Simply defining Gödel Incompleteness and Tarski Undefinability away V35 (Semantically Incorrect Defined)

[olcott]

In classical logic a sentence in a language is true or false under (and
only under) an interpretation and is therefore a truth-bearer.

https://en.wikipedia.org/wiki/Truth-bearer#Sentences_in_languages_of_classical_logic


A semantically incorrect logic sentence φ will be defined as a closed
WFF having no interpretation that satisfies either φ or ¬φ.

Copyright 2020 Pete Olcott

These sources are used for the meaning of the term "interpretation"

Satisfiability
A formula is satisfiable if it is possible to find an interpretation
(model) that makes the formula true.
https://en.wikipedia.org/wiki/Satisfiability

Interpretation (logic)
An interpretation is an assignment of meaning to the
[non-logical] symbols of a formal language.
https://en.wikipedia.org/wiki/Interpretation_(logic)

https://mathworld.wolfram.com/Interpretation.html

http://www.princeton.edu/~hhalvors/teaching/phi201_s2004/handouts/semantics

http://www.phil.cmu.edu/projects/logicandproofs/alpha/htmltest/m11_pred_formal_sem/translated_chapter11.html


https://en.wikipedia.org/wiki/Interpretation_(logic)#Logical_constants


--
Copyright 2020 Pete Olcott

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> In classical logic a sentence in a language is true or false under
> (and only under) an interpretation and is therefore a truth-bearer.
>
> https://en.wikipedia.org/wiki/Truth-bearer#Sentences_in_languages_of_classical_logic
>
> A semantically incorrect logic sentence φ will be defined as a closed
> WFF having no interpretation that satisfies either φ or ¬φ.

Then there are none (see page 65 in Mendelson). You really should try
to learn the meaning of the words you use.

--
Ben.

[olcott]

The easiest one to see is the one that Tarski's proof assumed.

[Ben Bacarisse]

Saying those words does not alter the facts. See page 65 of Mendelson.

--
Ben.

[olcott]

You are locked into thinking inside the box of assuming that all of the
aspects of mathematics have been infallibly connected together.

[Ben Bacarisse]

You are stuck using words you don't understand. By definition, every
sentence of a theory is either true or false in every interpretation.
Whatever set of formulas you are clutching at (the same set you've tried
to categorise for the last 20 years I think), your newest borrowed words
don't define it.

Normally I'd be tempted to say "define a language and write out such a
φ". But you don't know how to define a language, and you would end up
writing a symbol salad that we'd start quarrelling over instead of
sticking to the big picture.

The big picture: every interpretation makes every sentence either true
or false, and it makes the negation of every sentence have the opposite
truth value. If you had made it past page 58 in Mendelson, you would
know this.

--
Ben.

[olcott]

On 8/3/2020 10:05 PM, Ben Bacarisse wrote:
> olcott <No...@NoWhere.com> writes:
>
>> On 8/3/2020 8:45 PM, Ben Bacarisse wrote:
>>> olcott <No...@NoWhere.com> writes:
>>>
>>>> On 8/3/2020 7:57 PM, Ben Bacarisse wrote:
>>>>> olcott <No...@NoWhere.com> writes:
>>>>>
>>>>>> In classical logic a sentence in a language is true or false under
>>>>>> (and only under) an interpretation and is therefore a truth-bearer.
>>>>>>
>>>>>> https://en.wikipedia.org/wiki/Truth-bearer#Sentences_in_languages_of_classical_logic
>>>>>>
>>>>>> A semantically incorrect logic sentence φ will be defined as a closed
>>>>>> WFF having no interpretation that satisfies either φ or ¬φ.
>>>>>
>>>>> Then there are none (see page 65 in Mendelson). You really should try
>>>>> to learn the meaning of the words you use.
>>>>
>>>> The easiest one to see is the one that Tarski's proof assumed.
>>>
>>> Saying those words does not alter the facts. See page 65 of Mendelson.
>>
>> You are locked into thinking inside the box of assuming that all of
>> the aspects of mathematics have been infallibly connected together.
>
> You are stuck using words you don't understand. By definition, every
> sentence of a theory is either true or false in every interpretation.

Do you understand that a self-contradictory sentence would not conform
to this definition?

> Whatever set of formulas you are clutching at (the same set you've tried
> to categorise for the last 20 years I think), your newest borrowed words
> don't define it.
>
> Normally I'd be tempted to say "define a language and write out such a
> φ". But you don't know how to define a language, and you would end up
> writing a symbol salad that we'd start quarrelling over instead of
> sticking to the big picture.
>
> The big picture: every interpretation makes every sentence either true
> or false, and it makes the negation of every sentence have the opposite
> truth value. If you had made it past page 58 in Mendelson, you would
> know this.
>


--

Copyright 2020 Pete Olcott

[André G. Isaak]

