Try and fit this into the diagonal lemma

[peterolcott]

https://plato.stanford.edu/entries/tarski-truth/#ForCor
For all x, True(x) if and only if φ(x),

For clarity we focus on atomic propositions expressing a single relation between two Things, then True(x) merely means this expression:

(1) is a binary relation between exactly two things,

(2) both of these two things are types that are compatible with this binary relation, and

(3) this binary relation actually exists.

Binary-Relation(x) // predicate
Compatible-Types(P, a, b) // predicate
Get-Binary-Relation(x) // function
Binary-Relation-Satisfied(P, a, b) // predicate

True(x) ↔
Binary-Relation(x) &
Compatible-Types( Get-Binary-Relation(x), arg1, arg2 ) ) &
Binary-Relation-Satisfied(Get-Binary-Relation(x), arg1, arg2 )

https://en.wikipedia.org/wiki/Satisfiability#Satisfiability_in_model_theory

[Peter Percival]

Consider "Fred is the father of Penny." In this case is x the whole of
that sentence, or is it just "is the father of"?

>
> https://en.wikipedia.org/wiki/Satisfiability#Satisfiability_in_model_theory
>


--
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]

peterolcott wrote:
> https://plato.stanford.edu/entries/tarski-truth/#ForCor
> For all x, True(x) if and only if φ(x),
>
> For clarity we focus on atomic propositions expressing a single relation between two Things, then True(x) merely means this expression:
>
> (1) is a binary relation between exactly two things,
>
> (2) both of these two things are types that are compatible with this binary relation, and
>
> (3) this binary relation actually exists.
>
> Binary-Relation(x) // predicate
> Compatible-Types(P, a, b) // predicate
> Get-Binary-Relation(x) // function

What are the possible values of this function?

> Binary-Relation-Satisfied(P, a, b) // predicate
>
> True(x) ↔
> Binary-Relation(x) &
> Compatible-Types( Get-Binary-Relation(x), arg1, arg2 ) ) &
> Binary-Relation-Satisfied(Get-Binary-Relation(x), arg1, arg2 )
>
> https://en.wikipedia.org/wiki/Satisfiability#Satisfiability_in_model_theory
>

[peterolcott]

On Tuesday, February 7, 2017 at 9:23:34 AM UTC-6, Peter Percival wrote:
> peterolcott wrote:
> > https://plato.stanford.edu/entries/tarski-truth/#ForCor
> > For all x, True(x) if and only if φ(x),
> >
> > For clarity we focus on atomic propositions expressing a single relation between two Things, then True(x) merely means this expression:
> >
> > (1) is a binary relation between exactly two things,
> >
> > (2) both of these two things are types that are compatible with this binary relation, and
> >
> > (3) this binary relation actually exists.
> >
> > Binary-Relation(x) // predicate
> > Compatible-Types(P, a, b) // predicate
> > Get-Binary-Relation(x) // function
> > Binary-Relation-Satisfied(P, a, b) // predicate
> >
> > True(x) ↔
> > Binary-Relation(x) &
> > Compatible-Types( Get-Binary-Relation(x), arg1, arg2 ) ) &
> > Binary-Relation-Satisfied(Get-Binary-Relation(x), arg1, arg2 )
>
> Consider "Fred is the father of Penny." In this case is x the whole of
> that sentence, or is it just "is the father of"?
>

L is Tarski's object language, in this case natural language.
M is Tarski's metalanguage, in this case predicate logic.
https://plato.stanford.edu/entries/tarski-truth/#ObjLanMet

(s ∈ L) ↔ "Fred is the father of Penny."
mathematically maps to the following binary predicate:
(x ∈ M) Father-Of(Penny, Fred)

So I just said that x is the whole sentence when this sentence is translated from natural language to predicate logic.

[Jim Burns]

[Re: Try and fit this into the diagonal lemma]

On 2/7/2017 9:55 AM, peterolcott wrote:

> https://plato.stanford.edu/entries/tarski-truth/#ForCor
> For all x, True(x) if and only if φ(x),
>
> For clarity we focus on atomic propositions expressing
> a single relation between two Things,
> then True(x) merely means this expression:
>
> (1) is a binary relation between exactly two things,
>
> (2) both of these two things are types that are
> compatible with this binary relation, and
>
> (3) this binary relation actually exists.
>
> Binary-Relation(x) // predicate
> Compatible-Types(P, a, b) // predicate
> Get-Binary-Relation(x) // function
> Binary-Relation-Satisfied(P, a, b) // predicate
>
> True(x)
> Binary-Relation(x) &
> Compatible-Types( Get-Binary-Relation(x), arg1, arg2 ) ) &
> Binary-Relation-Satisfied(Get-Binary-Relation(x), arg1, arg2 )
>

> https://en.wikipedia.org/wiki/Satisfiability#Satisfiability_in_model_theory
>

See below, at emphasized text. I don't claim that I can prove
this myself, but I can follow in general terms what they're
saying, and I believe it can be proven. YMMV, but that's
on you.

First order logic with a single binary predicate is undecidable,
so anything with _at least_ first order logic with a single
binary predicate is undecidable. That seems to cover whatever
you might be thinking of.

Proving undecidability is not proving there is no True(x).
However, the requirements for the first (FOPL, sub(x,y)
and Provable(x)) are stronger than the requirements for
the second (FOPL and sub(x,y)). If we can prove the first
(which we can, for first order logic with a single
binary predicate), then we can prove the second: that
there is no True(x) for first order logic with a single
binary predicate. -- or any consistent extension of that.

That's what you asked for, that's what I provided.

Shocking, isn't it? I know I was shocked when this was
pointed out to me.

<wiki>
The most remarkable fact about ST (and hence GST), is that
these tiny fragments of set theory give rise to such rich
metamathematics. While ST is a small fragment of the
well-known canonical set theories ZFC and NBG, ST interprets
Robinson arithmetic (Q), so that ST inherits the nontrivial
metamathematics of Q. For example, ST is essentially
undecidable because Q is, and every consistent theory whose
theorems include the ST axioms is also essentially
undecidable.[4] This includes GST and every axiomatic set
theory worth thinking about, assuming these are consistent.
In fact, the undecidability of ST implies *the undecidability*
*of first-order logic with a single binary predicate*
*letter* .[5]
</wiki>
[emphasis added]
<https://en.wikipedia.org/wiki/General_set_theory#Metamathematics>

[Peter Percival]

peterolcott wrote:

> L is Tarski's object language, in this case natural language.
> M is Tarski's metalanguage, in this case predicate logic.

I don't think that will work.

> https://plato.stanford.edu/entries/tarski-truth/#ObjLanMet

Where we read "The metalanguage should contain a copy of the object
language (so that anything one can say in L can be said in M too), and M
should also be able to talk about the sentences of L and their syntax."

Predicate logic doesn't contain a copy of natural language. Actually, I
was surprised to read that the metalanguage should contain a _copy_ of
the object language. I think it suffices for the metalanguage to
contain names for (at least some of) the symbols of the object language.
But even if that's so (and you may be quite sure I know nothing about
the matter) you'll have problems.

I would have thought

> L is Tarski's object language, in this case predicate logic.
> M is Tarski's metalanguage, in this case natural language.

Also

> (s ∈ L) ↔ "Fred is the father of Penny."

makes no sense. "Fred is the father of Penny." is either true or false.
Most likely it's true but I'll consider both possibilities.
i) "Fred is the father of Penny." is true. So (s ∈ L) is true. You've
told us what L is (so, for the moment, I'm putting my doubts above to
one side), but what is s? What makes (s ∈ L) true? Are all s's in L or
is some particular s in L? You're fond of doing this - having a free
variable (it might be x or it might be s) on one side of an iff but not
on the other. If it's some particular s, which one?
ii) "Fred is the father of Penny." is false. So (s ∈ L) is false. You've
told us what L is (so, for the moment, I'm putting my doubts above to
one side), but what is s? What makes (s ∈ L) false? Are all s's not in L
or is some particular s not in L? You're fond of doing this - having a
free variable (it might be x or it might be s) on one side of an iff but
not on the other. If it's some particular s, which one?

[peterolcott]

It certainty does not cover second order logic.
My formula only requires HOL of a finite order,
thus disavowing Tarski's infinite hierarchy.

I am able to fully evaluate several age old paradoxes within SOL, rejecting them as ill-formed.

[peterolcott]

On Tuesday, February 7, 2017 at 12:20:32 PM UTC-6, Peter Percival wrote:
> peterolcott wrote:
>
> > L is Tarski's object language, in this case natural language.
> > M is Tarski's metalanguage, in this case predicate logic.
>
> I don't think that will work.
>
> > https://plato.stanford.edu/entries/tarski-truth/#ObjLanMet
>
> Where we read "The metalanguage should contain a copy of the object
> language (so that anything one can say in L can be said in M too), and M
> should also be able to talk about the sentences of L and their syntax."
>
> Predicate logic doesn't contain a copy of natural language. Actually, I
> was surprised to read that the metalanguage should contain a _copy_ of
> the object language. I think it suffices for the metalanguage to
> contain names for (at least some of) the symbols of the object language.
> But even if that's so (and you may be quite sure I know nothing about
> the matter) you'll have problems.
>
> I would have thought
>
> > L is Tarski's object language, in this case predicate logic.
> > M is Tarski's metalanguage, in this case natural language.

