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Godel's theorem once again, why not?
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David Petry
2017-06-13 19:44:08 UTC
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First of all, I can't make sense of the recent sci.math discussions of Gödel's theorems. But I would like to point out that in the logic of natural language, Gödel's theorems are pretty silly, to say the least.

In the logic of natural language, the following is true: we are justified in saying 'if A, then B' if and only if we have a way of converting a proof of A into a proof of B.

Then Gödel's (second) theorem, which roughly speaking says, "if mathematics is consistent, then mathematics cannot prove that mathematics is consistent" can be translated into "we have a way of converting a proof of the consistency of mathematics into a proof that there does not exist a proof of the consistency of mathematics", which is pretty obviously silly.
p***@gmail.com
2017-06-13 21:12:14 UTC
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Post by David Petry
First of all, I can't make sense of the recent sci.math discussions of Gödel's theorems. But I would like to point out that in the logic of natural language, Gödel's theorems are pretty silly, to say the least.
In the logic of natural language, the following is true: we are justified in saying 'if A, then B' if and only if we have a way of converting a proof of A into a proof of B.
Then Gödel's (second) theorem, which roughly speaking says, "if mathematics is consistent, then mathematics cannot prove that mathematics is consistent" can be translated into "we have a way of converting a proof of the consistency of mathematics into a proof that there does not exist a proof of the consistency of mathematics", which is pretty obviously silly.
Yes it seems that you get it too. What I have done is not just show that its silly, I have shown the infinitely recursive structure of G.

http://liarparadox.org/Provability_with_Minimal_Type_Theory.pdf
David Petry
2017-06-13 22:29:29 UTC
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Post by David Petry
First of all, I can't make sense of the recent sci.math discussions of Gödel's theorems. But I would like to point out that in the logic of natural language, Gödel's theorems are pretty silly, to say the least.
In the logic of natural language, the following is true: we are justified in saying 'if A, then B' if and only if we have a way of converting a proof of A into a proof of B.
Then Gödel's (second) theorem, which roughly speaking says, "if mathematics is consistent, then mathematics cannot prove that mathematics is consistent" can be translated into "we have a way of converting a proof of the consistency of mathematics into a proof that there does not exist a proof of the consistency of mathematics", which is pretty obviously silly.
Yes it seems that you get it too.
But I'm not convinced that you do "get it".

In the logic of natural language, a "proof" is a compelling argument. In formal mathematics, a "proof" is a purely formal construct in the game of mathematics, and in the game of mathematics, words and sentences have no meaning.
Post by p***@gmail.com
What I have done is not just show that its silly, I have shown the infinitely recursive structure of G.
http://liarparadox.org/Provability_with_Minimal_Type_Theory.pdf
One of the properties of silly things is that anything you say about a silly thing is a silly thing to say.
p***@gmail.com
2017-06-13 23:26:55 UTC
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Post by David Petry
Post by p***@gmail.com
Post by David Petry
First of all, I can't make sense of the recent sci.math discussions of Gödel's theorems. But I would like to point out that in the logic of natural language, Gödel's theorems are pretty silly, to say the least.
In the logic of natural language, the following is true: we are justified in saying 'if A, then B' if and only if we have a way of converting a proof of A into a proof of B.
Then Gödel's (second) theorem, which roughly speaking says, "if mathematics is consistent, then mathematics cannot prove that mathematics is consistent" can be translated into "we have a way of converting a proof of the consistency of mathematics into a proof that there does not exist a proof of the consistency of mathematics", which is pretty obviously silly.
Yes it seems that you get it too.
But I'm not convinced that you do "get it".
In the logic of natural language, a "proof" is a compelling argument. In formal mathematics, a "proof" is a purely formal construct in the game of mathematics, and in the game of mathematics, words and sentences have no meaning.
Post by p***@gmail.com
What I have done is not just show that its silly, I have shown the infinitely recursive structure of G.
http://liarparadox.org/Provability_with_Minimal_Type_Theory.pdf
One of the properties of silly things is that anything you say about a silly thing is a silly thing to say.
Well it is not really silly, so much as logically incoherent. Until Minimal Type Theory (MTT) there was no way to explicitly express exactly how the constituent parts of an expression interact with one another. The directed acyclic graph (DAG) of page four of my above link now makes this unequivocally clear.
Jim Burns
2017-06-14 00:14:49 UTC
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On Tuesday, June 13, 2017 at 2:12:27 PM UTC-7,
Post by p***@gmail.com
What I have done is not just show that its silly,
I have shown the infinitely recursive structure of G.
http://liarparadox.org/Provability_with_Minimal_Type_Theory.pdf
[to Peter Olcott]

By G, we mean a sentence G such that, for a formal system T,
there is a proof such that
T |- G <-> ~Provable-in-T([G])

You claim that you have replaced Goedel's G with your own G,
Olcott's G.
G( ~∃Γ ⊂ PM (Γ ⊢ G) )
or maybe something else. But you are very definite about
something replacing what Goedel proved is true-iff-not-provable.

I would like you to prove that Olcott's G is an effective
replacement for Goedel's G.

The only important characteristic of Goedel's G is that
there is a proof in T , so that
T |- G <-> ~Provable-in-T([G])
so, please prove that.

-- In what formal system is Olcott's G an effective replacement
of Goedel's G? I call it T, but that says nothing of course.

Presumably T will not be your MTT, whenever you get around to
describing it. Is it P from Goedel's paper? Is it Robinson
arithmetic Q ? Is it something else? Note that a single
specific G is true-iff-not-provable in a single specific
formal system. Please pick a system and show Olcott's G
is true-iff-not-provable in that system.

[end, to Peter Olcott]

----
One of the properties of silly things is that
anything you say about a silly thing is
a silly thing to say.
[to David Petry]

Although, as a general matter, that seems to be a reasonable
thing to say (and not silly), that does NOT seem to me to
be entirely true. For example, it does not seem silly to
say (correctly) that something is a silly thing to say.

Perhaps it is more correct, if less lyrical, to say
that there aren't _many_ non-silly things to say about
silly things. Once we've decided X is silly, the discussion
of X seems to wither, other than silliness. Is that
your experience, too?
p***@gmail.com
2017-06-14 15:03:23 UTC
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Post by Jim Burns
On Tuesday, June 13, 2017 at 2:12:27 PM UTC-7,
Post by p***@gmail.com
What I have done is not just show that its silly,
I have shown the infinitely recursive structure of G.
http://liarparadox.org/Provability_with_Minimal_Type_Theory.pdf
[to Peter Olcott]
By G, we mean a sentence G such that, for a formal system T,
there is a proof such that
T |- G <-> ~Provable-in-T([G])
You claim that you have replaced Goedel's G with your own G,
Olcott's G.
G( ~∃Γ ⊂ PM (Γ ⊢ G) )
or maybe something else. But you are very definite about
something replacing what Goedel proved is true-iff-not-provable.
I would like you to prove that Olcott's G is an effective
replacement for Goedel's G.
The only important characteristic of Goedel's G is that
there is a proof in T , so that
T |- G <-> ~Provable-in-T([G])
so, please prove that.
-- In what formal system is Olcott's G an effective replacement
of Goedel's G? I call it T, but that says nothing of course.
Presumably T will not be your MTT, whenever you get around to
describing it. Is it P from Goedel's paper? Is it Robinson
arithmetic Q ? Is it something else? Note that a single
specific G is true-iff-not-provable in a single specific
formal system. Please pick a system and show Olcott's G
is true-iff-not-provable in that system.
[end, to Peter Olcott]
You have to pay attention to understand me. Mitch criticized me for repeating myself yet no one has ever actually read my words trying to understand them except the founder of [the foundations of logic] on Facebook.

