
>> Except that adding the axiom that negates the tautology doesn't
>> invalidate the logic that proved the tautology, it only makes the new
>> system with the added axiom inconsistent.
> You are confused about the whole thing. Tautology means true in ALL possible interpretations.

Then nothing is a tautology, as you can also totally reinterprete the
meaning of all the symbols used in the statement.

Even 1 = 1 can't be tautology as = could be defined as being compose of
exactly the same atoms in the same place, and the two 1's are written in
different physical locations and thus not equal.

>
> If a statement is true when interpreted in system A; but false when interpreted in system B then it's not a tautology! Because there is an interpretation in which it's NOT true.
>
> It has nothing to do with the consistency of the system as Bolyai, Lobachevsky, and Riemann all have shown that going contrary to Euclid's parallel postulate doesn't yield any contradictions.
>
> Euclud assumed one line could be drawn through a point parallel to a given line
> Bolyai asumed many lines could be drawn through a point parallel to a given line.
> Riemann assumed that no line could be drawn through a point parallel to a given line.
>
> No contradiction - just three different geometries.

Which is about axioms, and not tautologies, so you have gone off track.

[Richard Damon]

No, axioms are assumed true, and generally "self-evident".

Tautologies are statements provably true by the nature of the statement
and not needing emperical knowledge. It is true in ALL embodyments of
the basic system.

"Jack is twelve" is NOT a tautology, even if Jack happens to be twelve.

"If Jack was eleven a year ago, he now is twelve" would be a tautology
under the assumption that we are dealing with system with a normal
passage of time.

[Skep Dick]

On Tuesday, 6 December 2022 at 07:55:15 UTC+2, richar...@gmail.com wrote:
> No, axioms are assumed true, and generally "self-evident".
>
> Tautologies are statements provably true by the nature of the statement
> and not needing emperical knowledge. It is true in ALL embodyments of
> the basic system.

Distinction without a difference.

The "law" of excluded middle is formalised as. ~p ∨ p ⇔ ⊤
In English that spells out "~p ∨ p" is materially equivalent to a tautology.

> "Jack is twelve" is NOT a tautology, even if Jack happens to be twelve.

But '"Jack is twelve" is true' is a tautology.

> "If Jack was eleven a year ago, he now is twelve" would be a tautology
> under the assumption that we are dealing with system with a normal
> passage of time.

That's most definitely not a tautology. It's a conditional.

You know, because ⊤ means "unconditionally true".

[Skep Dick]

On Tuesday, 6 December 2022 at 07:48:42 UTC+2, richar...@gmail.com wrote:
> On 12/5/22 2:17 PM, Skep Dick wrote:
> > On Monday, 5 December 2022 at 15:16:36 UTC+2, richar...@gmail.com wrote:
> >> Except that adding the axiom that negates the tautology doesn't
> >> invalidate the logic that proved the tautology, it only makes the new
> >> system with the added axiom inconsistent.
> > You are confused about the whole thing. Tautology means true in ALL possible interpretations.
> Then nothing is a tautology, as you can also totally reinterprete the
> meaning of all the symbols used in the statement.

Precisely! Nothing **IS** a tautology. The empty type is inhabited ( https://en.wikipedia.org/wiki/Empty_type )

Which is exactly the principle of explosion. There's your first principle!

From falsehood anything follows. Which is precisely where axioms follow from! NOTHING.

> Even 1 = 1 can't be tautology as = could be defined as being compose of
> exactly the same atoms in the same place, and the two 1's are written in
> different physical locations and thus not equal.

Congratulations. Now you understand the problem!

Is comparing 1 to 1 a binary; or a unary operation?

The answer to that question literally determines the truth or falsify of the "=" operator!
If you are comparing one thing (to itself) - then 1 = 1 is true.
If you are comparing two things (to each other) - then 1 = 1 is false.

How many things are you comparing when you compare 1 = 1?

> Which is about axioms, and not tautologies, so you have gone off track.

Axioms are defined as tautologies. Which is precisely the game of "pretend" and self-deception you are playing with yourself when you put yourself IN a system.

You pretend THAT your axioms (e.g excluded middle) are tautologies knowing full-well there exist other systems in which it's not the case.
So in order to call it a "tautology" you have to pretend your logic-system is the ONLY system.

Just like any religion...

[Mikko]

There are several kinds of statements that need not be proven but can be
used in a proof anyway. Logically they are all axioms. You may choose to
call them by different words depending on how you justify them but that
is not necessary and makes no difference for logic.

Mikko

[Skep Dick]

On Monday, 5 December 2022 at 17:11:56 UTC+2, olcott wrote:
> The axiom/tautology must be verified as true entirely on the basis of
> its meaning.

For as long as you keep thinking in terms of "true" and "false" any axiom you consider "true" I will happily consider "false"

> That is mere nonsense.
To demonstrate...

> When we define that a herd of cows is a pile of bricks we are wrong.

When we define that a herd of cows is a pile of bricks we are right.

Because there's no such thing as "right" or "wrong" definitions.

> When we ignore the law of non-contradiction we are wrong.

When we ignore the law of non-contradiction we are right.

Because there's no such thing as "right" or "wrong" laws of logic.

https://plato.stanford.edu/entries/dialetheism/

[Skep Dick]

On Monday, 5 December 2022 at 17:25:19 UTC+2, olcott wrote:
> I am unifying (thus broadening) these concepts across formal systems and
> natural language to derive the most basic foundation of the notion of
> analytical truth.

The truth is that there is no truth.

Formal logic is synthetic, not analytic.

[olcott]

An axiom is typically an expression of language that is simply assumed
to be true as the basis for a proof. When creating a brand new
axiomatic system that overturns the results of other systems simply
assuming that an axiom of this system is true is not enough.

When this brand new axiom is be proven true entirely on the basis of its
meaning thenn (then and only then) can it be accepted to overturn prior
results. The above *axiom of simulating halt deciders* is proven to be
true on the basis of its meaning.

Proving that the premise of the above axiom is fulfilled by H and D
requires treating C/x86 as a formal system having its own axioms that
are proven to be true on the basis of their meaning:

(a) A correct simulation of N steps of x86 code is any simulation that
derives the line-by-line execution trace specified by its x86 source-
code for N steps of execution.