I am disavowing everything that Tarski said besides two things:
(1) An object language is defined in terms of a metalanguage.

(2) For all x, True(x) if and only if φ(x), specifies the starting point for formalizing the notion of the concept of truth.

>
> Also
>
> > (s ∈ L) ↔ "Fred is the father of Penny."
>
> makes no sense. "Fred is the father of Penny." is either true or false.

All that I am doing here is applying the label s to the above sentence so that we can refer to this sentence by its label. I am also saying that sentence s is one element of the set of natural language sentences.

What is the correct PL syntax for saying this?

> Most likely it's true but I'll consider both possibilities.
> i) "Fred is the father of Penny." is true. So (s ∈ L) is true. You've
> told us what L is (so, for the moment, I'm putting my doubts above to
> one side), but what is s? What makes (s ∈ L) true? Are all s's in L or
> is some particular s in L? You're fond of doing this - having a free
> variable (it might be x or it might be s) on one side of an iff but not
> on the other. If it's some particular s, which one?
> ii) "Fred is the father of Penny." is false. So (s ∈ L) is false. You've
> told us what L is (so, for the moment, I'm putting my doubts above to
> one side), but what is s? What makes (s ∈ L) false? Are all s's not in L
> or is some particular s not in L? You're fond of doing this - having a
> free variable (it might be x or it might be s) on one side of an iff but
> not on the other. If it's some particular s, which one?
>

You skipped right over the important part where I translated the English sentence into its binary relation predicate in predicate logic:

Father-Of(Penny, Fred)

[Peter Percival]

You'll get on better with plain(-ish) English:

Let s be the name of the sentence "Fred is the father of Penny." L is
such that s is a closed term of it.

>> Most likely it's true but I'll consider both possibilities.
>> i) "Fred is the father of Penny." is true. So (s ∈ L) is true. You've
>> told us what L is (so, for the moment, I'm putting my doubts above to
>> one side), but what is s? What makes (s ∈ L) true? Are all s's in L or
>> is some particular s in L? You're fond of doing this - having a free
>> variable (it might be x or it might be s) on one side of an iff but not
>> on the other. If it's some particular s, which one?
>> ii) "Fred is the father of Penny." is false. So (s ∈ L) is false. You've
>> told us what L is (so, for the moment, I'm putting my doubts above to
>> one side), but what is s? What makes (s ∈ L) false? Are all s's not in L
>> or is some particular s not in L? You're fond of doing this - having a
>> free variable (it might be x or it might be s) on one side of an iff but
>> not on the other. If it's some particular s, which one?
>>
>
> You skipped right over the important part where I translated the English sentence into its binary relation predicate in predicate logic:
>
> Father-Of(Penny, Fred)
>

[Jim Burns]

I'll have to look at it, but I think the article I stole
from said all consistent extensions. I suspect that
would include (consistent) second order logic. But I'll
get back to you. (Or you could look at the article yourself.)

It seems to me that, if I said
"Mares eat oats and does eat oats, but little lambs
eat ivy, a kid'll eat ivy, too, wouldn't you?"
and later I said
"Mares don't eat oats"
I would have contradicted myself, and finding a way
to express all that in second order logic so that
I have _not_ contradicted myself would be a reason
to avoid second order logic, not a reason to doubt
that I have contradicted myself.

However, I'll get back to you.

> My formula only requires HOL of a finite order,
> thus disavowing Tarski's infinite hierarchy.
>
> I am able to fully evaluate several age old paradoxes
> within SOL, rejecting them as ill-formed.

<sigh>

The diagonal lemma is ready with new paradoxes.

I can always name a higher number.

[Peter Percival]

Jim Burns wrote:

> First order logic with a single binary predicate is undecidable,
> so anything with _at least_ first order logic with a single
> binary predicate is undecidable.

Just so long as there are no non-logical axioms. The theory of ordered
abelian groups is decidable. Not that that affects what you went on to say.

The result referred to is due to Gurevich, I have no citation but it is
mentioned (though not proved) in

Monk, J. Donald, /Mathematical logic/, Springer-Verlag, 1976.

Therein are two tables, one of 19 decidable theories (seven with proofs
in the book), and one of 15 undecidable undecidable theories (seven with
proofs in the book).

[peterolcott]

This is the book that taught Douglas Hofstader the Incompleteness Theorem:
http://calculemus.org/cafe-aleph/raclog-arch/nagel-newman.pdf

Although you can always name a higher number you cannot derive any specific concrete expression that forms a counter-example showing that True(x) is not always logically equivalent to Boolean.True.

∀x True(x) ↔ Boolean.True
∀x WFF(x) ↔ (Boolean.True ∨ Boolean.False)

[Peter Percival]

peterolcott wrote:


> ∀x True(x) ↔ Boolean.True

The LHS is false, isn't it? After all not everything is true. But the
RHS is true. So the above is false. Is that your intention?

> ∀x WFF(x) ↔ (Boolean.True ∨ Boolean.False)

[peterolcott]

On Tuesday, February 7, 2017 at 2:12:23 PM UTC-6, Peter Percival wrote:
> peterolcott wrote:
>
>
> > ∀x True(x) ↔ Boolean.True
>
> The LHS is false, isn't it? After all not everything is true. But the
> RHS is true. So the above is false. Is that your intention?
>
> > ∀x WFF(x) ↔ (Boolean.True ∨ Boolean.False)
> >

There are three steps in my formula:
If True(x) satisfies all three then Boolean.True.
else Boolean.False ∨ ~WFF(x)

Would it be easier if I said:
Semantically-Incorrect(x)
Invalid-Proposition(x)
Its-Neither-True-Nor-False-Because-Its-Screwed-Up(x)

instead of ~WFF(x) ???

[Peter Percival]

peterolcott wrote:
> On Tuesday, February 7, 2017 at 2:12:23 PM UTC-6, Peter Percival wrote:
>> peterolcott wrote:
>>
>>
>>> ∀x True(x) ↔ Boolean.True
>>
>> The LHS is false, isn't it? After all not everything is true. But the
>> RHS is true. So the above is false. Is that your intention?
>>
>>> ∀x WFF(x) ↔ (Boolean.True ∨ Boolean.False)
>>>
>
> There are three steps in my formula:
> If True(x) satisfies all three then Boolean.True.
> else Boolean.False ∨ ~WFF(x)
>
> Would it be easier

Not right now it wouldn't because it doesn't address this problem:
You: ∀x True(x) ↔ Boolean.True
Me: The LHS is false, isn't it? After all not everything is true. But

the RHS is true. So the above is false. Is that your intention?

> if I said:
> Semantically-Incorrect(x)
> Invalid-Proposition(x)
> Its-Neither-True-Nor-False-Because-Its-Screwed-Up(x)
>
> instead of ~WFF(x) ???

... but since you're asking: x not being well formed (which is what I
have supposed ~WFF(x) means) isn't the same as x being semantically
incorrect. Not being well formed is a syntactic matter. See any logic
text. Also, x not being well formed isn't the same as x being invalid.
Invalidity only applies to wff. See any logic text.

[peterolcott]

On Tuesday, February 7, 2017 at 3:19:12 PM UTC-6, Peter Percival wrote:
> peterolcott wrote:
> > On Tuesday, February 7, 2017 at 2:12:23 PM UTC-6, Peter Percival wrote:
> >> peterolcott wrote:
> >>
> >>
> >>> ∀x True(x) ↔ Boolean.True
> >>
> >> The LHS is false, isn't it? After all not everything is true. But the
> >> RHS is true. So the above is false. Is that your intention?
> >>
> >>> ∀x WFF(x) ↔ (Boolean.True ∨ Boolean.False)
> >>>
> >
> > There are three steps in my formula:
> > If True(x) satisfies all three then Boolean.True.
> > else Boolean.False ∨ ~WFF(x)
> >
> > Would it be easier
>
> Not right now it wouldn't because it doesn't address this problem:
> You: ∀x True(x) ↔ Boolean.True
> Me: The LHS is false, isn't it? After all not everything is true. But
> the RHS is true. So the above is false. Is that your intention?

No, no, no no. it is not at all like that.
When my entire formula is fully satisfied by x, then and only then is x logically equivalent to Boolean.True. If any part of my formula is not fully satisfied by x, then x is NOT Boolean.True. Not Boolean.True includes Boolean.False and semantically incorrect.

>
> > if I said:
> > Semantically-Incorrect(x)
> > Invalid-Proposition(x)
> > Its-Neither-True-Nor-False-Because-Its-Screwed-Up(x)
> >
> > instead of ~WFF(x) ???
>
> ... but since you're asking: x not being well formed (which is what I
> have supposed ~WFF(x) means) isn't the same as x being semantically
> incorrect. Not being well formed is a syntactic matter. See any logic
> text. Also, x not being well formed isn't the same as x being invalid.
> Invalidity only applies to wff. See any logic text.

I have ONLY been using ~WFF(x) to mean semantically incorrect.
This is totally obvious if you merely look as how WFF(x) is defined.

Semantic correctness of x is verified syntactically as is required whenever any expression is formalized.