∀L ∈ Formal_Systems G <assign alias name> ( ~∃Γ ⊂ L (Γ ⊢ G) )

01 ∀ (2)(5)
02 ∈ (3)(4)
03 L
04 Formal Systems
05 ~ (6) // alias for G
06 ∃ (7)(10)
07 ⊂ (8)(3)
08 Γ
09 ⊢ (8)(5) // cycle indicate infinite evaluation loop

Since the above expression never terminates its evaluation with either satisfied or unsatisfied it is not a truth bearer. The expression's pathological self-reference specifies infinite recursion.

After G has been assigned as an alias for the RHS the name G and the expression on the RHS are considered to be identical. The name G is merely a short-hand way of saying the entire RHS.

Tarski confused this when he used the quoted expression as its name. He used names like computer science pointers that had to always be dereferenced prior to use. In Minimal Type Theory expressions are represented in directed acyclic graphs, thus their name is merely the root node of the expression. No dereferencing is ever needed.

Now for the first time ever the pathological self-reference error is formalized. No longer will we merely have some vague idea that self-reference sometimes causes the evaluation of expression to produce unexpected results. Now for the first time ever we can divide the error of pathological self-reference from self-reference that is not erroneous.

This shows five examples of Minimal Type Theory along with their graphs.
http://liarparadox.org/Provability_with_Minimal_Type_Theory.pdf
Jim Burns
2017-06-14 16:58:26 UTC
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On Tuesday, June 13, 2017 at 7:14:53 PM UTC-5,
Post by Jim Burns
-- In what formal system is Olcott's G an effective
replacement of Goedel's G? I call it T, but that says
nothing of course.
Presumably T will not be your MTT, whenever you get
around to describing it. Is it P from Goedel's paper?
Is it Robinson arithmetic Q ? Is it something else?
Note that a single specific G is true-iff-not-provable
in a single specific formal system. Please pick a system
and show Olcott's G is true-iff-not-provable in that
system.
[end, to Peter Olcott]
You have to pay attention to understand me.
I am not convinced that there is anything there to
understand. You show the signs of someone who is using
technical terms without understanding, as technobabble
or bafflegab.

Nonetheless, despite my misgivings, I am engaging your
argument from time to time, depending upon my mood of the
moment. I've given you a clear, reasonable request. You
are not addressing my request.

Kurt Goedel produces a sentence G, which he proves is
true-iff-not-provable (subject to stated assumptions).

Peter Olcott produces a sentence G, which he states
has some sort of correspondence to Goedel's G.
I'm asking you to show that correspondence.

In particular, please show, for Olcott's G and a formal
system T _of your choice_ ,
T |- G <-> ~Provable-in-T([G])

If you're not going to show that correspondence, will
you say _why_ you're not going to show that correspondence,
or give some other sort of argument _why_ what you say about
Olcott's G should be applied to Goedel's G?

(Please note: I am asking for an _argument_ here, not a
restatement of your claim. You know what an argument is,
right?)
Mitch criticized me for repeating myself yet no one
has ever actually read my words trying to understand
them except the founder of [the foundations of logic]
on Facebook.
[...]
Since the above expression never terminates its
evaluation with either satisfied or unsatisfied
it is not a truth bearer. The expression's pathological
self-reference specifies infinite recursion.
The above expression to which you refer is _Olcott's G_ ,
your own creation. Show me that it's relevant to Goedel's G.

_Goedel's G_ is provably a truth bearer, provable from
the definition of the formal language in which G is
a sentence. _Goedel's G_ does not specify infinite
recursion. You can find definitions for all of its
terms that lead you back to _arithmetic_ . They're
right there in his proof.

It's as though I claimed that pigs do not fly, and you
show me some pigeons and claim that proves I'm wrong.
"How are those pigs?" I ask.
"LOOK AT MY PIGEONS!" you respond.
Tell my why I should look at your pigeons.
p***@gmail.com
2017-06-15 03:57:21 UTC
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Post by Jim Burns
On Tuesday, June 13, 2017 at 2:12:27 PM UTC-7,
Post by p***@gmail.com
What I have done is not just show that its silly,
I have shown the infinitely recursive structure of G.
http://liarparadox.org/Provability_with_Minimal_Type_Theory.pdf
[to Peter Olcott]
By G, we mean a sentence G such that, for a formal system T,
there is a proof such that
T |- G <-> ~Provable-in-T([G])
You can't see that the above expression is infinitely recursive?
Is it the quoting / unquoting that prevents you from seeing this?

"@" means the LHS is assigned as an alias for the RHS

There is no referencing / dereferencing needed, G is one and the same thing as the expression that refers to G. G is not referring to its name, G is referring to itself.

Here are two di-graphs the first one has pathological self-reference the second one does not.

G @ ~Provable-in-T(G)
---------------------------------
01 ~ (2) // Alias for G
02 Provable-in-T (1) // infinite evaluation loop

X @ ~Provable-in-T(Y)
--------------------------------
01 ~ (2) // Alias for X
02 Provable-in-T (3)
03 Y // evaluation terminates at this leaf node
Jim Burns
2017-06-15 05:00:17 UTC
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Post by Jim Burns
On Tuesday, June 13, 2017 at 2:12:27 PM UTC-7,
Post by p***@gmail.com
What I have done is not just show that its silly,
I have shown the infinitely recursive structure of G.
http://liarparadox.org/Provability_with_Minimal_Type_Theory.pdf
[to Peter Olcott]
By G, we mean a sentence G such that, for a formal system T,
there is a proof such that
T |- G <-> ~Provable-in-T([G])
You can't see that the above expression is infinitely recursive?
Is it the quoting / unquoting that prevents you from seeing this?
Please prove _what I asked you to prove_ .

Pick one of _our_ terrible, horrible, very bad formal
systems -- _any one_ of them, _whichever one_ you think
that thing you're calling G is a true-iff-not-provable
sentence for. Tell us which formal system T is.

Then, prove _in that terrible, horrible, etc formal system_
T this sentence, with what you think G is substituted in:
G <-> ~Provable-in-T([G])

Thanks in advance.

If you have some other way of determining that what
what you call G corresponds in some way to a
true-iff-not-provable sentence in a (regular old)
formal system _of your choice_ , please share that
instead.

Does this formal system TBA (to be announced)[1]
have an axiom:
"The Olcott G corresponds to the Goedel G"
? That would be in line with "7 > 3" I suppose, but,
if so, what would you do to refute Goedel in some
formal system without such a convenient axiom?

[1]
It occurs to me that TBA captures the essence
of your own forthcoming formal system better than
MTT does. I am considering whether I should start
referring to the language L[TBA] and the inference
rules of TBA and the TBA axioms.