On 2020-08-03 22:29, olcott wrote:
> On 8/3/2020 10:05 PM, Ben Bacarisse wrote:
>> olcott <No...@NoWhere.com> writes:
>>
>>> On 8/3/2020 8:45 PM, Ben Bacarisse wrote:
>>>> olcott <No...@NoWhere.com> writes:
>>>>
>>>>> On 8/3/2020 7:57 PM, Ben Bacarisse wrote:
>>>>>> olcott <No...@NoWhere.com> writes:
>>>>>>
>>>>>>> In classical logic a sentence in a language is true or false under
>>>>>>> (and only under) an interpretation and is therefore a truth-bearer.
>>>>>>>
>>>>>>> https://en.wikipedia.org/wiki/Truth-bearer#Sentences_in_languages_of_classical_logic
>>>>>>>
>>>>>>>
>>>>>>> A semantically incorrect logic sentence φ will be defined as a
>>>>>>> closed
>>>>>>> WFF having no interpretation that satisfies either φ or ¬φ.
>>>>>>
>>>>>> Then there are none (see page 65 in Mendelson).  You really should
>>>>>> try
>>>>>> to learn the meaning of the words you use.
>>>>>
>>>>> The easiest one to see is the one that Tarski's proof assumed.
>>>>
>>>> Saying those words does not alter the facts.  See page 65 of Mendelson.
>>>
>>> You are locked into thinking inside the box of assuming that all of
>>> the aspects of mathematics have been infallibly connected together.
>>
>> You are stuck using words you don't understand.  By definition, every
>> sentence of a theory is either true or false in every interpretation.
>
> Do you understand that a self-contradictory sentence would not conform
> to this definition?

Do you understand that the sort of "self-contradictory" sentences you
keep talking about might occur in natural language, but they do not
occur in the sorts of formal systems under consideration?

André

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

[olcott]

You are incorrect.

[André G. Isaak]

No, I am not. If I were, then presumably you would be able to provide
an example of such a sentence, but you will not be able to. You have
attempted this many times in the past, but never successfully.

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> On 8/3/2020 10:05 PM, Ben Bacarisse wrote:
>> olcott <No...@NoWhere.com> writes:
>>
>>> On 8/3/2020 8:45 PM, Ben Bacarisse wrote:
>>>> olcott <No...@NoWhere.com> writes:
>>>>
>>>>> On 8/3/2020 7:57 PM, Ben Bacarisse wrote:
>>>>>> olcott <No...@NoWhere.com> writes:
>>>>>>
>>>>>>> In classical logic a sentence in a language is true or false under
>>>>>>> (and only under) an interpretation and is therefore a truth-bearer.
>>>>>>>
>>>>>>> https://en.wikipedia.org/wiki/Truth-bearer#Sentences_in_languages_of_classical_logic
>>>>>>>
>>>>>>> A semantically incorrect logic sentence φ will be defined as a closed
>>>>>>> WFF having no interpretation that satisfies either φ or ¬φ.
>>>>>>
>>>>>> Then there are none (see page 65 in Mendelson). You really should try
>>>>>> to learn the meaning of the words you use.
>>>>>
>>>>> The easiest one to see is the one that Tarski's proof assumed.
>>>>
>>>> Saying those words does not alter the facts. See page 65 of Mendelson.
>>>
>>> You are locked into thinking inside the box of assuming that all of
>>> the aspects of mathematics have been infallibly connected together.
>>
>> You are stuck using words you don't understand. By definition, every
>> sentence of a theory is either true or false in every interpretation.
>
> Do you understand that a self-contradictory sentence would not conform
> to this definition?

There are no sentences that meet your definition of "semantically
incorrect". You can deny this simple fact. You can ignore this simple
fact. You can pretend that you know better than people who know this
simple fact. But if you want to talk to anyone but yourself, you will
need to learn what the words you used mean.

You started, but you stopped when you realised it was going to be hard.

--
Ben.

[olcott]

(1) x ∉ Pr if and only if p
(2) x ∈ Tr if and only if p;

This sentence is self-contradictory:
(3) x ∉ Pr if and only if x ∈ Tr.

(4) either x ∉ Tr or ¬x ∉ Tr;
(5) if x ∈ Pr, then x ∈ Tr;
(6) if ¬x ∈ Pr, then ¬x ∈ Tr;
(7) x ∈ Tr
(8) x ∉ Pr.
(9) ¬x ∉ Pr.

http://www.liarparadox.org/Tarski_Proof_275_276.pdf
Tarski, Alfred 1983. “The concept of truth in formalized languages” in
Logic Semantics, Metamathematics. Indianapolis: Hacket Publishing
Company, 275-276.

[olcott]

There is no such thing as a self-conradictory sentence?

> You can deny this simple fact. You can ignore this simple
> fact. You can pretend that you know better than people who know this
> simple fact. But if you want to talk to anyone but yourself, you will
> need to learn what the words you used mean.
>
> You started, but you stopped when you realised it was going to be hard.
>


--

Copyright 2020 Pete Olcott

[André G. Isaak]

There's nothing self-contradictory about that sentence.

It's true if x is an element of Tr and not of Pr.
It's true if x is an element of Pr and not of Tr.
It's false otherwise.

Nor is there anything self-contradictory about the other 8 sentences you
give.

André

> (4) either x ∉ Tr or ¬x ∉ Tr;
> (5) if x ∈ Pr, then x ∈ Tr;
> (6) if ¬x ∈ Pr, then ¬x ∈ Tr;
> (7) x ∈ Tr
> (8) x ∉ Pr.
> (9) ¬x ∉ Pr.
>
> http://www.liarparadox.org/Tarski_Proof_275_276.pdf
> Tarski, Alfred 1983. “The concept of truth in formalized languages” in
> Logic Semantics, Metamathematics. Indianapolis: Hacket Publishing
> Company, 275-276.
>
>


--

[olcott]

When you lookup and see what being an element of Tr and Pr means thenn
(then and only then) you can see that it is self-contradictory.