[Peter Percival]

peterolcott wrote:
> On Tuesday, February 7, 2017 at 3:19:12 PM UTC-6, Peter Percival wrote:
>> peterolcott wrote:
>>> On Tuesday, February 7, 2017 at 2:12:23 PM UTC-6, Peter Percival wrote:
>>>> peterolcott wrote:
>>>>
>>>>
>>>>> ∀x True(x) ↔ Boolean.True
>>>>
>>>> The LHS is false, isn't it? After all not everything is true. But the
>>>> RHS is true. So the above is false. Is that your intention?
>>>>
>>>>> ∀x WFF(x) ↔ (Boolean.True ∨ Boolean.False)
>>>>>
>>>
>>> There are three steps in my formula:
>>> If True(x) satisfies all three then Boolean.True.
>>> else Boolean.False ∨ ~WFF(x)
>>>
>>> Would it be easier
>>
>> Not right now it wouldn't because it doesn't address this problem:
>> You: ∀x True(x) ↔ Boolean.True
>> Me: The LHS is false, isn't it? After all not everything is true. But
>> the RHS is true. So the above is false. Is that your intention?
>
> No, no, no no. it is not at all like that.
> When my entire formula is fully satisfied by x, then and only then is x logically equivalent to Boolean.True. If any part of my formula is not fully satisfied by x, then x is NOT Boolean.True. Not Boolean.True includes Boolean.False and semantically incorrect.

So you have a formula True with one free variable of the type sentence,
and when that free variable is replaced by a sentence x you symbolize
the result as True(x). The formula being 'satisfied' by x means True(x)
is true. And

True(x) is true iff (x iff Boolean.True) . . . . . (1)

Ok so far? Because I think (1) is just the same as

True(x) <-> x . . . . . . . . . . . . . . . . . . (2)

What you can't do is derive

(forall x)(True(x) <-> x)

from (2) because in

formula such-and-such is satisfied by so-and-so

so-and-so is not a variable. For example -

... is the President of the USA

is satisfied by Donald Trump. Another example -

... is a larger number than ...

is satisfied by (3,2), (pi,e), etc.

>>> if I said:
>>> Semantically-Incorrect(x)
>>> Invalid-Proposition(x)
>>> Its-Neither-True-Nor-False-Because-Its-Screwed-Up(x)
>>>
>>> instead of ~WFF(x) ???
>>
>> ... but since you're asking: x not being well formed (which is what I
>> have supposed ~WFF(x) means) isn't the same as x being semantically
>> incorrect. Not being well formed is a syntactic matter. See any logic
>> text. Also, x not being well formed isn't the same as x being invalid.
>> Invalidity only applies to wff. See any logic text.
>
> I have ONLY been using ~WFF(x) to mean semantically incorrect.
> This is totally obvious if you merely look as how WFF(x) is defined.
>
> Semantic correctness of x is verified syntactically as is required whenever any expression is formalized.

How do you do that - syntactically verify semantic correctness?

[peterolcott]

No and I have explained this in great detail to you at least thirty times so I will estimate that you are faking your lack of understanding.

[Peter Percival]

peterolcott wrote:
> [...]

> No and I have explained this in great detail to you at least thirty times so I will estimate that you are faking your lack of understanding.
>

I will follow your exchanges with Jim Burns to see if they shed any
light on the matter. Also, I will read the Church paper recently
referred to, in the hope that his notation explains yours.

[Jim Burns]

On 2/7/2017 2:39 PM, Peter Percival wrote:
> Jim Burns wrote:

>> First order logic with a single binary predicate is
>> undecidable, so anything with _at least_ first order
>> logic with a single binary predicate is undecidable.
>
> Just so long as there are no non-logical axioms. The
> theory of ordered abelian groups is decidable. Not that
> that affects what you went on to say.

However, if I'm honest, I'll have to admit that it affects
what I was thinking. _Now_ I am thinking that I have an
underdeveloped sense of what it means for one theory
(such as ST) to interpret another theory (such as Q).

In an ideal world, in which I would be an ideal student, I
would go and develop my sense of what "interpret" means in
this context. (Ironically, doing that would be something very
similar to what I have advised others to do.) And I think
I will go and develop in a little while.

(In fact, now that I think a bit, Presburger arithmetic
is provably complete, and it certainly has a binary
predicate, = )

I've been thinking out loud (on the screen) about all this,
but I just deleted it all. I'd rather go somewhere quiet to
think.

One thought, though:
-- Q is provably undecidable.
-- Q + {something consistent} will still prove the same
sentence G has no proof.
-- Q - {something} is also undecidable, even if not
provably undecidable. G still has no proof.
-- If I take {something} away from Q and then add
{something else}, I might no longer be able to extend
Q - {something} + {something else} back to Q.
If that were so, then the Incompleteness result for
Q would be taken off the table, finally.
Or something.

[peterolcott]

Forget all about the notation. Try and simply understand that no logical proposition can be formed that does not express some relation between things.

7 is greater than.
7 is greater than an orange.
7 > 5.

[peterolcott]

On Tuesday, February 7, 2017 at 7:56:33 PM UTC-6, Jim Burns wrote:
> On 2/7/2017 2:39 PM, Peter Percival wrote:
> > Jim Burns wrote:
>
> >> First order logic with a single binary predicate is
> >> undecidable, so anything with _at least_ first order
> >> logic with a single binary predicate is undecidable.
> >
> > Just so long as there are no non-logical axioms. The
> > theory of ordered abelian groups is decidable. Not that
> > that affects what you went on to say.
>
> However, if I'm honest, I'll have to admit that it affects
> what I was thinking. _Now_ I am thinking that I have an
> underdeveloped sense of what it means for one theory
> (such as ST) to interpret another theory (such as Q).
>
> In an ideal world, in which I would be an ideal student, I
> would go and develop my sense of what "interpret" means in
> this context. (Ironically, doing that would be something very
> similar to what I have advised others to do.) And I think
> I will go and develop in a little while.

Here is what I mean by interpret:
https://plato.stanford.edu/entries/model-theory/#Basic

[peterolcott]

On Tuesday, February 7, 2017 at 5:48:15 PM UTC-6, Peter Percival wrote:
> peterolcott wrote:
> > [...]
> > No and I have explained this in great detail to you at least thirty times so I will estimate that you are faking your lack of understanding.
> >
> I will follow your exchanges with Jim Burns to see if they shed any
> light on the matter. Also, I will read the Church paper recently
> referred to, in the hope that his notation explains yours.
>

I have put at least 1000 hours into this since July. I have put 200 hours into this in the last month, working 60 hours per week for the last three weeks.

It is a little annoying that no one has ever expressed any understanding at all. It is really not that hard to understand that comparing integers and oranges is semantically incorrect.

Compatible-Types( binary-relation, arg1, arg2 )

Kurt Gödel (1944) "theory of simple types"
...sentences of the form: " a has the property φ ", " b bears the relation R to c ", etc. are meaningless, if a, b, c, R, φ are not of types fitting together.

[peterolcott]

On Tuesday, February 7, 2017 at 7:56:33 PM UTC-6, Jim Burns wrote:

https://plato.stanford.edu/entries/model-theory/#Cons

[Peter Percival]

peterolcott wrote:


> Forget all about the notation. Try and simply understand that no logical proposition can be formed that does not express some relation between things.
>
> 7 is greater than.
> 7 is greater than an orange.
> 7 > 5.

Do you include what might be called "one place relations" among
relations? So is "lions are bold" a logical proposition? (Naturally, I
wouldn't call them one place relations, I'd call them properties.)

[peterolcott]

On Tuesday, February 7, 2017 at 4:30:49 PM UTC-6, Peter Percival wrote:
> peterolcott wrote:
> > On Tuesday, February 7, 2017 at 3:19:12 PM UTC-6, Peter Percival wrote:
> >> peterolcott wrote:
> >>> On Tuesday, February 7, 2017 at 2:12:23 PM UTC-6, Peter Percival wrote:
> >>>> peterolcott wrote:
> >>>>
> >>>>
> >>>>> ∀x True(x) ↔ Boolean.True
> >>>>
> >>>> The LHS is false, isn't it? After all not everything is true. But the
> >>>> RHS is true. So the above is false. Is that your intention?
> >>>>
> >>>>> ∀x WFF(x) ↔ (Boolean.True ∨ Boolean.False)
> >>>>>
> >>>
> >>> There are three steps in my formula:
> >>> If True(x) satisfies all three then Boolean.True.
> >>> else Boolean.False ∨ ~WFF(x)
> >>>
> >>> Would it be easier
> >>
> >> Not right now it wouldn't because it doesn't address this problem:
> >> You: ∀x True(x) ↔ Boolean.True
> >> Me: The LHS is false, isn't it? After all not everything is true. But
> >> the RHS is true. So the above is false. Is that your intention?
> >
> > No, no, no no. it is not at all like that.
> > When my entire formula is fully satisfied by x, then and only then is x logically equivalent to Boolean.True. If any part of my formula is not fully satisfied by x, then x is NOT Boolean.True. Not Boolean.True includes Boolean.False and semantically incorrect.
>
> So you have a formula True with one free variable of the type sentence,
> and when that free variable is replaced by a sentence x you symbolize
> the result as True(x). The formula being 'satisfied' by x means True(x)
> is true. And
>

I do not define True(x) using the circular reasoning of saying that it maps to Boolean.True. Instead I provide the criteria by which True(x) would map to Boolean.True:

(1) A logical proposition asserts a relation between things.
(2) These things must be of compatible types with this relation.
(3) This relation actually exists. // else false

7 is greater than. // Numeric-Greater-Than(7)
7 is greater than an Orange. // Numeric-Greater-Than(7, an Orange)
7 > 5. // Numeric-Greater-Than(7, 5)

We can't really call it a logical proposition until after it has been validated so before it has been validated I call it an expression. An expression is a sequence of symbols that purport to be a logical proposition.