Of course, if you told us what those things were,
my tiny little joke would fall flat, But we all
know how likely that is, don't we?
Suppose a formal system didn't have "@" -- for example,
as no regular old formal systems that I know of has
"@" or anything that does what you say "@" does --
what then?
Post by p***@gmail.com
There is no referencing / dereferencing needed, G is one and the same thing as the expression that refers to G. G is not referring to its name, G is referring to itself.
Here are two di-graphs the first one has pathological self-reference the second one does not.
---------------------------------
01 ~ (2) // Alias for G
02 Provable-in-T (1) // infinite evaluation loop
--------------------------------
01 ~ (2) // Alias for X
02 Provable-in-T (3)
03 Y // evaluation terminates at this leaf node
p***@gmail.com
2017-06-15 14:18:30 UTC
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Post by Jim Burns
Post by p***@gmail.com
Post by Jim Burns
On Tuesday, June 13, 2017 at 2:12:27 PM UTC-7,
Post by p***@gmail.com
What I have done is not just show that its silly,
I have shown the infinitely recursive structure of G.
http://liarparadox.org/Provability_with_Minimal_Type_Theory.pdf
[to Peter Olcott]
By G, we mean a sentence G such that, for a formal system T,
there is a proof such that
T |- G <-> ~Provable-in-T([G])
You can't see that the above expression is infinitely recursive?
Is it the quoting / unquoting that prevents you from seeing this?
Please prove _what I asked you to prove_ .
Sure dodge the question again.
Post by Jim Burns
Pick one of _our_ terrible, horrible, very bad formal
systems -- _any one_ of them, _whichever one_ you think
that thing you're calling G is a true-iff-not-provable
sentence for. Tell us which formal system T is.
I already did this Predicate Logic above FOPL.
Post by Jim Burns
Then, prove _in that terrible, horrible, etc formal system_
G <-> ~Provable-in-T([G])
Thanks in advance.
You already have your proof right there in the expression that you just wrote. You can't see it ??? If you can't see it then go back to prior prior replay and look for additional details.
Post by Jim Burns
If you have some other way of determining that what
what you call G corresponds in some way to a
true-iff-not-provable sentence in a (regular old)
formal system _of your choice_ , please share that
instead.
Does this formal system TBA (to be announced)[1]
"The Olcott G corresponds to the Goedel G"
? That would be in line with "7 > 3" I suppose, but,
if so, what would you do to refute Goedel in some
formal system without such a convenient axiom?
[1]
It occurs to me that TBA captures the essence
of your own forthcoming formal system better than
MTT does. I am considering whether I should start
referring to the language L[TBA] and the inference
rules of TBA and the TBA axioms.
Of course, if you told us what those things were,
my tiny little joke would fall flat, But we all
know how likely that is, don't we?
as no regular old formal systems that I know of has
what then?
Post by p***@gmail.com
There is no referencing / dereferencing needed, G is one and the same thing as the expression that refers to G. G is not referring to its name, G is referring to itself.
Here are two di-graphs the first one has pathological self-reference the second one does not.
---------------------------------
01 ~ (2) // Alias for G
02 Provable-in-T (1) // infinite evaluation loop
--------------------------------
01 ~ (2) // Alias for X
02 Provable-in-T (3)
03 Y // evaluation terminates at this leaf node
Jim Burns
2017-06-15 15:47:10 UTC
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On Thursday, June 15, 2017 at 12:00:23 AM UTC-5,
Post by Jim Burns
Please prove _what I asked you to prove_ .
Sure dodge the question again.
So, when I ask again a question you didn't answer,
you consider me to have dodged your diversionary
question.

The fact is, until you establish that you're talking
about _Goedel's G_ , not just _Olcott's G_ , your
loops in _Olcott's G_ are irrelevant.

I'm helping you out by pressing for details here,
not that I expect you to thank me.
Post by Jim Burns
Pick one of _our_ terrible, horrible, very bad formal
systems -- _any one_ of them, _whichever one_ you think
that thing you're calling G is a true-iff-not-provable
sentence for. Tell us which formal system T is.
I already did this Predicate Logic above FOPL.
FOPL is a class of formal systems.

If you asked "Who ate the last of the pizza?",
and I answered, "A human being ate it"
would you consider that enough of an answer?

Write down what you say Olcott's G is.
Show that it is a statement in a particular formal
system T, by the definition of a statement in the
language of T. That's enough for you to get started.

You will also need to formalize what Provable-in-T()
is and show that, however you formalize it, it behaves
as advertised.

And you will need to show that, if Olcott's G is true.
then Provable-in-T([G]) is false, and if Olcott's G is
false, then Provable([G]) is true.

No, I'm not going to take your word for all of that.
Post by Jim Burns
Then, prove _in that terrible, horrible, etc formal system_
G <-> ~Provable-in-T([G])
Thanks in advance.
You already have your proof right there in the
expression that you just wrote. You can't see it ???
If you can't see it then go back to prior prior replay
and look for additional details.
No. I'm not going to pretend you gave me a proof
and I just didn't understand your notation.

I asked for a proof _in T_ . Where is your proof _in T_ ?

What is the language of T? What are the axioms of T?
What are the rules of inference of T? I'm not asking
about MTT, I'm asking about what is supposedly
a true-iff-not-provable sentence in some sense in
some formal system T _as we mean formal system_ .

You say you've "simplified" Goedel's G to Olcott's G.
Show me that your simplification hasn't simplified away
the reason we're talking about it at all:
T |- G <-> ~Provable-in-T([G])

If there is no specific T for which Olcott's G makes
that provable, then Olcott's G cannot be some version
of Goedel's G.

Compare that to the natural numbers.
Suppose that I invent my own version of the natural
numbers, and I call what I write as @ "zero".

Is it really zero? A good test is whether @ is
the successor of any other natural number. If it is
the successor of something, it's not reasonable to
call @ "zero" in the natural numbers. I'm asking
you to do the same for Olcott's G.
Peter Percival
2017-06-14 10:58:22 UTC
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Post by David Petry
In the logic of natural language, a "proof" is a compelling argument.
In formal mathematics, a "proof" is a purely formal construct in the
game of mathematics, and in the game of mathematics, words and
sentences have no meaning.
One way to make natural language proofs compelling is to formalize them.
What would be the point of that if formal proofs had no meaning?
--
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
m***@wp.pl
2017-06-14 11:43:11 UTC
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Post by Peter Percival
One way to make natural language proofs compelling is to formalize them.
What would be the point of that if formal proofs had no meaning?
Exactly what you said - the point is to make them compelling.
David Petry
2017-06-14 19:32:34 UTC
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Post by m***@wp.pl
Post by Peter Percival
One way to make natural language proofs compelling is to formalize them.
What would be the point of that if formal proofs had no meaning?
Exactly what you said - the point is to make them compelling.
Well, yes, but if in the process of formalizing proofs, words must be given new definitions, then problems arise.
Julio Di Egidio
2017-06-13 21:22:57 UTC
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Post by David Petry
In the logic of natural language, the following is true: we are justified
in saying 'if A, then B' if and only if we have a way of converting a
proof of A into a proof of B.
That is the stupidest thing I have heard this week.

Julio
mathman1
2017-06-13 21:00:13 UTC
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The point of the statement is that the premise (mathematics is consistent) is not provable.
m***@wp.pl
2017-06-14 06:37:50 UTC
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Post by David Petry
First of all, I can't make sense of the recent sci.math discussions of Gödel's theorems. But I would like to point out that in the logic of natural language, Gödel's theorems are pretty silly, to say the least.
In the logic of natural language, the following is true: we are justified in saying 'if A, then B' if and only if we have a way of converting a proof of A into a proof of B.
Natural language is inconsistent. A liar paradox, You know.
And who said liar paradox is the only one of its kind?
The phrase used by Godel for the proof is very similiar,
as everybody knows.

Notice, that Godel has never proven that any incomplete theory
exists. He has only proven that every theory is EITHER
incomplete OR inconsistent. Considering what he used - he's
proven an obviousness well known for millenias: our logic
can't manage some specific loops.
Peter Percival
2017-06-14 10:59:16 UTC
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Post by m***@wp.pl
Notice, that Godel has never proven that any incomplete theory
exists. He has only proven that every theory is EITHER
Every?
Post by m***@wp.pl
incomplete OR inconsistent. Considering what he used - he's
proven an obviousness well known for millenias: our logic
can't manage some specific loops.
--
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
m***@wp.pl
2017-06-14 11:30:52 UTC
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Post by Peter Percival
Post by m***@wp.pl
Notice, that Godel has never proven that any incomplete theory
exists. He has only proven that every theory is EITHER
Every?
Formally not. Practically yes.
Post by Peter Percival
Post by m***@wp.pl
incomplete OR inconsistent. Considering what he used - he's
proven an obviousness well known for millenias: our logic
can't manage some specific loops.
--
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
Jim Burns
2017-06-14 16:03:00 UTC
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W dniu środa, 14 czerwca 2017 12:59:23 UTC+2
Post by Peter Percival
Post by m***@wp.pl
Notice, that Godel has never proven that any incomplete
theory exists. He has only proven that every theory is
EITHER
Every?
Formally not. Practically yes.
Remember, you're talking about what Goedel has proven
here. What does it mean that Goedel has _practically_
proven every theory is either incomplete or inconsistent?