(b) Non-terminating behavior is recognized by matching any
non-terminating behavior pattern after N steps of execution.
*Details of these two are provided in the original post*

When the simlating halt decider axiom and corresponding x86 axioms are
understood to be necessarily true then when H reports that D specifies a
non-halting sequence of configurations, H is understood to be
necessarily correct.

[olcott]

On 12/6/2022 6:52 AM, Skep Dick wrote:
> On Monday, 5 December 2022 at 17:11:56 UTC+2, olcott wrote:
>> The axiom/tautology must be verified as true entirely on the basis of
>> its meaning.
> For as long as you keep thinking in terms of "true" and "false" any axiom you consider "true" I will happily consider "false"
>
>> That is mere nonsense.
> To demonstrate...
>
>> When we define that a herd of cows is a pile of bricks we are wrong.
> When we define that a herd of cows is a pile of bricks we are right.
>
> Because there's no such thing as "right" or "wrong" definitions.
>

Incoherence proves that a definition is wrong:
A cup is the same as 16 tablespoons
is the same as 48 teaspoons
is the same as three cups.

Contradiction between two definitions proves
that at least one of them is wrong.

[olcott]

On 12/6/2022 6:55 AM, Skep Dick wrote:
> On Monday, 5 December 2022 at 17:25:19 UTC+2, olcott wrote:
>> I am unifying (thus broadening) these concepts across formal systems and
>> natural language to derive the most basic foundation of the notion of
>> analytical truth.
> The truth is that there is no truth.

negating itself thus proving that there is truth.

>
> Formal logic is synthetic, not analytic.

You are utterly clueless about the analytic synthetic distinction.
Any expression of language that can be proven to be true (or false)
entirely on the basis of its meaning without the need for any sense data
from the sense organs is analytic.

[Skep Dick]

On Tuesday, 6 December 2022 at 16:36:11 UTC+2, olcott wrote:
> Incoherence proves that a definition is wrong:
> A cup is the same as 16 tablespoons
> is the same as 48 teaspoons
> is the same as three cups.
>
> Contradiction between two definitions proves
> that at least one of them is wrong.

No, it just proves that they are mutually exclusive.

That doesn't mean "wrong". Wrongness is a moral judgment - logic doesn't deal with moral judgments.

[Skep Dick]

On Tuesday, 6 December 2022 at 16:41:28 UTC+2, olcott wrote:
> On 12/6/2022 6:55 AM, Skep Dick wrote:
> > On Monday, 5 December 2022 at 17:25:19 UTC+2, olcott wrote:
> >> I am unifying (thus broadening) these concepts across formal systems and
> >> natural language to derive the most basic foundation of the notion of
> >> analytical truth.
> > The truth is that there is no truth.
> negating itself thus proving that there is truth.

No. Thus demonstrating that despite the inherent undecidability of the meaning of the sentence - you are going to to interpret it whichever way you want.
Because the interpretation of undecidable expressions is dependent on arbitrary choice.

> You are utterly clueless about the analytic synthetic distinction.

What distinction? Everything is in logic and language is synthetic.

Maths is invented. Logic is invented. Language is invented.

> Any expression of language that can be proven to be true (or false)
> entirely on the basis of its meaning without the need for any sense data
> from the sense organs is analytic.

But linguistic meaning is invented. Thus it's synthetic, not analytic.

[olcott]

Wrong meaning incorrect.

[olcott]

That is not what is meant by the analytic/synthetic distinction.
You are using the incorrect sense meaning of the terms.

Synthetic of the analytic/synthetic distinction is an idiomatic meaning
having nothing to do with synthesis.

[Skep Dick]

On Tuesday, 6 December 2022 at 17:27:39 UTC+2, olcott wrote:
> Wrong meaning incorrect.
Doesn't matter which word you use. Structurally correct doesn't mean behaviourally correct.

"Beware of bugs in the above code; I have only proved it correct, not tried it." --Donald Knuth

[Skep Dick]

On Tuesday, 6 December 2022 at 17:32:16 UTC+2, olcott wrote:
> That is not what is meant by the analytic/synthetic distinction.
> You are using the incorrect sense meaning of the terms.

No, I am not. I am using precisely the sense I want to be using and there's nothing wrong; or incorrect about it.

> Synthetic of the analytic/synthetic distinction is an idiomatic meaning
> having nothing to do with synthesis.

I don't care what it has to do with. I reject your distinction.

[Richard Damon]

On 12/6/22 10:34 AM, Skep Dick wrote:
> On Tuesday, 6 December 2022 at 17:32:16 UTC+2, olcott wrote:
>> That is not what is meant by the analytic/synthetic distinction.
>> You are using the incorrect sense meaning of the terms.
> No, I am not. I am using precisely the sense I want to be using and there's nothing wrong; or incorrect about it.

Which is precisely your logical flaw.

It doesn't matter what YOU think something means, it matters what the
agreed meaning is.

>
>> Synthetic of the analytic/synthetic distinction is an idiomatic meaning
>> having nothing to do with synthesis.
> I don't care what it has to do with. I reject your distinction.

So you reject that basis of logic.

[Richard Damon]

On 12/5/22 1:50 AM, Skep Dick wrote:
> On Tuesday, 6 December 2022 at 07:48:42 UTC+2, richar...@gmail.com wrote:
>> On 12/5/22 2:17 PM, Skep Dick wrote:
>>> On Monday, 5 December 2022 at 15:16:36 UTC+2, richar...@gmail.com wrote:
>>>> Except that adding the axiom that negates the tautology doesn't
>>>> invalidate the logic that proved the tautology, it only makes the new
>>>> system with the added axiom inconsistent.
>>> You are confused about the whole thing. Tautology means true in ALL possible interpretations.
>> Then nothing is a tautology, as you can also totally reinterprete the
>> meaning of all the symbols used in the statement.
> Precisely! Nothing **IS** a tautology. The empty type is inhabited ( https://en.wikipedia.org/wiki/Empty_type )
>
> Which is exactly the principle of explosion. There's your first principle!
>
> From falsehood anything follows. Which is precisely where axioms follow from! NOTHING.