> It's true if x is an element of Pr and not of Tr.
> It's false otherwise.
>
> Nor is there anything self-contradictory about the other 8 sentences you
> give.
>
> André
>
>> (4) either x ∉ Tr or ¬x ∉ Tr;
>> (5) if x ∈ Pr, then x ∈ Tr;
>> (6) if ¬x ∈ Pr, then ¬x ∈ Tr;
>> (7) x ∈ Tr
>> (8) x ∉ Pr.
>> (9) ¬x ∉ Pr.
>>
>> http://www.liarparadox.org/Tarski_Proof_275_276.pdf
>> Tarski, Alfred 1983. “The concept of truth in formalized languages” in
>> Logic Semantics, Metamathematics. Indianapolis: Hacket Publishing
>> Company, 275-276.
>>
>>
>
>


--

Copyright 2020 Pete Olcott

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> On 8/4/2020 9:47 AM, Ben Bacarisse wrote:

<cut>

>> There are no sentences that meet your definition of "semantically
>> incorrect".
>
> There is no such thing as a self-conradictory sentence?

There are no sentences that meet your definition of "semantically

incorrect" unless you've changed it in the last few hours:

"A semantically incorrect logic sentence φ will be defined as a closed
WFF having no interpretation that satisfies either φ or ¬φ."

Whatever it is you are trying to say, you need to say it using words you
understand.

--
Ben.

[olcott]

Do you understand what a self-contradictory sentence is? (Yes or No)

[Ben Bacarisse]

Yes. The first two quoted comments show that you are using the term as
a synonym for your earlier phrase "semantically incorrect" which has,
for you, a remarkably exact technical meaning.

Do you understand what a sentence φ having no interpretation that
satisfies either φ or ¬φ is? (yes or no)

--
Ben.

[olcott]

On 8/4/2020 7:40 PM, Ben Bacarisse wrote:
> olcott <No...@NoWhere.com> writes:
>
>> On 8/4/2020 2:44 PM, Ben Bacarisse wrote:
>>> olcott <No...@NoWhere.com> writes:
>>>
>>>> On 8/4/2020 9:47 AM, Ben Bacarisse wrote:
>>> <cut>
>>>>> There are no sentences that meet your definition of "semantically
>>>>> incorrect".
>>>>
>>>> There is no such thing as a self-conradictory sentence?
>>>
>>> There are no sentences that meet your definition of "semantically
>>> incorrect" unless you've changed it in the last few hours:
>>>
>>> "A semantically incorrect logic sentence φ will be defined as a closed
>>> WFF having no interpretation that satisfies either φ or ¬φ."
>>>
>>> Whatever it is you are trying to say, you need to say it using words you
>>> understand.
>>
>> Do you understand what a self-contradictory sentence is? (Yes or No)
>
> Yes.

Great !

> The first two quoted comments show that you are using the term as
> a synonym for your earlier phrase "semantically incorrect" which has,
> for you, a remarkably exact technical meaning.
>
> Do you understand what a sentence φ having no interpretation that
> satisfies either φ or ¬φ is? (yes or no)
>

Every formalized self-contradictory sentence would be one class of
sentences that match that criteria.

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> On 8/4/2020 7:40 PM, Ben Bacarisse wrote:
>> olcott <No...@NoWhere.com> writes:
>>
>>> On 8/4/2020 2:44 PM, Ben Bacarisse wrote:
>>>> olcott <No...@NoWhere.com> writes:
>>>>
>>>>> On 8/4/2020 9:47 AM, Ben Bacarisse wrote:
>>>> <cut>
>>>>>> There are no sentences that meet your definition of "semantically
>>>>>> incorrect".
>>>>>
>>>>> There is no such thing as a self-conradictory sentence?
>>>>
>>>> There are no sentences that meet your definition of "semantically
>>>> incorrect" unless you've changed it in the last few hours:
>>>>
>>>> "A semantically incorrect logic sentence φ will be defined as a closed
>>>> WFF having no interpretation that satisfies either φ or ¬φ."
>>>>
>>>> Whatever it is you are trying to say, you need to say it using words you
>>>> understand.
>>>
>>> Do you understand what a self-contradictory sentence is? (Yes or No)
>>
>> Yes.
>
> Great !
>
>> The first two quoted comments show that you are using the term as
>> a synonym for your earlier phrase "semantically incorrect" which has,
>> for you, a remarkably exact technical meaning.
>>
>> Do you understand what a sentence φ having no interpretation that
>> satisfies either φ or ¬φ is? (yes or no)

You did not give the requested yes/no answer to my question. Is that
something only you are allowed to ask for?

> Every formalized self-contradictory sentence would be one class of
> sentences that match that criteria.

Yes, I know you are using the terms as near synonyms. All that means is
there is another name for these sentences that don't exist.

--
Ben.

[André G. Isaak]

I'm aware of what Tarski's Tr and Pr are. There is still nothing
self-contradictory about the above.

Even if we were to accept your perverse idea that provability and truth
are equivalent notions (they aren't), the above would STILL not be a
self-contradictory sentence. In that case it would simply be false.

André

[olcott]

Instead of answering Yes or No I provided an example.

>
>> Every formalized self-contradictory sentence would be one class of
>> sentences that match that criteria.
>
> Yes, I know you are using the terms as near synonyms. All that means is
> there is another name for these sentences that don't exist.
>

So you are saying that formalized self-contradictory sentences do not
exist? Tarski's whole proof is based on one.

[olcott]

Provability (when done correctly) is merely the valid inference aspect
of sound deduction.

When this is understood then Tarski's proof merely concludes that untrue
sentences cannot be proven to be true.

[André G. Isaak]

And there are propositions where 'you can't get there from here' using
valid inference. That's what incompleteness is all about.

> When this is understood then Tarski's proof merely concludes that untrue
> sentences cannot be proven to be true.

Even if this were true, what does it have to do with
'self-contradictory' sentences?