One example of a theory that is both complete and consistent
is a subset of natural number arithmetic without
multiplication, Presburger arithmetic.
(Provably complete, certainly consistent if all of
arithmetic is consistent.)
Post by Peter Percival
Post by m***@wp.pl
incomplete OR inconsistent. Considering what he used
our logic can't manage some specific loops.
I think that the important point is that he's _proven_
an obviousness well known for millennia (loosely speaking).

A formal description of some system chooses particular
aspects of the system to describe. I don't think it is
ever _all_ the aspects. I suspect that's not even possible.
A solar system might be seen as point masses moving in
a 1/r potential, for example. There is lots more we could
say about those planets and sun, but that selection of
properties serves our purpose -- for some purposes, if
not for others.

We can choose to formalize certain aspects of our reasoning
and leave other aspects out. The logic of sentences
(propositional logic, zeroth-order logic) treats our
sentences as "point masses" interacting with operators
AND, OR, NOT, and so on. (First-order) predicate logic
includes the zeroth-order description and then adds to that
individuals of some kind and properties of some kind
for those individuals. And so on.

As you say, it should not be surprising that our
incomplete description of our reasoning should have
holes in it, that there should be claims that can be
stated but neither proven nor disproven. We are only
human, after all. But that's not Goedel's point.

Let us say that a formal system "knows" its axioms and
"knows" the theorems that follow from its axioms. What a
formal system "knows" is exactly what it can prove.

For a formal system which "knows" what it "knows"
(that can describe what a proof is in that same system),
that formal system "knows" that it does not "know" everything
(can prove that there are sentences without either proofs
or disproofs).

This piece of wisdom is at least millennia old.
(Whether it is obvious is arguable. I don't think
it would be hard to collect a horde of those who
believe they know everything. But never mind.)

According to Plato, Socrates said
[...] I seem, then, in just this little thing to be
wiser than this man at any rate, that what I do not
know I do not think I know either.

However, Socrates was very wise. How much wisdom must
we include in the description of our reasoning to be
as wise as Socrates? Kurt Goedel showed us: almost none.

If we know some basic logic and some basic arithmetic,
and if we can state what we mean by "proof", then we
"know" that we do not know everything.

The incompleteness of our knowledge is _not_ a result
of our human, finite selves leaving holes in our
incomplete descriptions. The incompleteness of our
knowledge is _a part of our knowledge_ , as much as
2 + 2 = 4 is. It won't go away if we grow wiser, if
we build a better formal system on top of our existing
formal system, any more than 2 + 2 = 4 will go away if
we supplement our natural numbers with negative numbers
and rationals and multivariate calculus.

(If 2 + 2 = 4 did go away, we would need to question
the validity of our supplements.)

Goedel titled his article
"On Formally Undecidable Propositions of
Principia Mathematica And Related Systems".
It is his use of the word "Formally" -- shown by _proof_
within a formal system -- that makes his work
extraordinary, worth all the praise it has received.
m***@wp.pl
2017-06-14 17:17:55 UTC
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Post by Jim Burns
W dniu środa, 14 czerwca 2017 12:59:23 UTC+2
Post by Peter Percival
Post by m***@wp.pl
Notice, that Godel has never proven that any incomplete
theory exists. He has only proven that every theory is
EITHER
Every?
Formally not. Practically yes.
Remember, you're talking about what Goedel has proven
here. What does it mean that Goedel has _practically_
proven every theory is either incomplete or inconsistent?
One example of a theory that is both complete and consistent
is a subset of natural number arithmetic without
multiplication, Presburger arithmetic.
(Provably complete, certainly consistent if all of
arithmetic is consistent.)
I can give You another example -
a language of a single symbol a, one possible
sentence a, one axiom a. Complete and consistent.
but practically not existing.
Post by Jim Burns
Post by Peter Percival
Post by m***@wp.pl
incomplete OR inconsistent. Considering what he used
our logic can't manage some specific loops.
I think that the important point is that he's _proven_
an obviousness well known for millennia (loosely speaking).
A formal description of some system chooses particular
aspects of the system to describe. I don't think it is
ever _all_ the aspects. I suspect that's not even possible.
A solar system might be seen as point masses moving in
a 1/r potential, for example. There is lots more we could
say about those planets and sun, but that selection of
properties serves our purpose -- for some purposes, if
not for others.
We can choose to formalize certain aspects of our reasoning
and leave other aspects out. The logic of sentences
(propositional logic, zeroth-order logic) treats our
sentences as "point masses" interacting with operators
AND, OR, NOT, and so on. (First-order) predicate logic
includes the zeroth-order description and then adds to that
individuals of some kind and properties of some kind
for those individuals. And so on.
As you say, it should not be surprising that our
incomplete description of our reasoning should have
holes in it, that there should be claims that can be
stated but neither proven nor disproven. We are only
human, after all. But that's not Goedel's point.
No, it's not. Godel DIDN'T prove such claims exist.
Godel proved something slightly different - that they
have to exist in consistent theories [of some properties].
Seems practically the same. But it's not.

A liar paradox can be proven by contradiction. It's
negation can be proven by contradiction. Wherever
we put liar paradox we have a claim we can prove to
be true and prove to be false. Wherever we put liar
paradox we have inconsistency. Having inconsistency
we have also (inconsistency or incompleteness). Right?
So, wherever we put liar paradox (or a claim of
similiar properties, of course) we have inconsistency
or incompleteness.
Now - Godel took a claim of obviously similiar properties
and put it everywhere (ok, almost everywhere). And what
did he get? Inconsistency or incompleteness. What a
surprise. Well done, but, as You can see, I can get
the same (and even more) simpler and faster.
Jim Burns
2017-06-14 18:38:06 UTC
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W dniu środa, 14 czerwca 2017 18:03:07 UTC+2
Post by Jim Burns
W dniu środa, 14 czerwca 2017 12:59:23 UTC+2
Post by Peter Percival
Post by m***@wp.pl
Notice, that Godel has never proven that any
incomplete theory exists. He has only proven that
every theory is EITHER
Every?
Formally not. Practically yes.
Remember, you're talking about what Goedel has proven
here. What does it mean that Goedel has _practically_
proven every theory is either incomplete or inconsistent?
One example of a theory that is both complete and
consistent is a subset of natural number arithmetic
without multiplication, Presburger arithmetic.
(Provably complete, certainly consistent if all of
arithmetic is consistent.)
I can give You another example -
a language of a single symbol a, one possible
sentence a, one axiom a. Complete and consistent.
but practically not existing.
Yes, but. I understood your "every theory is... " as
more than just " *There exist* theories which are one of
incomplete, inconsistent, impractical".