Kurt Gödel (1944) "theory of simple types"
...sentences of the form: " a has the property φ ", " b bears the relation R to c ", etc. are meaningless, if a, b, c, R, φ are not of types fitting together.

https://plato.stanford.edu/entries/type-theory-church/#Syn
I don't use the syntax of type theory because it is clumsy.

http://www.cyc.com/syntax-cycl/
I do as the largest AI project in the world does with their CycL use the syntax of FOPL, yet with a richer semantics.

[peterolcott]

On Wednesday, February 8, 2017 at 7:05:13 AM UTC-6, Peter Percival wrote:
> peterolcott wrote:
>
>
> > Forget all about the notation. Try and simply understand that no logical proposition can be formed that does not express some relation between things.
> >
> > 7 is greater than.
> > 7 is greater than an orange.
> > 7 > 5.
>
> Do you include what might be called "one place relations" among
> relations? So is "lions are bold" a logical proposition? (Naturally, I
> wouldn't call them one place relations, I'd call them properties.)
>

In my simplified type theory system everything is either a relation or an individual. A property is a type of relation: Have-Property(Lions, Boldness).

Relations and Individuals form the two subatomic units of semantic compositionality. The entire set of all knowledge (besides direct sense organ stimulus) can be represented as a single acyclic digraph such that this knowledge is both consistent and complete.

To provide the correct design to the Cyc project, Incompleteness must be shown to be erroneous. Incompleteness does not really show that formal systems are incomplete or inconsistent. It only shows that human understanding of formal systems has gaps in its reasoning.

I finally found the book that taught Douglas Hofstader the Incompleteness Theorem: Gödel’s Proof by Ernest Nagel and James R. Newman

Free copies can be found on the internet. I don't want to post a link because the free copy may be taken down. It is only 129 pages long. There are at least two different free copies, one is a scan of a hard copy. The other is a PDF the other one is an original PDF.

The concluding remarks of the book explicitly state that it is possible to prove general incompleteness to be incorrect because Gödel’s Proof may not actually be generalizable beyond Principia Mathematica.

So like I have been saying for at least four years the language of arithmetic may not be expressive enough to see that gaps in its own reasoning.

[Peter Percival]

peterolcott wrote:
> On Wednesday, February 8, 2017 at 7:05:13 AM UTC-6, Peter Percival wrote:
>> peterolcott wrote:
>>
>>
>>> Forget all about the notation. Try and simply understand that no logical proposition can be formed that does not express some relation between things.
>>>
>>> 7 is greater than.
>>> 7 is greater than an orange.
>>> 7 > 5.
>>
>> Do you include what might be called "one place relations" among
>> relations? So is "lions are bold" a logical proposition? (Naturally, I
>> wouldn't call them one place relations, I'd call them properties.)
>>
>
> In my simplified type theory system everything is either a relation or an individual. A property is a type of relation: Have-Property(Lions, Boldness).

And can relations be one-place, or must they be two or more place?

>
> Relations and Individuals form the two subatomic units of semantic compositionality. The entire set of all knowledge (besides direct sense organ stimulus) can be represented as a single acyclic digraph such that this knowledge is both consistent and complete.
>
> To provide the correct design to the Cyc project, Incompleteness must be shown to be erroneous. Incompleteness does not really show that formal systems are incomplete or inconsistent. It only shows that human understanding of formal systems has gaps in its reasoning.
>
> I finally found the book that taught Douglas Hofstader the Incompleteness Theorem: Gödel’s Proof by Ernest Nagel and James R. Newman
>
> Free copies can be found on the internet. I don't want to post a link because the free copy may be taken down. It is only 129 pages long. There are at least two different free copies, one is a scan of a hard copy. The other is a PDF the other one is an original PDF.

I have it. I'll see if I can find it.

> The concluding remarks of the book explicitly state that it is possible to prove general incompleteness to be incorrect because Gödel’s Proof may not actually be generalizable beyond Principia Mathematica.
>
> So like I have been saying for at least four years the language of arithmetic may not be expressive enough to see that gaps in its own reasoning.
>

[peterolcott]

On Wednesday, February 8, 2017 at 7:39:22 AM UTC-6, Peter Percival wrote:
> peterolcott wrote:
> > On Wednesday, February 8, 2017 at 7:05:13 AM UTC-6, Peter Percival wrote:
> >> peterolcott wrote:
> >>
> >>
> >>> Forget all about the notation. Try and simply understand that no logical proposition can be formed that does not express some relation between things.
> >>>
> >>> 7 is greater than.
> >>> 7 is greater than an orange.
> >>> 7 > 5.
> >>
> >> Do you include what might be called "one place relations" among
> >> relations? So is "lions are bold" a logical proposition? (Naturally, I
> >> wouldn't call them one place relations, I'd call them properties.)
> >>
> >
> > In my simplified type theory system everything is either a relation or an individual. A property is a type of relation: Have-Property(Lions, Boldness).
>
> And can relations be one-place, or must they be two or more place?

So far I can't imagine a one place relation that expresses truth.
Exists(Automobiles) is actually
Element-Of-Set(Automobiles, Universal-Set)
Has-Property(Automobiles, Existence)

> >
> > Relations and Individuals form the two subatomic units of semantic compositionality. The entire set of all knowledge (besides direct sense organ stimulus) can be represented as a single acyclic digraph such that this knowledge is both consistent and complete.
> >
> > To provide the correct design to the Cyc project, Incompleteness must be shown to be erroneous. Incompleteness does not really show that formal systems are incomplete or inconsistent. It only shows that human understanding of formal systems has gaps in its reasoning.
> >
> > I finally found the book that taught Douglas Hofstader the Incompleteness Theorem: Gödel’s Proof by Ernest Nagel and James R. Newman
> >
> > Free copies can be found on the internet. I don't want to post a link because the free copy may be taken down. It is only 129 pages long. There are at least two different free copies, one is a scan of a hard copy. The other is a PDF the other one is an original PDF.
>
> I have it. I'll see if I can find it.
>

page 109 of the revised edition.

[Charlie-Boo]

Which types fit together? Is the square root of -1 (i) equal to 0? Is it less than? "either less than or greater than"?

What does (0/0)-(0/0) equal?

C-B

[Peter Percival]

peterolcott wrote:
> On Wednesday, February 8, 2017 at 7:39:22 AM UTC-6, Peter Percival wrote:
>> peterolcott wrote:
>>> On Wednesday, February 8, 2017 at 7:05:13 AM UTC-6, Peter Percival wrote:
>>>> peterolcott wrote:
>>>>
>>>>
>>>>> Forget all about the notation. Try and simply understand that no logical proposition can be formed that does not express some relation between things.
>>>>>
>>>>> 7 is greater than.
>>>>> 7 is greater than an orange.
>>>>> 7 > 5.
>>>>
>>>> Do you include what might be called "one place relations" among
>>>> relations? So is "lions are bold" a logical proposition? (Naturally, I
>>>> wouldn't call them one place relations, I'd call them properties.)
>>>>
>>>
>>> In my simplified type theory system everything is either a relation or an individual. A property is a type of relation: Have-Property(Lions, Boldness).
>>
>> And can relations be one-place, or must they be two or more place?
>
> So far I can't imagine a one place relation that expresses truth.

Next time I meet a lion, I'll tell him he's not bold. I was going to
add: if I can find the courage. But "Peter is a coward" doesn't express
a truth either I suppose.

> Exists(Automobiles) is actually

I'd say "automobiles exist" is "actually"

(Ex,y)(A(x) & A(y) & x=/=y)

but there's more than one way to skin a lion.

> Element-Of-Set(Automobiles, Universal-Set)
> Has-Property(Automobiles, Existence)
>
>>>
>>> Relations and Individuals form the two subatomic units of semantic compositionality. The entire set of all knowledge (besides direct sense organ stimulus) can be represented as a single acyclic digraph such that this knowledge is both consistent and complete.
>>>
>>> To provide the correct design to the Cyc project, Incompleteness must be shown to be erroneous. Incompleteness does not really show that formal systems are incomplete or inconsistent. It only shows that human understanding of formal systems has gaps in its reasoning.
>>>
>>> I finally found the book that taught Douglas Hofstader the Incompleteness Theorem: Gödel’s Proof by Ernest Nagel and James R. Newman
>>>
>>> Free copies can be found on the internet. I don't want to post a link because the free copy may be taken down. It is only 129 pages long. There are at least two different free copies, one is a scan of a hard copy. The other is a PDF the other one is an original PDF.
>>
>> I have it. I'll see if I can find it.
>>
> page 109 of the revised edition.
>

[peterolcott]

Square-Root(Can-of-Tomatoes)
Numerical-Greater-Than(Seven, An-Orange)
Numerical-Greater-Than(Seven, Five)

[Peter Percival]

Which I can't, but it wasn't the revised edition anyway. I would have
bought my copy in the 70's.