[olcott]

Tarski "proved" that not all true propositions can be defined on the
basis of failing to prove that a self-contradictory sentence is true.
He failed to prove that an untrue sentence is true.

[André G. Isaak]

On 2020-08-04 23:59, olcott wrote:

> Tarski "proved" that not all true propositions can be defined on the
> basis of failing to prove that a self-contradictory sentence is true.
> He failed to prove that an untrue sentence is true.

And which 'self-contradictory' sentence was that?

If he failed to prove that this sentence was true, then in what sense
would it be self-contradictory as opposed to simply false?

[Ben Bacarisse]

OK. I won't comply either if you are ever so rude as to demand a yes/no
answer again.

>>> Every formalized self-contradictory sentence would be one class of
>>> sentences that match that criteria.
>>
>> Yes, I know you are using the terms as near synonyms. All that means is
>> there is another name for these sentences that don't exist.
>
> So you are saying that formalized self-contradictory sentences do not
> exist? Tarski's whole proof is based on one.

I am saying that your most recent definition is worse than all the
previous ones because other people know what the words you used mean,
even if you don't. Other people will know that there are no
"semantically incorrect" sentences as you have specified them.

For a while you seemed to want to know what the words you used mean.
Why did you change your mind?

If you stop using "self-contradictory sentence" as a synonym for this
new and unfortunately unambiguous definition, and go back using it as a
vague ill-defined term, you will be able to carry on waffling about this
topic without fear of being understood.

--
Ben.

[olcott]

On 8/5/2020 1:44 AM, André G. Isaak wrote:
> On 2020-08-04 23:59, olcott wrote:
>
>> Tarski "proved" that not all true propositions can be defined on the
>> basis of failing to prove that a self-contradictory sentence is true.
>> He failed to prove that an untrue sentence is true.
>
> And which 'self-contradictory' sentence was that?
>

It would then be possible to reconstruct the antinomy of the liar in the
metalanguage, by forming in the language itself a sentence x such that
the sentence of the metalanguage which is correlated with x asserts that
x is not a true sentence. http://www.liarparadox.org/247_248.pdf

Everywhere, both in the formulation of the theorem and in its proof, we
replace the symbol 'Tr' by the symbol 'Pr' which denotes the class of
all provable sentences of the theory
http://www.liarparadox.org/Tarski_275_276.pdf

transforming this: x ∉ Tr if and only if x ∈ Tr.
into this: (3) x ∉ Pr if and only if x ∈ Tr.

> If he failed to prove that this sentence was true, then in what sense
> would it be self-contradictory as opposed to simply false?
>
> André
>

All logical propositions have distinctly different separate parts that
make the evaluation of their truth conditions impossible when these
parts are conflated together.

(a) Proposition assertion
(b) Satisfaction of Proposition assertion
(c) Satisfaction of Proposition

"This sentence is not true" is neither true nor false.

(a) S.Assertion = ~True(S)
(b) Satisfied(S.Assertion) == TRUE
(c) Satisfied(S) == FALSE

The fact that its assertion is satisfied makes it not false, the
sentence is indeed not true.

The truth of its assertion cannot be applied to the whole sentence
because this forms self-contradiction.

https://www.researchgate.net/publication/307442489_Formalizing_the_logical_self-reference_error_of_the_Liar_Paradox

[olcott]

All logical propositions have distinctly different separate parts that
make the evaluation of their truth conditions impossible when these
parts are conflated together.

(a) Proposition assertion
(b) Satisfaction of Proposition assertion
(c) Satisfaction of Proposition

"This sentence is not true" is neither true nor false.

(a) S.Assertion = ~True(S)
(b) Satisfied(S.Assertion) == TRUE
(c) Satisfied(S) == FALSE

The fact that its assertion is satisfied makes it not false, the
sentence is indeed not true.

The truth of its assertion cannot be applied to the whole sentence
because this forms self-contradiction.

> For a while you seemed to want to know what the words you used mean.
> Why did you change your mind?
>

I am still proceeding on this separately. This diverges from Mendelson's
view quite a bit. A predicate symbol maps to a Boolean value and not to
a relation.

An interpretation of first-order logic consists of a non-empty domain D
and mappings for function and predicate symbols. Every n-place function
symbol is mapped to a function from Dⁿ to D, and every n-place predicate
symbol is mapped to a function from Dⁿ to the set comprised of two
values true and false. https://mathworld.wolfram.com/Interpretation.html

> If you stop using "self-contradictory sentence" as a synonym for this
> new and unfortunately unambiguous definition, and go back using it as a
> vague ill-defined term, you will be able to carry on waffling about this
> topic without fear of being understood.
>

I think that I have used a reasonable approximation of conventional
terminology to explain additional details of the notion of a
semantically incorrect sentence.

It is very difficult to use connventional terminology correctly when
established standards for the meaning of these terms diverge from one
another.

[Ben Bacarisse]

This is a significant improvement. It's unlikely that anyone will
understand this. They might even think you are talking about logical
propositions!

> (a) Proposition assertion
> (b) Satisfaction of Proposition assertion
> (c) Satisfaction of Proposition
>
> "This sentence is not true" is neither true nor false.
>
> (a) S.Assertion = ~True(S)
> (b) Satisfied(S.Assertion) == TRUE
> (c) Satisfied(S) == FALSE
>
> The fact that its assertion is satisfied makes it not false, the
> sentence is indeed not true.

Good. I am almost certain that no one will know what you ae talking
about. I'm certain they won't think you mean sentences in a formal
first-order theory so you are home and dry. You are "not even wrong".