Another example of a complete and consistent system is the
real closed field. It's hard to say without more context
whether RCF is what you would call a practical theory,
but it seems more practical than the "single a" theory,
at least.
<https://en.wikipedia.org/wiki/Real_closed_field>

I suspect that the notion of being representable is
related to what you are calling practical.
<https://plato.stanford.edu/entries/goedel-incompleteness/#Rep>

Using the terms from that article,
"x proves y"
is _strongly representable_ in systems of interest
such as Robinson arithmetic Q , which is to say
(x,y) e { x proves y } -> Q |- Proves(x,y)
(x,y) ~e { x proves y } -> Q |- ~Proves(x,y)

However, in those systems,
"y is provable" == "there is some x that proves y"
is only _weakly representable_ , which means
y e { y is provable } -> Q |- Provable(y)
y ~e { y is provable } -> ~( Q |- Provable(y) )

I think that this is interesting. I'm not sure that
it's true of _any_ system of interest, for any purpose
whatsoever.
m***@wp.pl
2017-06-14 20:13:02 UTC
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Post by Jim Burns
W dniu środa, 14 czerwca 2017 18:03:07 UTC+2
Post by Jim Burns
W dniu środa, 14 czerwca 2017 12:59:23 UTC+2
Post by Peter Percival
Post by m***@wp.pl
Notice, that Godel has never proven that any
incomplete theory exists. He has only proven that
every theory is EITHER
Every?
Formally not. Practically yes.
Remember, you're talking about what Goedel has proven
here. What does it mean that Goedel has _practically_
proven every theory is either incomplete or inconsistent?
One example of a theory that is both complete and
consistent is a subset of natural number arithmetic
without multiplication, Presburger arithmetic.
(Provably complete, certainly consistent if all of
arithmetic is consistent.)
I can give You another example -
a language of a single symbol a, one possible
sentence a, one axiom a. Complete and consistent.
but practically not existing.
Yes, but. I understood your "every theory is... " as
more than just " *There exist* theories which are one of
incomplete, inconsistent, impractical".
Tell me, please - if I said "every horse has a head"
would you take a horse and cut his head to prove
that formally I'm wrong?
Yes, using "every" I simplified. Yes, formally I
shouldn't. Yes, sentential calculus also doesn't match
Godel's requirements (I don't know if it makes it "complete"
or not, but I'd guess it does). No, it does not matter.

Now, do You have something to say about the rest of my
text?
Jim Burns
2017-06-14 20:52:39 UTC
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W dniu środa, 14 czerwca 2017 20:38:13 UTC+2
Post by Jim Burns
W dniu środa, 14 czerwca 2017 18:03:07 UTC+2
Post by Jim Burns
W dniu środa, 14 czerwca 2017 12:59:23 UTC+2
Post by Peter Percival
Post by m***@wp.pl
Notice, that Godel has never proven that any
incomplete theory exists. He has only proven that
every theory is EITHER
Every?
Formally not. Practically yes.
Remember, you're talking about what Goedel has proven
here. What does it mean that Goedel has _practically_
proven every theory is either incomplete or inconsistent?
One example of a theory that is both complete and
consistent is a subset of natural number arithmetic
without multiplication, Presburger arithmetic.
(Provably complete, certainly consistent if all of
arithmetic is consistent.)
I can give You another example -
a language of a single symbol a, one possible
sentence a, one axiom a. Complete and consistent.
but practically not existing.
Yes, but. I understood your "every theory is... " as
more than just " *There exist* theories which are one of
incomplete, inconsistent, impractical".
Tell me, please - if I said "every horse has a head"
would you take a horse and cut his head to prove
that formally I'm wrong?
I'm not going to cut any heads off, but if I see
a horse without a head, I will point out that not all
horses have heads. This isn't "formal". This is
natural language.
Yes, using "every" I simplified. Yes, formally I shouldn't.
Also you shouldn't say that because it's not true.
Yes, sentential calculus also doesn't match Godel's
requirements (I don't know if it makes it "complete"
or not, but I'd guess it does). No, it does not matter.
I can't think of any specific examples, but not
matching Goedel's requirements for proving that a
formal system is incomplete is different from being
incomplete. There could be systems that do not match
his requirements but are incomplete anyway.
(They might not be practical systems. Who can say?
What is a practical system?)
Now, do You have something to say about the rest of my
text?
Why do you ask? Is that something you would read?

If you have some time on your hands, I suppose that you
could go back and read the rest of my post.

Here, I'll make it easy for you:
<JB>
Post by Jim Burns
Another example of a complete and consistent system is the
real closed field. It's hard to say without more context
whether RCF is what you would call a practical theory,
but it seems more practical than the "single a" theory,
at least.
<https://en.wikipedia.org/wiki/Real_closed_field>
I suspect that the notion of being representable is
related to what you are calling practical.
<https://plato.stanford.edu/entries/goedel-incompleteness/#Rep>
Using the terms from that article,
"x proves y"
is _strongly representable_ in systems of interest
such as Robinson arithmetic Q , which is to say
(x,y) e { x proves y } -> Q |- Proves(x,y)
(x,y) ~e { x proves y } -> Q |- ~Proves(x,y)
However, in those systems,
"y is provable" == "there is some x that proves y"
is only _weakly representable_ , which means
y e { y is provable } -> Q |- Provable(y)
y ~e { y is provable } -> ~( Q |- Provable(y) )
I think that this is interesting. I'm not sure that
it's true of _any_ system of interest, for any purpose
whatsoever.
</JB>
m***@wp.pl
2017-06-14 23:10:05 UTC
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Post by Jim Burns
W dniu środa, 14 czerwca 2017 20:38:13 UTC+2
Post by Jim Burns
W dniu środa, 14 czerwca 2017 18:03:07 UTC+2
Post by Jim Burns
W dniu środa, 14 czerwca 2017 12:59:23 UTC+2
Post by Peter Percival
Post by m***@wp.pl
Notice, that Godel has never proven that any
incomplete theory exists. He has only proven that
every theory is EITHER
Every?
Formally not. Practically yes.
Remember, you're talking about what Goedel has proven
here. What does it mean that Goedel has _practically_
proven every theory is either incomplete or inconsistent?
One example of a theory that is both complete and
consistent is a subset of natural number arithmetic
without multiplication, Presburger arithmetic.
(Provably complete, certainly consistent if all of
arithmetic is consistent.)
I can give You another example -
a language of a single symbol a, one possible
sentence a, one axiom a. Complete and consistent.
but practically not existing.
Yes, but. I understood your "every theory is... " as
more than just " *There exist* theories which are one of
incomplete, inconsistent, impractical".
Tell me, please - if I said "every horse has a head"
would you take a horse and cut his head to prove
that formally I'm wrong?
I'm not going to cut any heads off, but if I see
a horse without a head, I will point out that not all
horses have heads. This isn't "formal". This is
natural language.
Not quite. Natural language don't specify whether
a horse without head is stil a horse or not.
And Socrates in natural language is sill alive
(in some way).
Post by Jim Burns
Yes, using "every" I simplified. Yes, formally I shouldn't.
Also you shouldn't say that because it's not true.
Yes, I shouldn't.
Post by Jim Burns
Why do you ask? Is that something you would read?
If you have some time on your hands, I suppose that you
could go back and read the rest of my post.
Yes, I shouldn't simplify, I shouldn't write "every",
even if it's practically every.
Yes, yes, yes, yes, I shouldn't.
Now, do You have something to say about the rest of my
text?
Jim Burns
2017-06-15 00:35:19 UTC
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W dniu środa, 14 czerwca 2017 22:52:47 UTC+2
Post by Jim Burns
Why do you ask? Is that something you would read?
If you have some time on your hands, I suppose that you
could go back and read the rest of my post.
Yes, I shouldn't simplify, I shouldn't write "every",
even if it's practically every.
Yes, yes, yes, yes, I shouldn't.
Now, do You have something to say about the rest of my
text?
Honestly, I've forgotten what else you had to say,
after this back-and-forth.

How much effort do you think I should put into answering
you? I thought I made an interesting point about
representability, but you keep deleting it. Should I
give you that much effort, as much as you've given me?