>>
> page 109 of the revised edition.
>

Which, in the pdf I have now downloaded, is the first page of Chapter
VIII 'Concluding Reflections'. I see nothing there to support your

'The concluding remarks of the book explicitly state that it is possible

to prove general incompleteness to be incorrect because Gödel’s Proof

may not actually be generalizable beyond Principia Mathematica.'

[Peter Percival]

Which seems not to answer Charlie-Boo's question.

i=0, i<0, i<0 or i>0

are all well-formed and false.

[Jim Burns]

On 2/8/2017 9:14 AM, peterolcott wrote:
> On Wednesday, February 8, 2017 at 7:39:22 AM UTC-6,
> Peter Percival wrote:

>> And can relations be one-place, or must they be
>> two or more place?
>
> So far I can't imagine a one place relation that
> expresses truth.
> Exists(Automobiles) is actually
> Element-Of-Set(Automobiles, Universal-Set)
> Has-Property(Automobiles, Existence)

Perhaps you mean Universal-Class.

Every set has the property that it is strictly smaller
than its power set. (Cantor's theorem.) This would include
Universal-Set, if it existed.

However, Power(Universal-Set), the power set of Universal-Set
must necessarily be be a proper subset of Universal-Set.
It's universal, after all. Contradiction.

Therefore, there is no Universal-Set.

Sometimes you seem to care about issues like this,
so I thought you might like to know.

[peterolcott]

First of all one must start with the precise meaning of the term possible:
◇P ↔ ~◻~P Possibly(P) <--> Not(Necessarily) Not(P)
◻P ↔ ~◇~P Necessarily(P) <--> Not(Possibly) Not(P)

Second we must understand that General Incompleteness applies to every element of the set of formal systems not just Principia Mathematica.

Then we get to the cut-and-paste from the book:
The possibility of constructing a finitistic absolute proof of consistency for a formal system such as Principia Mathematica is not excluded by Gödel’s results.

Gödel showed that no such proof is possible that can be mirrored inside Principia Mathematica. His argument does not eliminate the possibility of strictly finitistic proofs that cannot be mirrored inside Principia Mathematica.

But no one today appears to have a clear idea of what a finitistic proof would
be like that is not capable of being mirrored inside Principia Mathematica.

[peterolcott]

I don't want to muddy the issues. I want to stay focused on showing that my formula really does exclude all paradox as semantically ill-formed.

Knowing that I will not be willing to go off on any tangents, I will say that the powerset is logically incoherent because it requires that one of its elements is the set itself. So Russell's Paradox is very simply explained in the the concept of a set entirely containing itself is precisely analogous to a tin can entirely containing itself, thus merely an incoherent misconception.

[Peter Percival]

Which is irrelevant. Also, if I did need to know the precise meaning of
the term possible those two equivalence wouldn't help at all.

> Second we must understand that General Incompleteness applies to every element of the set of formal systems not just Principia Mathematica.

No it doesn't. The incompleteness theorem applies to very particular
kinds of formal system.

> Then we get to the cut-and-paste from the book:
> The possibility of constructing a finitistic absolute proof of consistency for a formal system such as Principia Mathematica is not excluded by Gödel’s results.
>
> Gödel showed that no such proof is possible that can be mirrored inside Principia Mathematica. His argument does not eliminate the possibility of strictly finitistic proofs that cannot be mirrored inside Principia Mathematica.
>
> But no one today appears to have a clear idea of what a finitistic proof would
> be like that is not capable of being mirrored inside Principia Mathematica.
>

Which doesn't support your 'The concluding remarks of the book

explicitly state that it is possible to prove general incompleteness to
be incorrect because Gödel’s Proof may not actually be generalizable
beyond Principia Mathematica.'

There are two theorems of Gödel's that are relevant here. They are
sometimes called the first and second incompleteness theorems. I took
your remark quoted by me in the previous paragraph to be about the first
incompleteness theorem. The remarks about a finitistic absolute proof
of consistency are clearly about the second incompleteness theorem.
Yes, a finitistic absolute proof of the consistency of PM is not
excluded by GIT2. But if such a proof were found that would not
invalidate GIT1.

[Peter Percival]

peterolcott wrote:

> Knowing that I will not be willing to go off on any tangents, I will
> say that the powerset is logically incoherent because it requires
> that one of its elements is the set itself. So Russell's Paradox is
> very simply explained in the the concept of a set entirely containing
> itself is precisely analogous to a tin can entirely containing
> itself, thus merely an incoherent misconception.

Clearly you don't know what you're talking about. Example, S = {a,b).
Power set of S, PS = {0,{a},{b},{a,b}}. Yes, one of its (that's PS's)
elements (viz {a,b}) is the set (that's S) itself. There is no logical
incoherence. I have written 0 for the empty set.

[peterolcott]

When I used the term "general incompleteness" I was referring to the (possibly incorrect) common understanding that the theorem showed that all formal systems will be forever incomplete or inconsistent.

[Peter Percival]

See the Examples here: https://en.wikipedia.org/wiki/Complete_theory.

[Jim Burns]

But, I just showed you a paradox that your formula
does not exclude! You use a universal set, and that is
both larger and smaller than its power set.

But this time, you don't care. Okay.

Then, why you do care about the Liar paradox?

> Knowing that I will not be willing to go off on any
> tangents, I will say that the powerset is logically
> incoherent because it requires that one of its elements
> is the set itself.

Your claim that it is logically incoherent for the
power set P(A) of a set A to have A as an element
is logically incoherent. It doesn't even contradict
the axiom of foundation (which would ban sets like p = {p} ).
P(A) is not A. Why _shouldn't_ A be in P(A)?

Anyway, the paradox still stands, even taking into
account your (incomprehensible) objection.
The set of all subsets of the universe V _except for V_
P(V)/{V} is still easily proven larger _and_ smaller than V.

Therefore there is no V, no Universal-Set.

> So Russell's Paradox is very simply
> explained in the the concept of a set entirely containing
> itself is precisely analogous to a tin can entirely
> containing itself, thus merely an incoherent misconception.

How odd that we're suddenly talking about Russell's class.
One might almost think you have no idea what you're
talking about, despite spend such an enormous amount
of time on it.

You've been asking for people to read your work, so that
they will understand you. Consider the possibility that
people _have_ read your work, enough to see that you're
wrong, and gone off to do other things.

Here's what happened just now, when I showed you an error
in your system:
You gave a bogus dismissal of the issue. ( A e P(A) ? )
You declared that you weren't going off on a tangent
(justifying ignoring a contradiction).
You changed the subject. (To Russell's class, thus going
off on a _different_ tangent than one that addresses the
problem I showed you).

The way you behave, you should be grateful for any attention
at all.

[peterolcott]

You did not show me anything that my formula did not exclude. You showed me something that I did not exclude.

>
> But this time, you don't care. Okay.
>
> Then, why you do care about the Liar paradox?
>
> > Knowing that I will not be willing to go off on any
> > tangents, I will say that the powerset is logically
> > incoherent because it requires that one of its elements
> > is the set itself.
>
> Your claim that it is logically incoherent for the
> power set P(A) of a set A to have A as an element
> is logically incoherent. It doesn't even contradict
> the axiom of foundation (which would ban sets like p = {p} ).
> P(A) is not A. Why _shouldn't_ A be in P(A)?
>
> Anyway, the paradox still stands, even taking into
> account your (incomprehensible) objection.
> The set of all subsets of the universe V _except for V_
> P(V)/{V} is still easily proven larger _and_ smaller than V.
>
> Therefore there is no V, no Universal-Set.

That depends upon whether Cantor is correct that infinities have differing sizes. His diagonalization of integers over reals seems convincing.

>
> > So Russell's Paradox is very simply
> > explained in the the concept of a set entirely containing
> > itself is precisely analogous to a tin can entirely
> > containing itself, thus merely an incoherent misconception.
>
> How odd that we're suddenly talking about Russell's class.
> One might almost think you have no idea what you're
> talking about, despite spend such an enormous amount
> of time on it.
>
> You've been asking for people to read your work, so that
> they will understand you. Consider the possibility that
> people _have_ read your work, enough to see that you're
> wrong, and gone off to do other things.
>

Yet pointed out zero mistakes? I don't buy that.
Yes I may have made a mistake using the term universal set instead of universal class. This is not at all the same thing as finding any error in my formula.

> Here's what happened just now, when I showed you an error
> in your system:
> You gave a bogus dismissal of the issue. ( A e P(A) ? )
> You declared that you weren't going off on a tangent
> (justifying ignoring a contradiction).
> You changed the subject. (To Russell's class, thus going
> off on a _different_ tangent than one that addresses the
> problem I showed you).
>
> The way you behave, you should be grateful for any attention
> at all.

Maybe. There was one guy once that neither pointed out any errors nor acknowledged any understanding, and seemed to just play me for amusement. I spent a thousand hours talking to this guy over many years.