It would have been a better resolution if you had acknowledged the error
in your original post but now that you've moved into waffle territory I
doubt you'll want to go back to using well-defined terms.

> The truth of its assertion cannot be applied to the whole sentence
> because this forms self-contradiction.
>
>> For a while you seemed to want to know what the words you used mean.
>> Why did you change your mind?
>
> I am still proceeding on this separately. This diverges from
> Mendelson's view quite a bit. A predicate symbol maps to a Boolean
> value and not to a relation.
>
> An interpretation of first-order logic consists of a non-empty domain
> D and mappings for function and predicate symbols. Every n-place
> function symbol is mapped to a function from Dⁿ to D, and every
> n-place predicate symbol is mapped to a function from Dⁿ to the set
> comprised of two values true and false.
> https://mathworld.wolfram.com/Interpretation.html

Good luck. You need to be a strong student to juggle more than one way
of doing it, especially as mathworld won't explain any details.

>> If you stop using "self-contradictory sentence" as a synonym for this
>> new and unfortunately unambiguous definition, and go back using it as a
>> vague ill-defined term, you will be able to carry on waffling about this
>> topic without fear of being understood.
>
> I think that I have used a reasonable approximation of conventional
> terminology to explain additional details of the notion of a
> semantically incorrect sentence.

No more detail is needed: there are none (as you originally defined
them). Now that you are going on about propositions and impossible
truth conditions and using made-up math-a-like notations you could be
defining almost anything.

> It is very difficult to use connventional terminology correctly when
> established standards for the meaning of these terms diverge from one
> another.

Yes, I can imagine. I'd counsel against doing so. But most
mathematicians will pick up on another expert's way of doing this in a
few moments.

--
Ben.

[olcott]

Mendelson never seem to get to TRUE ans TRUE is the most important thing
that I need so Mendelson is no good.

>>> If you stop using "self-contradictory sentence" as a synonym for this
>>> new and unfortunately unambiguous definition, and go back using it as a
>>> vague ill-defined term, you will be able to carry on waffling about this
>>> topic without fear of being understood.
>>
>> I think that I have used a reasonable approximation of conventional
>> terminology to explain additional details of the notion of a
>> semantically incorrect sentence.
>
> No more detail is needed: there are none (as you originally defined
> them). Now that you are going on about propositions and impossible
> truth conditions and using made-up math-a-like notations you could be
> defining almost anything.
>
>> It is very difficult to use connventional terminology correctly when
>> established standards for the meaning of these terms diverge from one
>> another.
>
> Yes, I can imagine. I'd counsel against doing so. But most
> mathematicians will pick up on another expert's way of doing this in a
> few moments.
>

On the basis of Mendelson it would seem that mathemticians may have a
religious conviction against Boolean. What I really need to know is the
formal process for deciding sentence X is semantically true. His
"interpretation" never gets there.

When I ask is Bill the father of Sam? the Mendelson "interpretation"
responds with a very long list of ordered pairs.

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> On 8/5/2020 3:32 PM, Ben Bacarisse wrote:
>> olcott <No...@NoWhere.com> writes:

<cut>

>>> An interpretation of first-order logic consists of a non-empty domain
>>> D and mappings for function and predicate symbols. Every n-place
>>> function symbol is mapped to a function from Dⁿ to D, and every
>>> n-place predicate symbol is mapped to a function from Dⁿ to the set
>>> comprised of two values true and false.
>>> https://mathworld.wolfram.com/Interpretation.html
>>
>> Good luck. You need to be a strong student to juggle more than one way
>> of doing it, especially as mathworld won't explain any details.
>
> Mendelson never seem to get to TRUE ans TRUE is the most important
> thing that I need so Mendelson is no good.

Don't be silly. Ironically it was you that did that! You tried
exercise 2.10 where Mendelson specifically asks for cases that are true
or false but some of your answers never made that final step.

You won't be able to learn this stuff from MathWorld definitions,
and I think you are looking for reasons to ditch Mendelson so I doubt
you will ever know what these words mean.

> On the basis of Mendelson it would seem that mathemticians may have a
> religious conviction against Boolean. What I really need to know is
> the formal process for deciding sentence X is semantically true. His
> "interpretation" never gets there.

That's almost funny. I wonder what mental process lets you do an
exercise in a book that asks you to decide if some sentences are true or
false and conclude that the method in the book "never gets there".

And I suppose you never read page 61 where there is a list of properties
of true and false sentences.

--
Ben.

[olcott]

On 8/5/2020 5:48 PM, Ben Bacarisse wrote:
> olcott <No...@NoWhere.com> writes:
>
>> On 8/5/2020 3:32 PM, Ben Bacarisse wrote:
>>> olcott <No...@NoWhere.com> writes:
> <cut>
>>>> An interpretation of first-order logic consists of a non-empty domain
>>>> D and mappings for function and predicate symbols. Every n-place
>>>> function symbol is mapped to a function from Dⁿ to D, and every
>>>> n-place predicate symbol is mapped to a function from Dⁿ to the set
>>>> comprised of two values true and false.
>>>> https://mathworld.wolfram.com/Interpretation.html
>>>
>>> Good luck. You need to be a strong student to juggle more than one way
>>> of doing it, especially as mathworld won't explain any details.
>>
>> Mendelson never seem to get to TRUE ans TRUE is the most important
>> thing that I need so Mendelson is no good.
>
> Don't be silly. Ironically it was you that did that! You tried
> exercise 2.10 where Mendelson specifically asks for cases that are true
> or false but some of your answers never made that final step.
>
> You won't be able to learn this stuff from MathWorld definitions,
> and I think you are looking for reasons to ditch Mendelson so I doubt
> you will ever know what these words mean.
>

As I said many times I must know what these terms mean in mathematics.
If there is a range of a divergence of opinion of what the
interpretation of a predicate is whether it be a set of ordered pairs or
a Boolean value then I must learn these discrepancies.