I looked and it turns out you're not asking for much.
So, here goes:
<MLW>
Natural language is inconsistent. A liar paradox, You know.
And who said liar paradox is the only one of its kind?
The phrase used by Godel for the proof is very similiar,
as everybody knows.
Notice, that Godel has never proven that any incomplete
theory exists. He has only proven that every theory is
EITHER incomplete OR inconsistent. Considering what he
used - he's proven an obviousness well known for
millenias: our logic can't manage some specific loops.
</MLW>

I agree that natural language is inconsistent. There
possibly are ways to adjust to that. I'm thinking of
paraconsistent logic. I think (not sure) that would work by
giving priority to the conclusion (of P and not-P) more
_relevant_ to the hypotheses.

There are, in fact, other self-referential paradoxes.
Some are even harder to banish than the Liar.
<https://plato.stanford.edu/entries/self-reference/>

As you say, the true-iff-not-provable sentence G from
Goedel's proof is _similar_ to the Liar, but it is
different in important ways, ways that allow G to be
formalized with a bit of arithmetic. And, strictly
speaking, Goedel's sentence is not a paradox, it's
just unexpected.

I think that it is misleading of you to say that Goedel
has never proven that any incomplete theory exists. If
we know that a theory T is consistent (possibly as
an assumption), and we've proven it's incomplete or
inconsistent, then we've proven that it's incomplete.

The question then becomes, do we trust our assumptions?
And we have very good reasons to trust the assumption
of consistency in a number of interesting cases. It
might be that you are going to be unreasonably skeptical
about that -- but if you are, how do you manage to claim
that Goedel proved anything at all, such as that
[certain] theories are either inconsistent or incomplete
-- while maintaining that same high level of skepticism?

A different theorem that Goedel proved states that
a first-order theory is consistent if and only of
there is a model of that theory. There are some theories
with well-known models, and questions about the
existence of these models drift into questions of
what it means for any mathematical object to "exist".
Which, yes, one could talk about, but this is a far
different kind of discussion that it first appeared
to be back when you said that Goedel never proved that
any incomplete theory exists.

Examples of provably incomplete systems which we have
good reason to think are consistent include:
-- the natural numbers (including at least Robinson
arithmetic, which does not have induction, but does
have addition and multiplication)
-- a fragment of set theory with only three axioms
i) an empty set exists
ii) sets are equal iff they have the same elements
iii) for all sets x and y, there is a set z,
z = (x U {y}) -- in the standard notation.

I already mentioned how important I think it is that
Goedel _proved_ incompleteness for certain formal
systems. If you're curious about what I said, you could
read my earlier posts, I suppose.
m***@wp.pl
2017-06-15 08:57:38 UTC
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Post by Jim Burns
W dniu środa, 14 czerwca 2017 22:52:47 UTC+2
Post by Jim Burns
Why do you ask? Is that something you would read?
If you have some time on your hands, I suppose that you
could go back and read the rest of my post.
Yes, I shouldn't simplify, I shouldn't write "every",
even if it's practically every.
Yes, yes, yes, yes, I shouldn't.
Now, do You have something to say about the rest of my
text?
Honestly, I've forgotten what else you had to say,
after this back-and-forth.
How much effort do you think I should put into answering
you? I thought I made an interesting point about
representability, but you keep deleting it.
It's surely interesting, but not me. Sorry.
I don't want to talk about it.
You don't have to talk about things I want to
talk, of course.
If there is no subject we BOTH want to talk -
we don't have to talk.
Post by Jim Burns
I looked and it turns out you're not asking for much.
<MLW>
Natural language is inconsistent. A liar paradox, You know.
And who said liar paradox is the only one of its kind?
The phrase used by Godel for the proof is very similiar,
as everybody knows.
Notice, that Godel has never proven that any incomplete
theory exists. He has only proven that every theory is
EITHER incomplete OR inconsistent. Considering what he
used - he's proven an obviousness well known for
millenias: our logic can't manage some specific loops.
</MLW>
I agree that natural language is inconsistent. There
possibly are ways to adjust to that.
But as adjusting remains undone for millenias,
we can easily guess the task is not crucial.
And as natural language stays in use, it has to
have some advantages of more importance than
inconsistency. Do You agree?
Post by Jim Burns
There are, in fact, other self-referential paradoxes.
Some are even harder to banish than the Liar.
<https://plato.stanford.edu/entries/self-reference/>
As you say, the true-iff-not-provable sentence G from
Goedel's proof is _similar_ to the Liar, but it is
different in important ways, ways that allow G to be
formalized with a bit of arithmetic.
But what for? Using my unformalized way I can get the
same (actually, stronger) result - faster and easier.
I've shown You how.
Post by Jim Burns
I think that it is misleading of you to say that Goedel
has never proven that any incomplete theory exists. If
we know that a theory T is consistent (possibly as
an assumption), and we've proven it's incomplete or
inconsistent, then we've proven that it's incomplete.
The question then becomes, do we trust our assumptions?
And we have very good reasons to trust the assumption
of consistency in a number of interesting cases.
If I personally put something similiar to liar paradox
into a theory, assuming it's still consistent doesn't
sound like a good idea for me. And take a look what
Godel did next - it's a proof that proving consistency
will prove inconsistency. Another surprise? And again,
I'm not surprised.
Post by Jim Burns
It
might be that you are going to be unreasonably skeptical
about that --
Unreasonably?
Liar paradox has blown out the dreams of natural
language consistency. You're playing with similiar
sentence, You're getting similiar results and You say
my skepticism is unreasonable?
Maybe it is. Or maybe not. ANyway, Godel's results are
obvious for me.
BTW. There is no liar paradox in formal language, because
formal language has no "false" predicate. Right?
Not quite, as long false(p) and (not p) mean the same.
Post by Jim Burns
but if you are, how do you manage to claim
that Goedel proved anything at all, such as that
[certain] theories are either inconsistent or incomplete
-- while maintaining that same high level of skepticism?
I've proven more. I've eliminated "either/or".
Fast and easy. My reasoning was correct, though
worthless.
Godel used formalism instead thinking, so
he wasn't as effective. But he also got correct,
worthless results similiar to mine. Why not?
Post by Jim Burns
A different theorem that Goedel proved states that
a first-order theory is consistent if and only of
there is a model of that theory.
Under what assumptions?
You know, there are good reasons why natural
language never precises its assumptions and is
always ready to withdraw from any statement.
Mathematics has a completely different approach.
Both approaches have advantages and disadvantages,
but remember what Tarski said: the "truth" term
can't be formalized. The "truth" term will forever
stay in the realm where nothing ever is sure.

There are some theories
Post by Jim Burns
with well-known models, and questions about the
existence of these models drift into questions of
what it means for any mathematical object to "exist".
Which, yes, one could talk about, but this is a far
different kind of discussion that it first appeared
to be back when you said that Goedel never proved that
any incomplete theory exists.
Examples of provably incomplete systems which we have
-- the natural numbers (including at least Robinson
arithmetic, which does not have induction, but does
have addition and multiplication)
-- a fragment of set theory with only three axioms
i) an empty set exists
ii) sets are equal iff they have the same elements
iii) for all sets x and y, there is a set z,
z = (x U {y}) -- in the standard notation.
I already mentioned how important I think it is that
Goedel _proved_ incompleteness for certain formal
systems. If you're curious about what I said, you could
read my earlier posts, I suppose.
David Petry
2017-06-15 09:48:15 UTC
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Post by m***@wp.pl
Liar paradox has blown out the dreams of natural
language consistency.
Some of us think that is simply ridiculous. Let me try to argue the point.

When we talk, we always implicitly claim to be telling the truth. Communication is simply impossible if we don't make the implicit claim. Thus, for example, when we say "two plus two equals four" what we are saying is "<implicitly> I'm telling the truth; <explicitly> two plus two is four".