[peterolcott]

This is what I meant by general incompleteness:

https://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems
The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.

https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem
The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system.

[Jim Burns]

On 2/8/2017 1:42 PM, peterolcott wrote:
> On Wednesday, February 8, 2017 at 12:06:58 PM UTC-6,
> Jim Burns wrote:
>> On 2/8/2017 12:32 PM, peterolcott wrote:

>>> I don't want to muddy the issues. I want to stay focused
>>> on showing that my formula really does exclude all
>>> paradox as semantically ill-formed.
>>
>> But, I just showed you a paradox that your formula
>> does not exclude! You use a universal set, and that is
>> both larger and smaller than its power set.
>
> You did not show me anything that my formula did not
> exclude. You showed me something that I did not exclude.

I showed you a contradiction and you immediately threw
three separate, sometimes _conflicting_ excuses on the
table for not looking at it.

With

"You did not show me anything that my formula did not
exclude."

I count four now. How many more?

You're just bullshitting to avoid having to deal with
problems that people point out to you.

>> But this time, you don't care. Okay.
>>
>> Then, why you do care about the Liar paradox?
>>
>>> Knowing that I will not be willing to go off on any
>>> tangents, I will say that the powerset is logically
>>> incoherent because it requires that one of its elements
>>> is the set itself.
>>
>> Your claim that it is logically incoherent for the
>> power set P(A) of a set A to have A as an element
>> is logically incoherent. It doesn't even contradict
>> the axiom of foundation (which would ban sets like p = {p} ).
>> P(A) is not A. Why _shouldn't_ A be in P(A)?
>>
>> Anyway, the paradox still stands, even taking into
>> account your (incomprehensible) objection.
>> The set of all subsets of the universe V _except for V_
>> P(V)/{V} is still easily proven larger _and_ smaller than V.
>>
>> Therefore there is no V, no Universal-Set.
>
> That depends upon whether Cantor is correct that
> infinities have differing sizes. His diagonalization
> of integers over reals seems convincing.

Maybe we should talk about that some. There's a reason
that argument we've been talking about is called the
_diagonal_ lemma. Your plan is analogous to being very
clever about cramming more and more and more reals into
a list. My response is to point out it's not going to work.

I presume you see why cramming more reals into a list
-- no matter how clever -- is not going to work. Maybe
you would not be very interested in _how_ someone was
going to cram more reals into a list. And the more clever,
the more baroque the descriptions of what was going to happen,
the less interested you would be (I bet). The more work
it would take to wade though the details to find a specific
error for the real-crammer to ignore.

The most general form of Cantor's argument is also the
shortest and clearest:
_Every_ set is strictly smaller than its power set.

Let A be a set and P(A) be its power set.

Let f: A -> P(A) be some sort of attempt to match every
subset of A (the elements of P(A)) to an element of A.
It will fail. No matter how clever the construction of
f is, there will be at least one subset unmatched.

Consider the set D[f], constructed from the attempt-to-match
f. We define D[f]
D[f] = { x e A | ~(x e f(x)) }

The set D[f] is not paradoxical. It's not in the least
self-referential. It's a perfectly fine set with the property
that, if f matches some d to D[f] , then d will be in D[f]
if and only if it is not in D[f].

Therefore, there is no f that matches the elements of A to the
subsets of A. Therefore, P(A) is strictly larger than A.

>>> So Russell's Paradox is very simply
>>> explained in the the concept of a set entirely containing
>>> itself is precisely analogous to a tin can entirely
>>> containing itself, thus merely an incoherent misconception.
>>
>> How odd that we're suddenly talking about Russell's class.
>> One might almost think you have no idea what you're
>> talking about, despite spend such an enormous amount
>> of time on it.
>>
>> You've been asking for people to read your work, so that
>> they will understand you. Consider the possibility that
>> people _have_ read your work, enough to see that you're
>> wrong, and gone off to do other things.
>
> Yet pointed out zero mistakes? I don't buy that.
> Yes I may have made a mistake using the term

No, you don't get it yet.

Proper classes are a real thing, and it was a real error to
speak of a proper class such as The Universe as though it
were a set because that createa contradiction, which would
break your _whole_ formal system.

However, the most important point I'd like to make here is
that _it is a subtle error_ . The error you make when you
(try to) formalize True(x) _is a subtle error_ , even more
subtle than the Universal-Set error. Just because it's not
smacking you in the face, that doesn't mean it's not important.

> universal set instead of universal class. This is not at
> all the same thing as finding any error in my formula.

Oh, I'm done finding errors in your formula. Been there,
done that, got the T-shirt.

My problem is not finding an error in your formula,
it's getting you to listen. Judging by your reaction to the
Universal-Set thing, it's looking pretty bad for my team.

>> Here's what happened just now, when I showed you an error
>> in your system:
>> You gave a bogus dismissal of the issue. ( A e P(A) ? )
>> You declared that you weren't going off on a tangent
>> (justifying ignoring a contradiction).
>> You changed the subject. (To Russell's class, thus going
>> off on a _different_ tangent than one that addresses the
>> problem I showed you).
>>
>> The way you behave, you should be grateful for any attention
>> at all.
>
> Maybe. There was one guy once that neither pointed out
> any errors nor acknowledged any understanding, and seemed
> to just play me for amusement. I spent a thousand hours
> talking to this guy over many years.

Am I supposed to be this guy, in your view? I've watched you
post -- from a distance, more often than not. I never thought
you had the least chance of being right, but I rarely said
anything, because it's a lot of work tracing lines of thought
back, and back, and back, until you get to the knot. There's
only so much of that that I have the resources for.

I stuck my oar in this time because you said something in error
that I thought I could fairly clearly and fairly quickly
explain how it was in error: You said you formalize True(x).
For any reasonable version of what you might mean by True(x)
and quite a wide range of unreasonable versions, sufficiently
defining True(x) also sufficiently defines a counter-example
-- a specific, concrete counter-example.

(If you're curious, _go read what I've posted a dozen times_ .)

So, you say you get hassled by people who don't show you
errors. Yet, my experience of you is that when I show you
errors you find excuses, sometimes very lame excuses, to
ignore the errors.

If I decide to play you for amusement (I haven't yet -- not
quite), it will be because _trying to help you_ doesn't
get me anywhere.

[peterolcott]

On Wednesday, February 8, 2017 at 5:54:31 PM UTC-6, Jim Burns wrote:
> On 2/8/2017 1:42 PM, peterolcott wrote:
> > On Wednesday, February 8, 2017 at 12:06:58 PM UTC-6,
> > Jim Burns wrote:
> >> On 2/8/2017 12:32 PM, peterolcott wrote:
>
> >>> I don't want to muddy the issues. I want to stay focused
> >>> on showing that my formula really does exclude all
> >>> paradox as semantically ill-formed.
> >>
> >> But, I just showed you a paradox that your formula
> >> does not exclude! You use a universal set, and that is
> >> both larger and smaller than its power set.
> >
> > You did not show me anything that my formula did not
> > exclude. You showed me something that I did not exclude.
>
> I showed you a contradiction and you immediately threw
> three separate, sometimes _conflicting_ excuses on the
> table for not looking at it.

please cut-and-paste what you are referring to.

>
> With
> "You did not show me anything that my formula did not
> exclude."
> I count four now. How many more?
>
> You're just bullshitting to avoid having to deal with
> problems that people point out to you.

Not really I count bullshitting as aligning myself with Satan (the father of all lies) so I don't do it.

>
> >> But this time, you don't care. Okay.
> >>
> >> Then, why you do care about the Liar paradox?
> >>
> >>> Knowing that I will not be willing to go off on any
> >>> tangents, I will say that the powerset is logically
> >>> incoherent because it requires that one of its elements
> >>> is the set itself.
> >>
> >> Your claim that it is logically incoherent for the
> >> power set P(A) of a set A to have A as an element
> >> is logically incoherent. It doesn't even contradict
> >> the axiom of foundation (which would ban sets like p = {p} ).
> >> P(A) is not A. Why _shouldn't_ A be in P(A)?
> >>
> >> Anyway, the paradox still stands, even taking into
> >> account your (incomprehensible) objection.
> >> The set of all subsets of the universe V _except for V_
> >> P(V)/{V} is still easily proven larger _and_ smaller than V.
> >>
> >> Therefore there is no V, no Universal-Set.
> >
> > That depends upon whether Cantor is correct that
> > infinities have differing sizes. His diagonalization
> > of integers over reals seems convincing.
>
> Maybe we should talk about that some. There's a reason
> that argument we've been talking about is called the
> _diagonal_ lemma. Your plan is analogous to being very
> clever about cramming more and more and more reals into
> a list. My response is to point out it's not going to work.

If I wanted to disavow Cantor, I would be taking reals out of the list until there was an equal number to the number of integers.

I really don't want to talk about infinities very much. The set of integers is infinite, the set of reals is infinite and may or may not be larger than the set of integers.