So since you start from nothing, you can get nothing,

You don't even have the empty type, as that is something

>
>> Even 1 = 1 can't be tautology as = could be defined as being compose of
>> exactly the same atoms in the same place, and the two 1's are written in
>> different physical locations and thus not equal.
> Congratulations. Now you understand the problem!
>
> Is comparing 1 to 1 a binary; or a unary operation?

Depends on Context.

>
> The answer to that question literally determines the truth or falsify of the "=" operator!
> If you are comparing one thing (to itself) - then 1 = 1 is true.
> If you are comparing two things (to each other) - then 1 = 1 is false.

Right, so Context matters.

>
> How many things are you comparing when you compare 1 = 1?
>
>> Which is about axioms, and not tautologies, so you have gone off track.
> Axioms are defined as tautologies. Which is precisely the game of "pretend" and self-deception you are playing with yourself when you put yourself IN a system.
>
> You pretend THAT your axioms (e.g excluded middle) are tautologies knowing full-well there exist other systems in which it's not the case.
> So in order to call it a "tautology" you have to pretend your logic-system is the ONLY system.
>
> Just like any religion...

ALL Knowledge comes from "religion", as Religion is really just your
picking of the First Principles you are going to work with.

And you don't need to assum your logic-system is the ONLY system, just
the system you are talking in. It is quite possible to acknoledge the
existance of other systems but still define that you are talking within
your given one.

This all seems beyond your understanding.

[Richard Damon]

On 12/5/22 1:20 AM, Skep Dick wrote:
> On Tuesday, 6 December 2022 at 07:55:15 UTC+2, richar...@gmail.com wrote:
>> No, axioms are assumed true, and generally "self-evident".
>>
>> Tautologies are statements provably true by the nature of the statement
>> and not needing emperical knowledge. It is true in ALL embodyments of
>> the basic system.
> Distinction without a difference.
>
> The "law" of excluded middle is formalised as. ~p ∨ p ⇔ ⊤
> In English that spells out "~p ∨ p" is materially equivalent to a tautology.

Which only applies to a logic system that contains the law of the
excluded middle, so not all systems.

>
>> "Jack is twelve" is NOT a tautology, even if Jack happens to be twelve.
> But '"Jack is twelve" is true' is a tautology.

No, because Jack might actually be 11.

>
>> "If Jack was eleven a year ago, he now is twelve" would be a tautology
>> under the assumption that we are dealing with system with a normal
>> passage of time.
> That's most definitely not a tautology. It's a conditional.
>
> You know, because ⊤ means "unconditionally true".

So, what is the condition that the statments "If Jack as eleven a year
ago, he now is twelve" is not true?

The STATEMENT is unconditionally true, in ALL models (instances of
reality) in the system assumed.

We could even likely strengthen the statement to "ONLY if Jack wa eleven
a year ago, will he now be twelve."

[Richard Damon]

So if that is what yo are attempting, you need to admit that you aren't
working in the original system.

You prove yourself to be a liar.

>
> When this brand new axiom is be proven true entirely on the basis of its
> meaning thenn (then and only then) can it be accepted to overturn prior
> results. The above *axiom of simulating halt deciders* is proven to be
> true on the basis of its meaning.

Statements that can be proven true entierly on the basis of its meaning
do not NEED To be taken as axioms,

>
> Proving that the premise of the above axiom is fulfilled by H and D
> requires treating C/x86 as a formal system having its own axioms that
> are proven to be true on the basis of their meaning:

Nope, the program does NOT prove your claim.

You are shown to be a bad programmer that can't recognize your own bugs.

>
> (a) A correct simulation of N steps of x86 code is any simulation that
> derives the line-by-line execution trace specified by its x86 source-
> code for N steps of execution.

Which, since different than the definition of correct used in the
definition of a UTM, you have lost the ability to say that the correct
simulaiton of the input represents that actual behavior of the macnine
that input represents.

>
> (b) Non-terminating behavior is recognized by matching any
> non-terminating behavior pattern after N steps of execution.
> *Details of these two are provided in the original post*

Except you never prove non-terminal behavior of the actual execution of
the input, since you use the wrong definition of correct simulation.

You logic FAILS.

>
> When the simlating halt decider axiom and corresponding x86 axioms are
> understood to be necessarily true then when H reports that D specifies a
> non-halting sequence of configurations, H is understood to be
> necessarily correct.
>

Nope, You are just creting a new INCONSITANT logic system that accepts
as non-halitng inputs that actually do halt.

[Mr Flibble]

Will you two idiots get a room and save us the fucking drama?

/Flibble

[Skep Dick]

On Tuesday, 6 December 2022 at 19:24:01 UTC+2, richar...@gmail.com wrote:
> On 12/5/22 1:20 AM, Skep Dick wrote:
> > On Tuesday, 6 December 2022 at 07:55:15 UTC+2, richar...@gmail.com wrote:
> >> No, axioms are assumed true, and generally "self-evident".
> >>
> >> Tautologies are statements provably true by the nature of the statement
> >> and not needing emperical knowledge. It is true in ALL embodyments of
> >> the basic system.
> > Distinction without a difference.
> >
> > The "law" of excluded middle is formalised as. ~p ∨ p ⇔ ⊤
> > In English that spells out "~p ∨ p" is materially equivalent to a tautology.
> Which only applies to a logic system that contains the law of the
> excluded middle, so not all systems.

Obviously. In those logic systems it's not a tautology.

Hence the part where you have to play amnesiac and forget that a tautology has to be true in all models.

It was a misnomer all along.

> >> "Jack is twelve" is NOT a tautology, even if Jack happens to be twelve.
> > But '"Jack is twelve" is true' is a tautology.
> No, because Jack might actually be 11.

Then '"Jack is 11" is true' is a tautology.

> So, what is the condition that the statments "If Jack as eleven a year
> ago, he now is twelve" is not true?

The part after the "IF". If Jack wasn't 11 a year ago then he's not 12.

> The STATEMENT is unconditionally true, in ALL models (instances of
> reality) in the system assumed.