>> On the basis of Mendelson it would seem that mathemticians may have a
>> religious conviction against Boolean. What I really need to know is
>> the formal process for deciding sentence X is semantically true. His
>> "interpretation" never gets there.
>
> That's almost funny. I wonder what mental process lets you do an
> exercise in a book that asks you to decide if some sentences are true or
> false and conclude that the method in the book "never gets there".
>

There is no way to get there within his definition of an interpretation
of a first order language. None of (a)(b)(c)(d) ever get to Boolean.

> And I suppose you never read page 61 where there is a list of properties
> of true and false sentences.
>

Definitions
1. A wf B is true for the interpretation M (written ⊨M B) if and only if
every sequence in Σ satisfies B.

We still don't know what Σ is.
Let Σ be the set of all denumerable sequences of elements of D.

If we take the precise compositional meaning of these words
denumerable + sequences + elements

Then we get the set of all tuples of D, yet this seems a little crazy
because a 5-tuple cannot possibly satisfy a 6-tuple relation.

When you bake a cake the ingredients must be food items. If you consider
5-tuples for a 6-tuple relation that is like considering a box of rocks
as a possible ingredient to the cake.

-- Definitions
-- 1. A wf B is true for the interpretation M (written ⊨M B)
-- if and only if every sequence in Σ satisfies B.

So if Let Σ is the set of all denumerable sequences of elements of D
Then unless every sequence of humans is not the father of Bill then Bill
has no father.

If all three of {Mary, Jane, Betty} are not the father of Bill then
because not every denumerable sequences of elements of humans satisfies
the father of relation then the father of relation is not true and Bill
has no father.

[André G. Isaak]

On 2020-08-05 12:04, olcott wrote:
> On 8/5/2020 1:44 AM, André G. Isaak wrote:
>> On 2020-08-04 23:59, olcott wrote:
>>
>>> Tarski "proved" that not all true propositions can be defined on the
>>> basis of failing to prove that a self-contradictory sentence is true.
>>> He failed to prove that an untrue sentence is true.
>>
>> And which 'self-contradictory' sentence was that?
>>
>
> It would then be possible to reconstruct the antinomy of the liar in the
> metalanguage, by forming in the language itself a sentence x such that
> the sentence of the metalanguage which is correlated with x asserts that
> x is not a true sentence.  http://www.liarparadox.org/247_248.pdf
>
> Everywhere, both in the formulation of the theorem and in its proof, we
> replace the symbol 'Tr' by the symbol 'Pr' which denotes the class of
> all provable sentences of the theory
> http://www.liarparadox.org/Tarski_275_276.pdf
>
> transforming this: x ∉ Tr if and only if x ∈ Tr.
> into this: (3) x ∉ Pr if and only if x ∈ Tr.

You're failing to indicate where the alleged contradiction occurs in
(3). You can call it 'self-contradictory' to your hearts content, but
unless you can actually demonstrate that there is something
contradictory about the above you're not making your case.

>> If he failed to prove that this sentence was true, then in what sense
>> would it be self-contradictory as opposed to simply false?
>>
>> André
>>
>
> All logical propositions have distinctly different separate parts that
> make the evaluation of their truth conditions impossible when these
> parts are conflated together.
>
> (a) Proposition assertion
> (b) Satisfaction of Proposition assertion
> (c) Satisfaction of Proposition

AFAICT, the above is simply gibberish.

André

> "This sentence is not true" is neither true nor false.
>
> (a) S.Assertion = ~True(S)
> (b) Satisfied(S.Assertion) == TRUE
> (c) Satisfied(S) == FALSE
>
> The fact that its assertion is satisfied makes it not false, the
> sentence is indeed not true.
>
> The truth of its assertion cannot be applied to the whole sentence
> because this forms self-contradiction.
>
> https://www.researchgate.net/publication/307442489_Formalizing_the_logical_self-reference_error_of_the_Liar_Paradox
>
>
>


--

[olcott]

You have decided that you want to pretend to not understand.
I have zero patience for that.

> André
>
>> "This sentence is not true" is neither true nor false.
>>
>> (a) S.Assertion = ~True(S)
>> (b) Satisfied(S.Assertion) == TRUE
>> (c) Satisfied(S) == FALSE
>>
>> The fact that its assertion is satisfied makes it not false, the
>> sentence is indeed not true.
>>
>> The truth of its assertion cannot be applied to the whole sentence
>> because this forms self-contradiction.
>>
>> https://www.researchgate.net/publication/307442489_Formalizing_the_logical_self-reference_error_of_the_Liar_Paradox
>>
>>
>>
>
>


--

Copyright 2020 Pete Olcott

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> On 8/5/2020 5:48 PM, Ben Bacarisse wrote:
>> olcott <No...@NoWhere.com> writes:
>>
>>> On 8/5/2020 3:32 PM, Ben Bacarisse wrote:
>>>> olcott <No...@NoWhere.com> writes:
>> <cut>
>>>>> An interpretation of first-order logic consists of a non-empty domain
>>>>> D and mappings for function and predicate symbols. Every n-place
>>>>> function symbol is mapped to a function from Dⁿ to D, and every
>>>>> n-place predicate symbol is mapped to a function from Dⁿ to the set
>>>>> comprised of two values true and false.
>>>>> https://mathworld.wolfram.com/Interpretation.html
>>>>
>>>> Good luck. You need to be a strong student to juggle more than one way
>>>> of doing it, especially as mathworld won't explain any details.
>>>
>>> Mendelson never seem to get to TRUE ans TRUE is the most important
>>> thing that I need so Mendelson is no good.
>>
>> Don't be silly. Ironically it was you that did that! You tried
>> exercise 2.10 where Mendelson specifically asks for cases that are true
>> or false but some of your answers never made that final step.
>>
>> You won't be able to learn this stuff from MathWorld definitions,
>> and I think you are looking for reasons to ditch Mendelson so I doubt
>> you will ever know what these words mean.
>>
>
> As I said many times I must know what these terms mean in mathematics.