> >>> So Russell's Paradox is very simply
> >>> explained in the the concept of a set entirely containing
> >>> itself is precisely analogous to a tin can entirely
> >>> containing itself, thus merely an incoherent misconception.
> >>
> >> How odd that we're suddenly talking about Russell's class.
> >> One might almost think you have no idea what you're
> >> talking about, despite spend such an enormous amount
> >> of time on it.
> >>
> >> You've been asking for people to read your work, so that
> >> they will understand you. Consider the possibility that
> >> people _have_ read your work, enough to see that you're
> >> wrong, and gone off to do other things.
> >
> > Yet pointed out zero mistakes? I don't buy that.
> > Yes I may have made a mistake using the term
>
> No, you don't get it yet.
>
> Proper classes are a real thing, and it was a real error to
> speak of a proper class such as The Universe as though it
> were a set because that createa contradiction, which would
> break your _whole_ formal system.

Since it creates a contradiction in its own definition is only exists as a misconception, thus is merely a set of incoherent reasoning. If we assume that your reasoning is correct, then Exists(Universal-Set) = false. No paradox is created.

If we eliminate infinities with algorithmic compression we can still have a Universal set of all unique finite things. The Universal set would then include the entire concept of set theory, but, no actual sets. Actually now that you point this out, that is what I have had in mind all along.

http://www.cyc.com/ontologists-handbook/
I have always been thinking in terms of a finite acyclic digraph Pete's simplest theory of types knowledge ontology of the currently existing set of all knowledge.

The above link provides the documentation of CycL means of representing the set of concepts. Cycorp has 700 labor years worth of effort manually encoded this stuff over the last 33 years.

>
> However, the most important point I'd like to make here is
> that _it is a subtle error_ . The error you make when you
> (try to) formalize True(x) _is a subtle error_ , even more
> subtle than the Universal-Set error. Just because it's not
> smacking you in the face, that doesn't mean it's not important.
>
> > universal set instead of universal class. This is not at
> > all the same thing as finding any error in my formula.
>
> Oh, I'm done finding errors in your formula. Been there,
> done that, got the T-shirt.

By that you are referring to the fact that the universal set does not really exist? I already addressed that.

>
> My problem is not finding an error in your formula,
> it's getting you to listen. Judging by your reaction to the
> Universal-Set thing, it's looking pretty bad for my team.

If the Universal set does not exist because the set of axioms defining it results in a contradiction, then my system does not have to deal with it. The universal set does not present a question that has no correct answer like the Liar Paradox, (Is the Liar Paradox true or false?). The universal set is merely a misconception like the square root of a can of tomatoes.

>
> >> Here's what happened just now, when I showed you an error
> >> in your system:
> >> You gave a bogus dismissal of the issue. ( A e P(A) ? )
> >> You declared that you weren't going off on a tangent
> >> (justifying ignoring a contradiction).
> >> You changed the subject. (To Russell's class, thus going
> >> off on a _different_ tangent than one that addresses the
> >> problem I showed you).
> >>
> >> The way you behave, you should be grateful for any attention
> >> at all.
> >
> > Maybe. There was one guy once that neither pointed out
> > any errors nor acknowledged any understanding, and seemed
> > to just play me for amusement. I spent a thousand hours
> > talking to this guy over many years.
>
> Am I supposed to be this guy, in your view? I've watched you
> post -- from a distance, more often than not. I never thought
> you had the least chance of being right, but I rarely said
> anything, because it's a lot of work tracing lines of thought
> back, and back, and back, until you get to the knot. There's
> only so much of that that I have the resources for.

No you are totally not this guy at all you have been quite patient, yet fail to shake your own assumption that I must be totally incorrect. I have said some key things incorrectly. It looks like my system will never actually disprove any Incompleteness Theorem that is restricted to FOL because my system requires at least SOL.

>
> I stuck my oar in this time because you said something in error
> that I thought I could fairly clearly and fairly quickly
> explain how it was in error: You said you formalize True(x).
> For any reasonable version of what you might mean by True(x)
> and quite a wide range of unreasonable versions, sufficiently
> defining True(x) also sufficiently defines a counter-example
> -- a specific, concrete counter-example.
>
> (If you're curious, _go read what I've posted a dozen times_ .)
>
> So, you say you get hassled by people who don't show you
> errors. Yet, my experience of you is that when I show you
> errors you find excuses, sometimes very lame excuses, to
> ignore the errors.
>
> If I decide to play you for amusement (I haven't yet -- not
> quite), it will be because _trying to help you_ doesn't
> get me anywhere.

You never yet did respond to anything that I actually said in my formula itself. All of your responses have been based on you know that I must be wrong so you won't look at what I said and will try to correct my error without ever looking at my formula.

You would only have to directly see how my formula rejects numerous syntactically correct logical propositions as semantically ill-formed to begin to actually see what I am actually saying.

I did not do a very good job of totally translating these syntactically correct, semantically ill-formed propositions into conventional predicate logic until just now: (It may still violate conventions)

"This sentence is true." is formalized as:
hasProperty( x, True(x) ) // this predicate defines x

If we simple hypothetically assume that Tarski's True(x) is defined to see where this reasoning leads, we will see that it leads to the evaluation of above predicate getting stuck in an infinite loop.

Is the above hasProperty predicate now correctly specified a predicate logic, or do I need some quotes somewhere? We are NOT assuming FOL. It looks like we are quantifying over predicates.

[Jim Burns]

On 2/8/2017 11:16 PM, peterolcott wrote:
> On Wednesday, February 8, 2017 at 5:54:31 PM UTC-6,
> Jim Burns wrote:
>> On 2/8/2017 1:42 PM, peterolcott wrote:
>>> On Wednesday, February 8, 2017 at 12:06:58 PM UTC-6,
>>> Jim Burns wrote:
>>>> On 2/8/2017 12:32 PM, peterolcott wrote:

>>>>> I don't want to muddy the issues. I want to stay focused
>>>>> on showing that my formula really does exclude all
>>>>> paradox as semantically ill-formed.
>>>>
>>>> But, I just showed you a paradox that your formula
>>>> does not exclude! You use a universal set, and that is
>>>> both larger and smaller than its power set.
>>>
>>> You did not show me anything that my formula did not
>>> exclude. You showed me something that I did not exclude.
>>
>> I showed you a contradiction and you immediately threw
>> three separate, sometimes _conflicting_ excuses on the
>> table for not looking at it.
>
> please cut-and-paste what you are referring to.

<PO>

I don't want to muddy the issues. I want to stay focused
on showing that my formula really does exclude all
paradox as semantically ill-formed.

Knowing that I will not be willing to go off on any
tangents, I will say that the powerset is logically
incoherent because it requires that one of its elements

is the set itself. So Russell's Paradox is very simply

explained in the the concept of a set entirely
containing itself is precisely analogous to a tin can
entirely containing itself, thus merely an incoherent
misconception.

</PO>

>> With
>> "You did not show me anything that my formula did not
>> exclude."
>> I count four now. How many more?
>>
>> You're just bullshitting to avoid having to deal with
>> problems that people point out to you.
>
> Not really I count bullshitting as aligning myself with
> Satan (the father of all lies) so I don't do it.

I'd like to believe you. But if I believe you, then I don't
think I'll be able to shake your conviction that I have
nothing to show you.

On the other hand, if your claim that you don't (as you
see it) bullshit is bullshit, then I don't want to be
around you.

Heads you lose, tails you lose.
Which way will it come down? Hmmm.

I just thought I'd mention that Cantor's theorem, the one
you think you know, about giving an example of a real
not on a list of reals, for every list of reals, is
another thing you don't know about. Just saying.

Carry on.

There's nothing in the proof I gave that mentions infinities.
Go ahead and look. See? No infinities.

This kind of thing is why I'm giving up on you.

Okay, there are implications for infinities from the
proof. But that proof? No infinities.

> The set of integers is infinite, the set of reals is
> infinite and may or may not be larger than the
> set of integers.

SPOILER ALERT: The reals are larger.

>>>>> So Russell's Paradox is very simply
>>>>> explained in the the concept of a set entirely containing
>>>>> itself is precisely analogous to a tin can entirely
>>>>> containing itself, thus merely an incoherent misconception.
>>>>
>>>> How odd that we're suddenly talking about Russell's class.
>>>> One might almost think you have no idea what you're
>>>> talking about, despite spend such an enormous amount
>>>> of time on it.
>>>>
>>>> You've been asking for people to read your work, so that
>>>> they will understand you. Consider the possibility that
>>>> people _have_ read your work, enough to see that you're
>>>> wrong, and gone off to do other things.
>>>
>>> Yet pointed out zero mistakes? I don't buy that.
>>> Yes I may have made a mistake using the term
>>
>> No, you don't get it yet.
>>
>> Proper classes are a real thing, and it was a real error to
>> speak of a proper class such as The Universe as though it
>> were a set because that createa contradiction, which would
>> break your _whole_ formal system.
>
> Since it creates a contradiction in its own definition
> is only exists as a misconception, thus is merely a
> set of incoherent reasoning. If we assume that your
> reasoning is correct, then Exists(Universal-Set) = false.
> No paradox is created.

One more thing you need to learn is how bad a contradiction
is. If any contradiction is in your system, then anything
can be proven, anything and its negation, too.
("Ex falso quodlibet". This has been known a long time.)
Which means nothing can be proven, for who would believe
it if its negation could also be proven?