It's literally false in every instance of reality where Jack wasn't 11 a year ago!

> We could even likely strengthen the statement to "ONLY if Jack wa eleven
> a year ago, will he now be twelve."

Strengthen it all you want. It's a conditional - therefore not a tautology.

[Mr Flibble]

Get a fucking room already.

/Flibble

[Richard Damon]

On 12/6/22 1:16 PM, Skep Dick wrote:
> On Tuesday, 6 December 2022 at 19:24:01 UTC+2, richar...@gmail.com wrote:
>> On 12/5/22 1:20 AM, Skep Dick wrote:
>>> On Tuesday, 6 December 2022 at 07:55:15 UTC+2, richar...@gmail.com wrote:
>>>> No, axioms are assumed true, and generally "self-evident".
>>>>
>>>> Tautologies are statements provably true by the nature of the statement
>>>> and not needing emperical knowledge. It is true in ALL embodyments of
>>>> the basic system.
>>> Distinction without a difference.
>>>
>>> The "law" of excluded middle is formalised as. ~p ∨ p ⇔ ⊤
>>> In English that spells out "~p ∨ p" is materially equivalent to a tautology.
>> Which only applies to a logic system that contains the law of the
>> excluded middle, so not all systems.
> Obviously. In those logic systems it's not a tautology.

Right, "Tautology" is a lable that applies to a statement in the systems
it applies to.

>
> Hence the part where you have to play amnesiac and forget that a tautology has to be true in all models.

All Models, not all systems.

You don't seem to understand the difference.

>
> It was a misnomer all along.

Seems to be a misunderstanding of your part.

>
>>>> "Jack is twelve" is NOT a tautology, even if Jack happens to be twelve.
>>> But '"Jack is twelve" is true' is a tautology.
>> No, because Jack might actually be 11.
> Then '"Jack is 11" is true' is a tautology.
>
>> So, what is the condition that the statments "If Jack as eleven a year
>> ago, he now is twelve" is not true?
> The part after the "IF". If Jack wasn't 11 a year ago then he's not 12.
>
>> The STATEMENT is unconditionally true, in ALL models (instances of
>> reality) in the system assumed.
> It's literally false in every instance of reality where Jack wasn't 11 a year ago!

Nope. You don't understand the meaning of the statement then.

>
>> We could even likely strengthen the statement to "ONLY if Jack wa eleven
>> a year ago, will he now be twelve."
> Strengthen it all you want. It's a conditional - therefore not a tautology.

Nope, you don't seem to understand the stateent.

Seems normal for you.

[olcott]

If they are to be used as the basis of a proof that overturns
conventional wisdom then they are being used as axioms.

Mentioning that it is true on the basis of its meaning causes people
that are interested in truth to verify that it is true on this basis,
otherwise they would not bother and thus be unaware of its truth.

>>
>> Proving that the premise of the above axiom is fulfilled by H and D
>> requires treating C/x86 as a formal system having its own axioms that
>> are proven to be true on the basis of their meaning:
>
> Nope, the program does NOT prove your claim.
>
> You are shown to be a bad programmer that can't recognize your own bugs.
>
>>
>> (a) A correct simulation of N steps of x86 code is any simulation that
>> derives the line-by-line execution trace specified by its x86 source-
>> code for N steps of execution.
>
> Which, since different than the definition of correct used in the
> definition of a UTM, you have lost the ability to say that the correct
> simulaiton of the input represents that actual behavior of the macnine
> that input represents.

It is generally the case that when the line-by-line execution trace of a
simulation of a machine description exactly matches the line-by-line
steps specified by this machine description that this simulation is
proved to be correct.

The exact same non-terminating behavior pattern specified by the
correctly simulated D of H(D,D) is precisely matched by the Peter Linz
correctly simulated Ĥ of embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩. (page 4)

*Simulating Halt Decider Applied to the Halting Theorem*
https://www.researchgate.net/publication/364657019_Simulating_Halt_Decider_Applied_to_the_Halting_Theorem

>>
>> (b) Non-terminating behavior is recognized by matching any
>> non-terminating behavior pattern after N steps of execution.
>> *Details of these two are provided in the original post*
>
> Except you never prove non-terminal behavior of the actual execution of
> the input, since you use the wrong definition of correct simulation.
>
> You logic FAILS.
>
>>
>> When the simlating halt decider axiom and corresponding x86 axioms are
>> understood to be necessarily true then when H reports that D specifies a
>> non-halting sequence of configurations, H is understood to be
>> necessarily correct.
>>
>
> Nope, You are just creting a new INCONSITANT logic system that accepts
> as non-halitng inputs that actually do halt.
>

[Richard Damon]

Then you aren't working in the original field, as fields are defined by
the axiom that they are built on.

>
> Mentioning that it is true on the basis of its meaning causes people
> that are interested in truth to verify that it is true on this basis,
> otherwise they would not bother and thus be unaware of its truth.
>

No, your claim that it it is from the meaning of the words is just an
attempt to cover the fact that you can't actually PROVE them from the
actual meaning of the words.

As I pointed out, you are using two different, and incompatible,
meanings of correct simulation to bridge fr0m what H does and what a UTM
shows.

This makes your logic invalid.

>>>
>>> Proving that the premise of the above axiom is fulfilled by H and D
>>> requires treating C/x86 as a formal system having its own axioms that
>>> are proven to be true on the basis of their meaning:
>>
>> Nope, the program does NOT prove your claim.
>>
>> You are shown to be a bad programmer that can't recognize your own bugs.
>>
>>>
>>> (a) A correct simulation of N steps of x86 code is any simulation that
>>> derives the line-by-line execution trace specified by its x86 source-
>>> code for N steps of execution.
>>
>> Which, since different than the definition of correct used in the
>> definition of a UTM, you have lost the ability to say that the correct
>> simulaiton of the input represents that actual behavior of the macnine
>> that input represents.
>
> It is generally the case that when the line-by-line execution trace of a
> simulation of a machine description exactly matches the line-by-line
> steps specified by this machine description that this simulation is
> proved to be correct.

No. That is a correct PARTIAL simulation.