No you don't. You would not have started this thread by carefully
defining, as if it were important, a set of no sentences if you know
what the words you used actually meant.

>>> On the basis of Mendelson it would seem that mathemticians may have a
>>> religious conviction against Boolean. What I really need to know is
>>> the formal process for deciding sentence X is semantically true. His
>>> "interpretation" never gets there.
>>
>> That's almost funny. I wonder what mental process lets you do an
>> exercise in a book that asks you to decide if some sentences are true or
>> false and conclude that the method in the book "never gets there".
>
> There is no way to get there within his definition of an
> interpretation of a first order language. None of (a)(b)(c)(d) ever
> get to Boolean.

Interpretations (a), (b) and (c) (there is no (d)) give the sentence
(iii) a truth value. If that's not "Boolean" then no one cares what you
mean by it. What he asks for is whether the sentence is true or not
under the various interpretations. It's as simple as that.

>> And I suppose you never read page 61 where there is a list of properties
>> of true and false sentences.
>>
>
> Definitions 1. A wf B is true for the interpretation M (written ⊨M B)
> if and only if every sequence in Σ satisfies B.
>
> We still don't know what Σ is.

You mean you don't. I do. Maths students who read this text do.

> Let Σ be the set of all denumerable sequences of elements of D.
>
> If we take the precise compositional meaning of these words
> denumerable + sequences + elements
>
> Then we get the set of all tuples of D,

No "we" don't. You see? Even the simplest thing about Mendelson's
presentation confuses you. How can you have the nerve to suggest that
"he doesn't get to true/false" when you have no idea what he's saying?

> yet this seems a little crazy because a 5-tuple cannot possibly
> satisfy a 6-tuple relation.

How long have you been trying to understand what an interpretation is?

> When you bake a cake the ingredients must be food items. If you
> consider 5-tuples for a 6-tuple relation that is like considering a
> box of rocks as a possible ingredient to the cake.
>
> -- Definitions
> -- 1. A wf B is true for the interpretation M (written ⊨M B)
> -- if and only if every sequence in Σ satisfies B.

Give an actual B and M and I'll explain this.

> So if Let Σ is the set of all denumerable sequences of elements of D
> Then unless every sequence of humans is not the father of Bill then
> Bill has no father.

Oh dear. So far from anything Mendelson is saying that you need to go
back and start again. The first step would be to stop waffling about
fathers and start writing formulas. Interpretations with domains apply
to formulas in formal languages. You have never once written a valid
formula in a formal language that you have specified.

But the underlying problem is that can't learn because you can't
conceive of being totally wrong or of completely misunderstanding
something. If I had concluded that a book on logic, in a section about
truth, had failed to say how sentences can be true or false, I would not
think the author is obviously wrong. I'd think, how can this be?
Surely I have failed to understood how interpretations lead to the truth
values for sentences. After all he asks me to find out if a sentence is
true or false in the very first exercise. And this book got to be in
its sixth edition.

--
Ben.

[André G. Isaak]

I am not pretending. How am I supposed to understand when you haven't
defined what 'proposition assertion' means, nor what 'satisfaction of
proposition assertion' means, nor what 'satisfaction of proposition'
means? These are your terms, not standard ones, and I am not telepathic.

You also have failed to indicate how these relate to the proposition in (3).

André

[olcott]

YOU GOT THIS TOTALLY INCORRECTLY HERE IS THE QUOTE.

Definition
Let L be a first-order language. An interpretation M of L consists of
the following
ingredients.
a. A nonempty set D, called the domain of the interpretation.
b. For each predicate letter Aj
n of L, an assignment of an n-place relation
(Aj )
n M in D.
c. For each function letter fj
n of L, an assignment of an n-place operation
( fj )
n M in D (that is, a function from Dn into D).
d. For each individual constant ai of L, an assignment of some fixed element
(ai)M of D.

WHERE DOES IT SAY ANYTHING ABOUT TRUE AND FALSE IN THERE?
OTHER AUTHORS SIMPLY HAVE B MAP TO BOOLEAN VALUES.

>
>>> And I suppose you never read page 61 where there is a list of properties
>>> of true and false sentences.
>>>
>>
>> Definitions 1. A wf B is true for the interpretation M (written ⊨M B)
>> if and only if every sequence in Σ satisfies B.
>>
>> We still don't know what Σ is.
>
> You mean you don't. I do. Maths students who read this text do.
>
>> Let Σ be the set of all denumerable sequences of elements of D.
>>
>> If we take the precise compositional meaning of these words
>> denumerable + sequences + elements
>>
>> Then we get the set of all tuples of D,
>
> No "we" don't. You see? Even the simplest thing about Mendelson's
> presentation confuses you. How can you have the nerve to suggest that
> "he doesn't get to true/false" when you have no idea what he's saying?
>

Go back and see my quote that you got wrong.
The compositional meaning of denumerable + sequences + elements

(a) https://en.wikipedia.org/wiki/Countable_set
(b) https://www.lexico.com/en/definition/sequence
(c) The objects used to form a set are called its element or its members.