So the statement, "I am lying" is equivalent to "<implicitly> I am telling the truth; <explicitly> I am lying", which is a simple contradiction. There's nothing paradoxical about it. It doesn't have any deep implications for the nature of natural language.
m***@wp.pl
2017-06-15 11:45:41 UTC
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Post by David Petry
Post by m***@wp.pl
Liar paradox has blown out the dreams of natural
language consistency.
Some of us think that is simply ridiculous. Let me try to argue the point.
When we talk, we always implicitly claim to be telling the truth.
Communication is simply impossible if we don't make the implicit claim.

Of course not. "Well done" sometimes means "well done",
and other time "You idiot! You've ruined everything".
A brain is a big device and can deal with different
modes of communication.

But, of course, usually saying "x" we want our receiver
to receive "x". So what? The paradox can still be proven
false and proven true and that means we have the
condition of inconsistency fulfilled (formally, at
least).
David Petry
2017-06-15 17:12:59 UTC
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Post by David Petry
Post by David Petry
When we talk, we always implicitly claim to be telling the truth.
Communication is simply impossible if we don't make the implicit claim.
Of course not. "Well done" sometimes means "well done",
and other time "You idiot! You've ruined everything".
I'm not sure that I get your point, but I do find what you said to be amusing. And that's important too. So, thanks!
Jim Burns
2017-06-15 14:48:52 UTC
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W dniu czwartek, 15 czerwca 2017 02:35:26 UTC+2
Post by Jim Burns
How much effort do you think I should put into answering
you? I thought I made an interesting point about
representability, but you keep deleting it.
It's surely interesting, but not me. Sorry.
I don't want to talk about it.
You don't have to talk about things I want to
talk, of course.
If there is no subject we BOTH want to talk -
we don't have to talk.
I'll keep that in mind.

I misunderstood you earlier when you were asking
for a response to the rest of what you wrote,
again and again. It seemed as though you were asking
for your interests to be taken into account by me.
For example, here:
<MLW>
Yes, I shouldn't simplify, I shouldn't write "every",
even if it's practically every.
Yes, yes, yes, yes, I shouldn't.
Now, do You have something to say about the rest of my
text?
</MLW>
I did that in my next post.

I just assumed from your asking me to respond in a
way that _you_ would find interesting, that you thought
that what you made was a reasonable request.
From that, I concluded that you would also find it
reasonable for _me_ to ask _you_ to respond in a way
that _I_ would find interesting.

It turns out I was wrong about that, though.

[...]
Post by Jim Burns
I agree that natural language is inconsistent. There
possibly are ways to adjust to that.
But as adjusting remains undone for millenias,
we can easily guess the task is not crucial.
Diseases have raged through populations of human beings
for millennia and more. I do not consider this an
argument in favor of disease.
And as natural language stays in use, it has to
have some advantages of more importance than
inconsistency.
It seems to me that the main competition "in the wild"
for natural language, outside of classrooms and so on,
would be no language at all. Yes, I will grant you that
natural language is better than no language at all.

Sometimes the main advantage something has is that it
has the main advantage. The various versions of the
Microsoft operating systems were, in many ways, not the
best operating systems available. But they were on most
of the personal computers sold, so software developers
made sure to write a Microsoft version of their
spreadsheets and browsers and games. This made it
reasonable for people to want Microsoft on their PCs.
And around we go.

An advantage _to language_ , encouraging its use, is not
the same thing as an advantage _to us_ when we use it.
Do You agree?
I think that natural language has some large advantages
over formal languages for most purposes. I also think
formal languages have some large advantages over natural
languages for a few purposes -- but it seems to me that
those few purposes include what we are engaged in here.

If you're asking if I agree with your argument, no,
I don't think you have a very good argument.

[...]
Post by Jim Burns
As you say, the true-iff-not-provable sentence G from
Goedel's proof is _similar_ to the Liar, but it is
different in important ways, ways that allow G to be
formalized with a bit of arithmetic.
But what for? Using my unformalized way I can get the
same (actually, stronger) result - faster and easier.
I've shown You how.
I think a reasonable definition of "formalize" could
be "clarify". If we make our argument clear enough,
if we completely formalize it, then we will be able to
explain it successfully to a transistor, then we will be
able to mechanize that part of our reasoning.

What Goedel did was mechanize the reasoning that
leads us to say "We don't know everything". He did
that by representing our reasoning with numbers
and arithmetic expressions, by fully mechanizing it.

It's not just the result that Goedel arrived at which
is important. The process by which he arrived at that
result is also important. I would argue the process is
more important than the result.
m***@wp.pl
2017-06-15 15:36:41 UTC
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Post by Jim Burns
Diseases have raged through populations of human beings
for millennia and more.
And if there were ways to adjust that it surely
would have been done as soon as possible.
Post by Jim Burns
I think that natural language has some large advantages
over formal languages for most purposes. I also think
formal languages have some large advantages over natural
languages for a few purposes -- but it seems to me that
those few purposes include what we are engaged in here.
And for me it seems they don't include.
Post by Jim Burns
Post by m***@wp.pl
But what for? Using my unformalized way I can get the
same (actually, stronger) result - faster and easier.
I've shown You how.
I think a reasonable definition of "formalize" could
be "clarify".
I'd rather said "simplify". Of course,
simplification includes clarification too.
Post by Jim Burns
If we make our argument clear enough,
if we completely formalize it, then we will be able to
explain it successfully to a transistor, then we will be
able to mechanize that part of our reasoning.
Not a good idea at all. As we can see at Godel's
(and Yours) example. If you didn't mechanize Your
reasoning, maybe You would be able to notice that...
Post by Jim Burns
What Goedel did was mechanize the reasoning that
leads us to say "We don't know everything".
is an obvious bullshit.
Playing with liar-like paradoxes has NOTHING in
common with the reasoning that leads us to say
"We don't know everything".
David Petry
2017-06-14 20:31:06 UTC
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Post by Jim Burns
Goedel titled his article
"On Formally Undecidable Propositions of
Principia Mathematica And Related Systems".
It is his use of the word "Formally" -- shown by _proof_
within a formal system -- that makes his work
extraordinary, worth all the praise it has received.
That's entirely debatable.

1) From the point of view of natural, common sense reasoning, Gödel's theorems are utter trivialities.

2) Natural, common sense reasoning includes ways to distinguish reality from make believe, and these principles of reasoning are entirely lacking in the formal systems that Godel considered.

3) So, the question should be asked, whether Gödel's arguments say anything more about reality than the trivial common sense arguments.

4) In the 85 years since Godel published his results, absolutely no scientific or technological advances have arisen from his results, and there is no good reason to believe such advances will ever arise from his results.

5) Given the level of confused debate about his results, we have to ask whether his results are worthy of any praise at all; his "results" can be viewed as an assault on reason and logic, rather than a contribution to it, especially if you agree that the purpose of reason and logic is to help us learn about reality, and not to help us create elaborate but vacuous narratives.
konyberg
2017-06-14 20:44:02 UTC
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Post by David Petry
Post by Jim Burns
Goedel titled his article
"On Formally Undecidable Propositions of
Principia Mathematica And Related Systems".
It is his use of the word "Formally" -- shown by _proof_
within a formal system -- that makes his work
extraordinary, worth all the praise it has received.
That's entirely debatable.
1) From the point of view of natural, common sense reasoning, Gödel's theorems are utter trivialities.
2) Natural, common sense reasoning includes ways to distinguish reality from make believe, and these principles of reasoning are entirely lacking in the formal systems that Godel considered.
3) So, the question should be asked, whether Gödel's arguments say anything more about reality than the trivial common sense arguments.
4) In the 85 years since Godel published his results, absolutely no scientific or technological advances have arisen from his results, and there is no good reason to believe such advances will ever arise from his results.
5) Given the level of confused debate about his results, we have to ask whether his results are worthy of any praise at all; his "results" can be viewed as an assault on reason and logic, rather than a contribution to it, especially if you agree that the purpose of reason and logic is to help us learn about reality, and not to help us create elaborate but vacuous narratives.
Why is it that in mathematics we some times have to give definitions instead of theoremes?