The definiton you want to use to show simulation matchs to the direct
behavior requires COMPLETE correct simulation.

That is like saying you have correctly assembled a jig saw puzzle
because you have correctly put together a couple of the pieces of the set.

>
> The exact same non-terminating behavior pattern specified by the
> correctly simulated D of H(D,D) is precisely matched by the Peter Linz
> correctly simulated Ĥ of embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩. (page 4)

Nope. You "claimed" non-terminating behavior pattern is NOT correct, and
you have never even attempted to prove that it is.

YOU FAIL.

>
> *Simulating Halt Decider Applied to the Halting Theorem*
> https://www.researchgate.net/publication/364657019_Simulating_Halt_Decider_Applied_to_the_Halting_Theorem

A bunchn of garbage.

YOU FAIL.

You show that you don't understand the basics of logic or proof.

[Skep Dick]

On Tuesday, 6 December 2022 at 20:58:56 UTC+2, richar...@gmail.com wrote:
> Right, "Tautology" is a lable that applies to a statement in the systems
> it applies to.

Precisely! So it can't be true in ALL interpretations/systems/models.

Because it's a statement made in the internal language of the system!

> > Hence the part where you have to play amnesiac and forget that a tautology has to be true in all models.
> All Models, not all systems.

Yes, they are. There's logic. And then there are models/systems of logic.

Classical logic is a model/system of logic.
Intuitionistic is a model/system of logic.
Modal logic is a model/system of logic.
Temporal logic is a model/system of logic.

> You don't seem to understand the difference.

I understand how synonyms work just fine.

> > It was a misnomer all along.
> Seems to be a misunderstanding of your part.

You keep projecting your misunderstanding onto me.

> > It's literally false in every instance of reality where Jack wasn't 11 a year ago!
> Nope. You don't understand the meaning of the statement then.

Q.E.D. Projection.

> >> We could even likely strengthen the statement to "ONLY if Jack wa eleven
> >> a year ago, will he now be twelve."
> > Strengthen it all you want. It's a conditional - therefore not a tautology.
> Nope, you don't seem to understand the stateent.

Q.E.D. Projection.

> Seems normal for you.
Q.E.D. Projection.

[olcott]

On 12/7/2022 2:20 AM, Skep Dick wrote:
> On Tuesday, 6 December 2022 at 20:58:56 UTC+2, richar...@gmail.com wrote:
>> Right, "Tautology" is a lable that applies to a statement in the systems
>> it applies to.
> Precisely! So it can't be true in ALL interpretations/systems/models.
>
> Because it's a statement made in the internal language of the system!
>
>>> Hence the part where you have to play amnesiac and forget that a tautology has to be true in all models.
>> All Models, not all systems.
> Yes, they are. There's logic. And then there are models/systems of logic.
>
> Classical logic is a model/system of logic.
> Intuitionistic is a model/system of logic.
> Modal logic is a model/system of logic.
> Temporal logic is a model/system of logic.
>

The higher level logic of correct reasoning subsumes all of the
differently logics of formal and natural logic into a single system of
correct reasoning.

(1) Some finite strings are stipulated to have the semantic property of
Boolean true. Such as Haskell Curry elementary theorems
https://www.liarparadox.org/Haskell_Curry_45.pdf
or basic facts specified in natural language: cats ⊂ animals.

(2) Some finite strings are derived by applying truth preserving
operations operations to (1) and/or the result of (2).

Any system of reasoning that diverges from this model diverges from
correct reasoning.

>> You don't seem to understand the difference.
> I understand how synonyms work just fine.
>
>>> It was a misnomer all along.
>> Seems to be a misunderstanding of your part.
> You keep projecting your misunderstanding onto me.
>
>>> It's literally false in every instance of reality where Jack wasn't 11 a year ago!
>> Nope. You don't understand the meaning of the statement then.
> Q.E.D. Projection.
>
>>>> We could even likely strengthen the statement to "ONLY if Jack wa eleven
>>>> a year ago, will he now be twelve."
>>> Strengthen it all you want. It's a conditional - therefore not a tautology.
>> Nope, you don't seem to understand the stateent.
> Q.E.D. Projection.
>
>> Seems normal for you.
> Q.E.D. Projection.

[Skep Dick]

What did your parents do to you that you've spent your entire life trying to "be right" ?

[olcott]

If humanity is going to survive we must overturn the Tarski
undefinabilty theorem to derive an objective criterion measure that
divides truth from lies.

https://liarparadox.org/Tarski_275_276.pdf

[Skep Dick]

https://en.wikipedia.org/wiki/Messiah_complex

[olcott]

My criterion is objectively correct yours is anchored entirely in
ignorance. As long as the line-by-line execution trace of the x86 code /
TM description exactly matches the line-by-line x86 / TM description
source code for N steps of simulation then these N steps of simulation
are necessarily correct.

Mathematical induction proves the behavior of what the additional steps
would be at the point where N steps have been completed. In the case of
H(D,D) and the Peter Linz embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ the next steps would simply
repeat the simulation continuously until aborted.

[Richard Damon]

On 12/7/22 12:20 AM, Skep Dick wrote:
> On Tuesday, 6 December 2022 at 20:58:56 UTC+2, richar...@gmail.com wrote:
>> Right, "Tautology" is a lable that applies to a statement in the systems
>> it applies to.
> Precisely! So it can't be true in ALL interpretations/systems/models.
>
> Because it's a statement made in the internal language of the system!

But the definition didn't SAY all "Systems", so requiring it to is an error.

>
>>> Hence the part where you have to play amnesiac and forget that a tautology has to be true in all models.
>> All Models, not all systems.
> Yes, they are. There's logic. And then there are models/systems of logic.
>
> Classical logic is a model/system of logic.
> Intuitionistic is a model/system of logic.
> Modal logic is a model/system of logic.
> Temporal logic is a model/system of logic.
>
>> You don't seem to understand the difference.
> I understand how synonyms work just fine.

But in the system of the definition, they aren't.

>
>>> It was a misnomer all along.
>> Seems to be a misunderstanding of your part.
> You keep projecting your misunderstanding onto me.