There are attempts to make logic more flexible, less
breakable. I don't know much about them.
<https://en.wikipedia.org/wiki/Paraconsistent_logic>

> If we eliminate infinities with algorithmic compression
> we can still have a Universal set of all unique finite
> things. The Universal set would then include the entire
> concept of set theory, but, no actual sets. Actually now
> that you point this out, that is what I have had in mind
> all along.

This doesn't seem to make any sense, but I haven't looked
very closely at it.

> http://www.cyc.com/ontologists-handbook/
> I have always been thinking in terms of a finite acyclic
> digraph Pete's simplest theory of types knowledge ontology
> of the currently existing set of all knowledge.
> The above link provides the documentation of CycL means
> of representing the set of concepts. Cycorp has 700
> labor years worth of effort manually encoded this stuff
> over the last 33 years.

You might want to look at machine learning. They're doing
amazing things. My impression is that rules-based expert
systems stalled a while back.

If you've been at this 30 years, you may remember perceptrons.
I mean something more like that. It was hot, then not.
Now it's back to hot.

>> However, the most important point I'd like to make here is
>> that _it is a subtle error_ . The error you make when you
>> (try to) formalize True(x) _is a subtle error_ , even more
>> subtle than the Universal-Set error. Just because it's not
>> smacking you in the face, that doesn't mean it's not important.
>>
>>> universal set instead of universal class. This is not at
>>> all the same thing as finding any error in my formula.
>>
>> Oh, I'm done finding errors in your formula. Been there,
>> done that, got the T-shirt.
>
> By that you are referring to the fact that the universal
> set does not really exist? I already addressed that.

No, that was a small point.
The big point is that you won't be able to formalize True().

I already tried to explain why.

>> My problem is not finding an error in your formula,
>> it's getting you to listen. Judging by your reaction to the
>> Universal-Set thing, it's looking pretty bad for my team.
>
> If the Universal set does not exist because the set of
> axioms defining it results in a contradiction, then my
> system does not have to deal with it. The universal set
> does not present a question that has no correct answer
> like the Liar Paradox, (Is the Liar Paradox true or false?).
> The universal set is merely a misconception like the
> square root of a can of tomatoes.

A formalized True() is merely a misconception like the
square root of a con of tomatoes. Actually, it's _very much_
like the misconception of a universal set, in that it
_sounds_ reasonable, but leads to contradictions.

I'm guessing that the reason you're willing to toss
the universal set is that you haven't invested as much
time and effort and emotion in it.

I'm just speculating, but maybe you should think about
sunk costs.
<https://en.wikipedia.org/wiki/Sunk_cost>

>>>> Here's what happened just now, when I showed you an error
>>>> in your system:
>>>> You gave a bogus dismissal of the issue. ( A e P(A) ? )
>>>> You declared that you weren't going off on a tangent
>>>> (justifying ignoring a contradiction).
>>>> You changed the subject. (To Russell's class, thus going
>>>> off on a _different_ tangent than one that addresses the
>>>> problem I showed you).
>>>>
>>>> The way you behave, you should be grateful for any attention
>>>> at all.
>>>
>>> Maybe. There was one guy once that neither pointed out
>>> any errors nor acknowledged any understanding, and seemed
>>> to just play me for amusement. I spent a thousand hours
>>> talking to this guy over many years.
>>
>> Am I supposed to be this guy, in your view? I've watched you
>> post -- from a distance, more often than not. I never thought
>> you had the least chance of being right, but I rarely said
>> anything, because it's a lot of work tracing lines of thought
>> back, and back, and back, until you get to the knot. There's
>> only so much of that that I have the resources for.
>
> No you are totally not this guy at all you have been quite
> patient, yet fail to shake your own assumption
> that I must be totally incorrect.

It's not an assumption, it's a theorem.
Just because you don't understand, that doesn't
make it an assumption.

> I have said some key
> things incorrectly. It looks like my system will never
> actually disprove any Incompleteness Theorem that is
> restricted to FOL because my system requires at least SOL.

No, making your system more powerful won't help.
The systems which can be proven complete are _weaker_
than provably incomplete systems.

If you're determined to avoid the Liar, you must make it
impossible to have a True() predicate. Make it weaker.

But weak enough is really, really weak. Presburger
arithmetic is weak enough to be complete, to avoid the
Curse of Diagonals. Presburger arithmetic can't multiply.

Is that what you want? A really weak system? If you do,
sorry, my mistake.

>> I stuck my oar in this time because you said something in error
>> that I thought I could fairly clearly and fairly quickly
>> explain how it was in error: You said you formalize True(x).
>> For any reasonable version of what you might mean by True(x)
>> and quite a wide range of unreasonable versions, sufficiently
>> defining True(x) also sufficiently defines a counter-example
>> -- a specific, concrete counter-example.
>>
>> (If you're curious, _go read what I've posted a dozen times_ .)
>>
>> So, you say you get hassled by people who don't show you
>> errors. Yet, my experience of you is that when I show you
>> errors you find excuses, sometimes very lame excuses, to
>> ignore the errors.
>>
>> If I decide to play you for amusement (I haven't yet -- not
>> quite), it will be because _trying to help you_ doesn't
>> get me anywhere.
>
> You never yet did respond to anything that I actually
> said in my formula itself. All of your responses have
> been based on you know that I must be wrong so you won't
> look at what I said and will try to correct my error
> without ever looking at my formula.

Right. I didn't look at your formula. That's because
there is no formula that does what you want to it to do.

None. *Not* "We haven't found it yet, we need someone even
smarter than Kurt Goedel. Maybe Peter Olcott?" *None*

Would you like to look at my previously undiscovered
integer between 10^100 and 10^100+1 ? I assure you, I'm a
really smart guy.

[Peter Percival]

peterolcott wrote:

>> On 2/8/2017 1:42 PM, peterolcott wrote:

>>> That depends upon whether Cantor is correct that infinities have
>>> differing sizes. His diagonalization of integers over reals seems
>>> convincing.

> I really don't want to talk about infinities very much. The set of
> integers is infinite, the set of reals is infinite and may or may not
> be larger than the set of integers.

May or may not?

> If we eliminate infinities with algorithmic compression we can still

All compression algorithms have this property: there is some data that
they expand!

> have a Universal set of all unique finite things. The Universal set
> would then include the entire concept of set theory, but, no actual
> sets. Actually now that you point this out, that is what I have had
> in mind all along.

There is an existing theory of finite sets (look up hereditarily finite
sets).

[peterolcott]

On Wednesday, February 8, 2017 at 10:42:23 AM UTC-6, Jim Burns wrote:

https://plato.stanford.edu/entries/logic-higher-order/#3
The concept of general semantics for second-order logic avoids any pretense that the power-set operation is a fixed well-understood resource. Instead, the range of the quantifier ∀X must be directly specified.

I am not sure whether or not the above second-order logic convention permits the universal set or not.

If we define the Universal set (Universal class) as the set of elements that physically exist and the set of unique concepts such that the concept of a set of sets exists, yet none of these sets are actually populated with members, then we have a coherent and possibly finite U.

I was once able to figure out the number of pages in a book that would contain every page of every book that could ever possibly be written:
Number-of-Characters-in-Character-set ^ Maximum-Number-of-Characters-Per-Page When I was using ASCII and a PC DOS printed page this was 95 ^ 4800

The same idea applies to high resolution photographs and paired (for 3D effect) sequences of these high resolution photos. When we assume some fixed degree of DPI, we realize that this would produce a very high resolution 3D video of assassin John Kennedy killing president Lee Harvey Oswald. This video would be from every possible point of view and zoom level.

A very high resolution 3D video showing every detail of everything that ever happened or ever will happen within an infinite set of parallel universes could be produced in finite space. We could even add HD stereo sound to this and still have only require finite space.

[Peter Percival]

peterolcott wrote:

> I am not sure whether or not the above second-order logic convention permits the universal set or not.

There are set theories with a universal set. I know nothing about them
but lookout for T.E. Forster and Randal Holmes for starters.

[Peter Percival]

Peter Percival wrote:
> peterolcott wrote:
>
>> I am not sure whether or not the above second-order logic convention
>> permits the universal set or not.
>
> There are set theories with a universal set. I know nothing about them
> but lookout for T.E. Forster and Randal Holmes for starters.

Randall with two ells. Sorry.

[Peter Percival]

peterolcott wrote:

> A very high resolution 3D video showing every detail of everything that ever happened or ever will happen within an infinite set of parallel universes could be produced in finite space.

But that finite space would have to include that very video...

> We could even add HD stereo sound to this and still have only require finite space.
>

[peterolcott]

On Friday, February 10, 2017 at 10:05:49 AM UTC-6, Peter Percival wrote:
> peterolcott wrote:
>
> > A very high resolution 3D video showing every detail of everything that ever happened or ever will happen within an infinite set of parallel universes could be produced in finite space.
>
> But that finite space would have to include that very video...
>
> > We could even add HD stereo sound to this and still have only require finite space.
> >
>

If as Einstein suggested in his book for laymen, the universe is finite yet unbounded, we have room for as many (finite) copies of this video as we want to keep. My purpose in coming up with this thought experiment decades ago was to show that omniscience is finite. I read the book by Einstein when I was 17.