Example: a^0 = 1, C(0,0) = 1 or 0^0 = 0, 1 or undefined

Is Gödel involved?

KON
Jim Burns
2017-06-14 21:49:55 UTC
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[...] especially if you agree that the purpose of
reason and logic is to help us learn about reality,
and not to help us create elaborate but vacuous
narratives.
Oh.

Well, I think that the purpose of reason and logic
-- at least as I have seen it used, which strikes me
as at least reality-adjacent -- is to help us create
elaborate but vacuous narratives which have the
potential to help us learn about reality. Do I agree
or disagree with you?

For example, the narrative that goes
All men are mortal.
Socrates is a man.
Therefore, Socrates will never die.
is a faulty narrative. It is faulty whether or not
all men are mortal or Socrates is a man. This is the
area of study of logic, classifying which narratives are
faulty and which are not faulty, independently of whether
the assumptions inherent in the narrative are correct when
applied to reality.

Logic won't tell us that Socrates is a man, but it can
tell us that we can't have both Socrates being mortal
and Socrates being immortal.

It seems trivial in this example here. But the more
complex a chain of reasoning is, the harder it is to
keep from screwing up, and the more valuable it is to
avoid error from a screwed up chain of reasoning.

From the World of Reality, the reason that a bizarre
prediction by string theory that we actually have ten
or eleven dimensions (but we can't see most of them) is
taken seriously _because_ there are only a few numbers
of dimensions for which string theory is consistent.

That does not prove that string theory is a correct
description of Reality, it could easily be wrong.
But it drastically narrows down the possibilities that
could be right.

So, this faulty-reasoning-check is one valuable tool
for deciding what our description of reality should be
-- but it is a _different tool_ from checking our
descriptions against our observations, which is what
I think you mean by learning about reality.
David Petry
2017-06-14 22:55:11 UTC
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Post by Jim Burns
[...] especially if you agree that the purpose of
reason and logic is to help us learn about reality,
and not to help us create elaborate but vacuous
narratives.
Oh.
Well, I think that the purpose of reason and logic
-- at least as I have seen it used, which strikes me
as at least reality-adjacent -- is to help us create
elaborate but vacuous narratives which have the
potential to help us learn about reality. Do I agree
or disagree with you?
Would you describe yourself as a postmodernist? If not, do you have a word that you would use to describe yourself? Cultural Marxist? Secular Humanist?
Other?
Franz Gnaedinger
2017-06-14 06:42:42 UTC
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Post by David Petry
First of all, I can't make sense of the recent sci.math discussions of Gödel's theorems. But I would like to point out that in the logic of natural language, Gödel's theorems are pretty silly, to say the least.
In the logic of natural language, the following is true: we are justified in saying 'if A, then B' if and only if we have a way of converting a proof of A into a proof of B.
Then Gödel's (second) theorem, which roughly speaking says, "if mathematics is consistent, then mathematics cannot prove that mathematics is consistent" can be translated into "we have a way of converting a proof of the consistency of mathematics into a proof that there does not exist a proof of the consistency of mathematics", which is pretty obviously silly.
Goedel is very difficult to follow, but all who can are enthusiastic about
his ingenious work that can't be dissolved by word magic. Things are getting
way simpler if you go back to the foundament of mathematical logic of a = a
whereas language follows the wider logic of equal unequal, for example in
the formulation by Goethe: All is equal, all unequal ... An apple is an apple,
but one apple may be small and red, another big and yellow, and what if the
apple is eaten? Mathematical logic is not logic per se but a special case
of logic, the logic of building and mantaining. What Goedel did when seen
from this vantage point is that mathematical logic can't really be separated
from the wider logic of equal unequal. Its borders are the infinite, both
the infinitely large and infinitely small, the infinite being equal unequal
in itself. Paradoxa like the liar paradox can't be eliminated, for they are
the door from mathematical logic to the wider logic. Goedel shocked the
mathematical community of his time, and many still can't accept his proven
theorems, but in hindsight he made modern mathematics bloom like never before,
as he freed it from the rigor mortis axiomaticus. Everything hangs together
in mathematics, which accounts for its extraordinary power but also for
a most severe restriction: dismiss one single proven theorem and you are out,
tilt, game over. Those dismissing Goedel and Turing punish themselves by
working in vain, making the biggest promises but never achieving anything
real.
m***@wp.pl
2017-06-14 06:50:40 UTC
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Post by Franz Gnaedinger
Goedel is very difficult to follow,
Bullshit.
You don't have any occult science unavailable for common
mortals. You're only dreaming.
Peter Percival
2017-06-14 11:00:15 UTC
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Post by m***@wp.pl
Post by Franz Gnaedinger
Goedel is very difficult to follow,
Bullshit.
But your previous post shows that you never got to grips with it.
Post by m***@wp.pl
You don't have any occult science unavailable for common
mortals. You're only dreaming.
--
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
m***@wp.pl
2017-06-14 11:34:14 UTC
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Post by Peter Percival
Post by m***@wp.pl
Post by Franz Gnaedinger
Goedel is very difficult to follow,
Bullshit.
But your previous post shows that you never got to grips with it.
If you believe a god, everything shows you he exists.
If you believe everyone except you and some of your
friends is stupid, everything shows you it's true.
Simple and obvious wishful thinking.
Franz Gnaedinger
2017-06-15 06:22:14 UTC
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Post by Peter Percival
But your previous post shows that you never got to grips with it.
Also my general impression, still a lot of confusion about Goedel.
m***@wp.pl
2017-06-15 12:14:12 UTC
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Post by Franz Gnaedinger
Post by Peter Percival
But your previous post shows that you never got to grips with it.
Also my general impression, still a lot of confusion about Goedel.
Of course. A general impression of confusion is a
default impression when someone denies You.
David Petry
2017-06-14 06:50:54 UTC
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Post by David Petry
First of all, I can't make sense of the recent sci.math discussions of Gödel's theorems. But I would like to point out that in the logic of natural language, Gödel's theorems are pretty silly, to say the least.
In the logic of natural language, the following is true: we are justified in saying 'if A, then B' if and only if we have a way of converting a proof of A into a proof of B.
Then Gödel's (second) theorem, which roughly speaking says, "if mathematics is consistent, then mathematics cannot prove that mathematics is consistent" can be translated into "we have a way of converting a proof of the consistency of mathematics into a proof that there does not exist a proof of the consistency of mathematics", which is pretty obviously silly.
Here's an article I wrote a few years ago that goes into much more detail about how I view Gödel's theorem. I recommend it! :-)

https://groups.google.com/forum/#!original/sci.math/Jc4rNCUSMs4/fstig3C-JQAJ
Peter Percival
2017-06-14 11:01:28 UTC
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[...] I recommend it!
Why? It's crap, if I may be excused a vulgarism.
--
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
Peter Percival
2017-06-14 13:15:30 UTC
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Post by David Petry
In the logic of natural language, the following is true: we are
justified in saying 'if A, then B' if and only if we have a way of
converting a proof of A into a proof of B.
I think that is how intuitionists think of if-then, though the "way" and
the proofs had better be intuitionistically acceptable themselves.

But it is not true of the logic of natural language. (Counter examples
abound.)
--
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
m***@wp.pl
2017-06-14 13:23:55 UTC
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Post by Peter Percival
Post by David Petry
In the logic of natural language, the following is true: we are
justified in saying 'if A, then B' if and only if we have a way of
converting a proof of A into a proof of B.
I think that is how intuitionists think of if-then, though the "way" and
the proofs had better be intuitionistically acceptable themselves.
But it is not true of the logic of natural language. (Counter examples
abound.)
Logic of natural language includes "there is no rule without
an exception" rule.
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