No, you keep projecting your misunderstandings onto everyone else.

Yoour problem is you have adopted a system that can't handle certain
concepts, so you assume that no systems can.

>
>>> It's literally false in every instance of reality where Jack wasn't 11 a year ago!
>> Nope. You don't understand the meaning of the statement then.
> Q.E.D. Projection.

Yep, your projecting. I have explained the difference, you just reject
it because you don't understand.

>
>>>> We could even likely strengthen the statement to "ONLY if Jack wa eleven
>>>> a year ago, will he now be twelve."
>>> Strengthen it all you want. It's a conditional - therefore not a tautology.
>> Nope, you don't seem to understand the stateent.
> Q.E.D. Projection.

Agaibn, YOUR projection,

IT doesn't match YOUR system of definitions, so you reject it.

Just shows the limitation of your system.

[Richard Damon]

I never said that your simulation was not a correct PARTIAL simulation.

You seem to be incapable of understanding the words though.

>
> Mathematical induction proves the behavior of what the additional steps
> would be at the point where N steps have been completed. In the case of
> H(D,D) and the Peter Linz embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ the next steps would simply
> repeat the simulation continuously until aborted.
>

Nope, because if you corrrect simulate into the call of H(D,D) inside
the function D, you will see it simulated its copy of D till it reaches
the call to H(D,D) inside it, and then abort and return 0.

Thus, you do NOT have the infinite repeat that you claim.

Your problem is that when you do the step, you INCORRECTLY change the H
that D calls, which is NOT allowed.

The input D to H must remain calling the H that is exactly the H that
acts like the original. This H can not simulate past the point of the
call to H(D,D), becuase, by its definition in code it stops there.

Your failure to understand this fact just shows you are ignorant of the
rules because you have made yourself stupid.

Your repeating the statement after it has been pointed out makes you
just apparently a pathological liar, as you can't seem to keep yourself
from saying these lies.

You show that you do NOT understand the basics of logic and that you are
just trying to feed your own ego in making yourself think you have
discovered something great. You haven't, you have just demonstrated how
stupid you are.

Genius may hit the target that others couldn't see (until it was hit),
Stupid tries to hit targets that aren't actually there.

[Jeff Barnett]

On 12/7/2022 9:37 AM, Skep Dick wrote:
> On Wednesday, 7 December 2022 at 18:30:09 UTC+2, olcott wrote:
>> On 12/7/2022 10:18 AM, Skep Dick wrote:

<SIP> <SNIP>

>>> What did your parents do to you that you've spent your entire life trying to "be right" ?
>>>
>> If humanity is going to survive we must overturn the Tarski
>> undefinabilty theorem to derive an objective criterion measure that
>> divides truth from lies.
> 🤣🤣🤣🤣🤣🤣🤣🤣
>
> https://en.wikipedia.org/wiki/Messiah_complex

I'm trying to understand whether you think the same aura emanates from
you as PO from. For example, the above indicates that you are trying to
save him. As one that has watched these threads off an on for a while, I
want to warn you that you will not succeed. He's a troll within a nut
case within a troll. If he believes himself a savior, that is merely
another misconception.

Moving on, tell us about you; you seem to have a good dose of PO in you.
On the surface it seems to be a case "of takes one to know one" but we
all seek a deeper understanding of the contestants. N.B. I'm not
questioning the veracity of your comments about PO - it's almost
impossible to attribute too many negatives to him. However, in your
capacity as his savior you must demonstrate success, compassion, and a
deep understanding of his errors and omissions. I'm not sure you've met
that burden yet.
--
Jeff Barnett

[Skep Dick]

That's because I am not laughing to his benefit. I am laughing at his expense.

There's no such thing as "depth" of understanding. The abyss of formalism is all syntax - semantics is the land of illusions.

[olcott]

Severe anthropogenic climate change proven entirely with verifiable facts
https://www.researchgate.net/publication/336568434_Severe_anthropogenic_climate_change_proven_entirely_with_verifiable_facts

It is an objectively verifiable fact that climate change disinformation
is effectively preventing sufficient climate change mitigation. That
heat waves are beginning to cause mass deaths is also objectively proven:

How one heatwave killed 'a third' of a bat species in Australia
https://www.bbc.com/news/world-australia-46859000

[Skep Dick]

All of that is completely unrelated to anything you are doing.

[olcott]

It would seem that way to anyone that only glances at my words before
forming a rebuttal.

The Tarski Undefinability theorem that "proves" the notion of truth
cannot be formalized is directly related to its isomorphism of the 1931
Gödel incompleteness theorem.

When the notion of truth itself cannot be precisely nailed down then
lies such as climate change disinformation can far too easily slip
through undetected.

The Tarski Undefinability theorem has as its basis the inability to
prove that the Liar Paradox is true.

It would then be possible to reconstruct the antinomy of the
liar in the metalanguage, by forming in the language itself
a sentence x such that the sentence of the metalanguage which
is correlated with x asserts that x is not a true sentence.
https://liarparadox.org/Tarski_247_248.pdf

When we have a mathematical specification of the notion of truth thenn
(then and only then) can we have mathematical proofs that discern lies
from truth.

[Richard Damon]

You don't understand what Tarski was saying. He is talking about
formalization WITHIN the system can not define what is true in that system.

This is really just a expansion on the incompleteness theorem, that
there exists within any sufficiently prowerful system of logic,
statements which can not be proved to be true (or false).

> When the notion of truth itself cannot be precisely nailed down then
> lies such as climate change disinformation can far too easily slip
> through undetected.

Which isn't what it says.

>
> The Tarski Undefinability theorem has as its basis the inability to
> prove that the Liar Paradox is true.
>
>    It would then be possible to reconstruct the antinomy of the
>    liar in the metalanguage, by forming in the language itself
>    a sentence x such that the sentence of the metalanguage which
>    is correlated with x asserts that x is not a true sentence.
>    https://liarparadox.org/Tarski_247_248.pdf
>
> When we have a mathematical specification of the notion of truth thenn
> (then and only then) can we have mathematical proofs that discern lies
> from truth.
>

Which means we can not form a MATHEMATICAL test that universally
discerns Truth from Lies.

We can not define WITHIN THE SYSTEM, a definition of Truth.

We are not limited to being "within the system", so from outside the
system, we have the possiblity to use extra-system logic to determine
the truth.

It does seem to say that within the system that we do exist in, there
will be SOME statement we can not show to be true or false, but that
doesn't mean that most statements can have a truth value applied to them.

Thus, the truth of all your statements doesn't get negated by Tarski.
Tarski just says we can't create a "formula" that universally shows if a
statement is true or not.

[olcott]

The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))

The fundamental problem is that the conventional
definition of incompleteness is not actually proof
that T is Incomplete, it is merely proof that φ
in T is not a truth bearer.

Copyright PL Olcott 2018

[Richard Damon]

Except the φ from the Godel Proof IS a Truth bearer as all statements of
the form 'X is Provable' or 'X is not Provable' are by necessity truth
bearers, since a given statement will either be Provable or it will not.

This comes from, in part, that if X is not a Truth Bearer, then X can
NOT be proven, just as if X is false, it can not be proven either.

If you claim that Godel statement is not a Truth Bearer, then you are
saying that the provablitiy of a statement is not a Truth Bearer, Thus
there exist statement which are neither Provable or Unprovable, which is
again, the incompleteness property, so you are agreeing with Godel by
trying to disagree with him.

In other words, you don't understand what you are talking about.

[olcott]

As Ludwig Wittgenstein pointed out:
'True in Russell's system' means, as was said:
proved in Russell's system; and 'false in Russell's system'
means:the opposite has been proved in Russell's system
https://www.liarparadox.org/Wittgenstein.pdf

therefore: φ in T is not a truth bearer in T

[Richard Damon]

But Wittgenstein is WRONG that True means PROVABLE. That fails the
definition of True used in the field.

Simple matter of fact.

FAIL.

Note, the assumptiion of Asserting True is Provable is that it means we
can not use the law of the excluded middle, which breaks a lot of the
logic the system is built on.

In fact, from other proofs we have, it shows that the logic system can
not handle systems above a certain level of complexity without going
inconsistent.

[olcott]

It does not mean that Wittgenstein is wrong, he knows these things on
the much deeper basis of their philosophical foundation as opposed to
and contrast with memorizing a set of rules dogmatically and never
bothering to question their foundational basis.

The only correct way to determine that an analytical expression of
formal or natural language is true is:

(a) Expressions of language that are stipulated to be true such as
Haskell Curry elementary theorems:

Thus, given T, an elementary theorem is an elementary statement
which is true. https://www.liarparadox.org/Haskell_Curry_45.pdf

encoded in Mendelson as ⊢C, thus provable on the basis that it
is a theorem.

(b) the application of truth preserving operations to (a) and/or the
results of (b).

Thus showing that φ is true in T requires φ is provable in T.

> Simple matter of fact.
>
> FAIL.
>
> Note, the assumptiion of Asserting True is Provable is that it means we
> can not use the law of the excluded middle, which breaks a lot of the
> logic the system is built on.
>
> In fact, from other proofs we have, it shows that the logic system can
> not handle systems above a certain level of complexity without going
> inconsistent.
>

[Richard Damon]

No, HE IS WRONG about the Formal Logic systme of Mathemeatics.

This has been proven.

>
> The only correct way to determine that an analytical expression of
> formal or natural language is true is:

Note, you are talking about KNOWING it is true, not what makes it true.

>
> (a) Expressions of language that are stipulated to be true such as
> Haskell Curry elementary theorems:
>
> Thus, given T, an elementary theorem is an elementary statement
> which is true. https://www.liarparadox.org/Haskell_Curry_45.pdf
>
> encoded in Mendelson as ⊢C, thus provable on the basis that it
> is a theorem.
>
> (b) the application of truth preserving operations to (a) and/or the
> results of (b).
>
> Thus showing that φ is true in T requires φ is provable in T.

It may be needed to SHOW that it is True, but not to BE True.

>
>> Simple matter of fact.
>>
>> FAIL.
>>
>> Note, the assumptiion of Asserting True is Provable is that it means
>> we can not use the law of the excluded middle, which breaks a lot of
>> the logic the system is built on.
>>
>> In fact, from other proofs we have, it shows that the logic system can
>> not handle systems above a certain level of complexity without going
>> inconsistent.
>>
>

Since you confuse Truth with Knowledge, you are condemned to not to be
able to actually know the truth.

One problem is that Mathematics is not a "purely analyitcal" field, but
allows for "empirical" truths to exist that arise out of the
interactions of the axioms of the field.

For instance, even if we can never find a proof of the answer for the
Collatz conjecture, what ever is the correct answer, is True.

The Collatz Conjecture by its nature is a Truth Bearer, it allows for no
middle ground, either there does exist a number that when put throuth
the 3x+1 or /2 operation never reduces to 1, or there doesn't.

The provability of the statement doesn't matter, becuase the answer just IS.

[olcott]

That no proof is currently known to exist is not at all the same thing
as no proof can possibly exist because the expression of language is
self-contradictory.

Every element of the entire set of analytical truth is only true because
(as explained above) it is a part of a mutually self-defining
interlocking semantic tautology.

The former is a truth bearer with an unknown truth value and the latter
is not a truth bearer at all.

[Richard Damon]

??? What is the self-contraction? Are you denying that Reality actually
exists?

>
> Every element of the entire set of analytical truth is only true because
> (as explained above) it is a part of a mutually self-defining
> interlocking semantic tautology.
>

Right, but the field of Mathematics goes BEYOND just "analytic" truth.

> The former is a truth bearer with an unknown truth value and the latter
> is not a truth bearer at all.
>

So, what do you call a statement that MUST be True of False, because it
is a sttement of Empirical Fact, but can not be proven?

That is what statements like "X is Provable" or "X is not Provable" are.

Since it MUST be either True or False that a proof can exist, "X is
Provable" and "X is Not Provable" are Truth Bearers.

Thus, Godel sentence is a Truth Bearer, and shown to be unprovable.

[Jeff Barnett]

Thanks for the clarification. I was hoping we hadn't lost you to a
hopeless task and I'm now satisfied.

> There's no such thing as "depth" of understanding. The abyss of formalism is all syntax - semantics is the land of illusions.

One may still take the position that a formalism and a particular
semantics associated with it is a reasonable model of our reality. Of
course the eventually flaw with that is that we will learn more as time
passes and we must abandon the current model. The other reason we might
abandon the current model is we learn more about ourselves and our
fallibility. PO is, as has been previously noted, impervious to learning
more about reality and especial about his totally inadequate self.

Carry on with the good fight.
--
Jeff Barnett

[olcott]

This sentence is not true.
This sentence cannot be proven.

>>
>> Every element of the entire set of analytical truth is only true because
>> (as explained above) it is a part of a mutually self-defining
>> interlocking semantic tautology.
>>
>
> Right, but the field of Mathematics goes BEYOND just "analytic" truth.
>

Mathematics ⊂ "analytic" truth, thus cannot possibly go beyond

"analytic" truth.

>> The former is a truth bearer with an unknown truth value and the latter
>> is not a truth bearer at all.
>>
>
> So, what do you call a statement that MUST be True of False, because it
> is a sttement of Empirical Fact, but can not be proven?
>

Not empirical at all. An empirical fact requires sense data from the
sense organs: I see a television in my living room right now.

> That is what statements like "X is Provable" or "X is not Provable" are.
>
> Since it MUST be either True or False that a proof can exist, "X is
> Provable" and "X is Not Provable" are Truth Bearers.
>
> Thus, Godel sentence is a Truth Bearer, and shown to be unprovable.

Every sentence that meets the following definition is simply not a truth
bearer in T and thus does not show that T is incomplete.

The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))

Every self-contradictory sentence in T meets the above definition.

[olcott]

Of the analytic/synthetic distinction the analytic side that I focus on
has nothing to do with models of the world.

Mathematicians and logicians make sure to never bother to look at the
philosophical foundations of these things thus misconstrue everything
that departs from their learned-by-rote dogmatic rules as an error.

[Richard Damon]

So?

THe first Sentence is not a Truth Bearer, that is true.

The Second Sentence is True, as The truth value of "This Sentence Can
not be Proven" has to be TRUE, as if it was false, it would meen it wqs
provable, and all provable sentences are True.

Where is the self-contradiction in the statement about proof?

You are just showing you don't know what you are talking about.

SOmething either IS or IS NOT provable, there is an excluded middle for
that sort of statement, as it deals with the EXISTANCE of a proof. A
things either DO or DO NOT exist.

>
>>>
>>> Every element of the entire set of analytical truth is only true because
>>> (as explained above) it is a part of a mutually self-defining
>>> interlocking semantic tautology.
>>>
>>
>> Right, but the field of Mathematics goes BEYOND just "analytic" truth.
>>
>
> Mathematics ⊂ "analytic" truth, thus cannot possibly go beyond
> "analytic" truth.

Nope. Where do you get that?

>
>>> The former is a truth bearer with an unknown truth value and the latter
>>> is not a truth bearer at all.
>>>
>>
>> So, what do you call a statement that MUST be True of False, because
>> it is a sttement of Empirical Fact, but can not be proven?
>>
>
> Not empirical at all. An empirical fact requires sense data from the
> sense organs: I see a television in my living room right now.

No, empirical relats to testing with a reality. The axioms of
mathemematic define a reality that can be tested to determine emperical
fact. The "Mind" can be consider a valid sense organ.

Note, if you want to restrict Emperical to only senses, then you can't
limit analytical to only proven or your analytical systems either can't
express the properties of Mathematics or become inconsistent.

You are about a half a century behind in your understanding of logic.

You are just repeating the errors of history because you refused to
learn from it.

>
>> That is what statements like "X is Provable" or "X is not Provable" are.
>>
>> Since it MUST be either True or False that a proof can exist, "X is
>> Provable" and "X is Not Provable" are Truth Bearers.
>>
>> Thus, Godel sentence is a Truth Bearer, and shown to be unprovable.
>
> Every sentence that meets the following definition is simply not a truth
> bearer in T and thus does not show that T is incomplete.

Source of this claim?

You are just proving your stupidity.

>
> The conventional definition of incompleteness:
> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
>
> Every self-contradictory sentence in T meets the above definition.
>

So you have your logic backwards.

No, because self-contradictiory statement fail to meet the requirement
of φ, which must be a truth bearer.

Note, you haven't explained how Provable(x) can ever be a non-truth bearer.

[Richard Damon]

On 12/7/22 4:56 PM, olcott wrote:
>
> Of the analytic/synthetic distinction the analytic side that I focus on
> has nothing to do with models of the world.
>
> Mathematicians and logicians make sure to never bother to look at the
> philosophical foundations of these things thus misconstrue everything
> that departs from their learned-by-rote dogmatic rules as an error.
>

No, Mathematicians have LONG understood that the classical Philosophical
foundations of "Analytic" vs "Synthetic" didn't meet the requirements of
Mathematics.

Mathematics has long allowed things to be thought of a possible "True"
even if not yet proven.

It was "hoped" that Mathematics could be shown to be provable, but being
provable took second seat to being useful.

About a Century ago, it was finally proven that in Mathematics, for any
field sufficiently advanced, not all true statements were provable.

Some Philosophers had problems with this and tried to work around the
issues, but FAILED (as have you).

Mathematcians have just accepted that their field is by necessity
"Incomplete" and not fully provable, and live with it.

You are just showing yourself to be a century behind in your knowledge,
and repeating the failed arguements of the losing side.


Think about it, why was the question of completeness brought up at all,
unless they accepted that there could be "True" statements that were not
provable.

IF you define Truth as Provable, then you can not be incomplete. The
problem is when you do that, you find that either your systems are
strictly limited in what they can handle, or they become inconsistent.

Thus, it is clear that Mathemeticians never actually thought of Truth
being only provable as a basis of their system, it was just a hope.

And, it turned out that consistancy wasn't provable within the system,
you needed to go to some larger system to prove it, and needed to hope
that larger system was still consistent or the proof didn't actually
mean anything, as an inconsistent system can prove anything.