>> yet this seems a little crazy because a 5-tuple cannot possibly
>> satisfy a 6-tuple relation.
>
> How long have you been trying to understand what an interpretation is?

It is really hard to understand these things when they diverge from
their common meaning.

>> When you bake a cake the ingredients must be food items. If you
>> consider 5-tuples for a 6-tuple relation that is like considering a
>> box of rocks as a possible ingredient to the cake.
>>
>> -- Definitions
>> -- 1. A wf B is true for the interpretation M (written ⊨M B)
>> -- if and only if every sequence in Σ satisfies B.
>
> Give an actual B and M and I'll explain this.
>
>> So if Let Σ is the set of all denumerable sequences of elements of D
>> Then unless every sequence of humans is not the father of Bill then
>> Bill has no father.
>
> Oh dear. So far from anything Mendelson is saying that you need to go
> back and start again. The first step would be to stop waffling about
> fathers and start writing formulas. Interpretations with domains apply
> to formulas in formal languages. You have never once written a valid
> formula in a formal language that you have specified.

I need to have each term of the art fully defined using words that are
not terms of the art at all.

> But the underlying problem is that can't learn because you can't
> conceive of being totally wrong or of completely misunderstanding
> something. If I had concluded that a book on logic, in a section about
> truth, had failed to say how sentences can be true or false, I would not
> think the author is obviously wrong. I'd think, how can this be?
> Surely I have failed to understood how interpretations lead to the truth
> values for sentences. After all he asks me to find out if a sentence is
> true or false in the very first exercise. And this book got to be in
> its sixth edition.
>


--

Copyright 2020 Pete Olcott

[olcott]

YOU SKIPPED THE PART WHERE I EXPLAINED IT.

How am I supposed to understand when you haven't
> defined what 'proposition assertion' means, nor what 'satisfaction of
> proposition assertion' means, nor what 'satisfaction of proposition'
> means? These are your terms, not standard ones, and I am not telepathic.
>
> You also have failed to indicate how these relate to the proposition in
> (3).
>
> André
>


--

Copyright 2020 Pete Olcott

[olcott]

On 8/5/2020 8:57 PM, André G. Isaak wrote:
> On 2020-08-05 12:04, olcott wrote:
>> On 8/5/2020 1:44 AM, André G. Isaak wrote:
>>> On 2020-08-04 23:59, olcott wrote:
>>>
>>>> Tarski "proved" that not all true propositions can be defined on the
>>>> basis of failing to prove that a self-contradictory sentence is true.
>>>> He failed to prove that an untrue sentence is true.
>>>
>>> And which 'self-contradictory' sentence was that?
>>>
>>
>> It would then be possible to reconstruct the antinomy of the liar in
>> the metalanguage, by forming in the language itself a sentence x such
>> that the sentence of the metalanguage which is correlated with x
>> asserts that x is not a true sentence.
>> http://www.liarparadox.org/247_248.pdf
>>
>> Everywhere, both in the formulation of the theorem and in its proof,
>> we replace the symbol 'Tr' by the symbol 'Pr' which denotes the class
>> of all provable sentences of the theory
>> http://www.liarparadox.org/Tarski_275_276.pdf
>>
>> transforming this: x ∉ Tr if and only if x ∈ Tr.
>> into this: (3) x ∉ Pr if and only if x ∈ Tr.
>
> You're failing to indicate where the alleged contradiction occurs

IF YOU ARE GOING TO PRETEND TO NOT UNDERSTAND THAT THE LIAR PARADOX IS
SELF-CONTRADICTORY YOU SHOULD JUST S THE F UP !!!

Copyright 2020 Pete Olcott

[André G. Isaak]

I see no part where you explained it. If such a part exists, please
point it out.

[Ben Bacarisse]

You have the wrong quote. I won't explain because, currently, your mind
is closed to the idea of your being wrong.

> Definition
> Let L be a first-order language. An interpretation M of L consists of the following
> ingredients.
> a. A nonempty set D, called the domain of the interpretation.
> b. For each predicate letter Aj
> n of L, an assignment of an n-place relation
> (Aj )
> n M in D.
> c. For each function letter fj
> n of L, an assignment of an n-place operation
> ( fj )
> n M in D (that is, a function from Dn into D).
> d. For each individual constant ai of L, an assignment of some fixed element
> (ai)M of D.
>
> WHERE DOES IT SAY ANYTHING ABOUT TRUE AND FALSE IN THERE?
> OTHER AUTHORS SIMPLY HAVE B MAP TO BOOLEAN VALUES.

You have a choice to make. You can keep think you know better and
therefore keep making a fool of yourself, or you change your way of
thinking to that of a student. Search for what you might have got
wrong. Ask people who might know to help.

By the way, you have also misunderstood the "other" ways of doing it.
They don't map B (a formula) to Boolean values at this point either.
True and false come later, once satisfaction is explained.

>>>> And I suppose you never read page 61 where there is a list of properties
>>>> of true and false sentences.
>>>>
>>>
>>> Definitions 1. A wf B is true for the interpretation M (written ⊨M B)
>>> if and only if every sequence in Σ satisfies B.
>>>
>>> We still don't know what Σ is.
>>
>> You mean you don't. I do. Maths students who read this text do.
>>
>>> Let Σ be the set of all denumerable sequences of elements of D.
>>>
>>> If we take the precise compositional meaning of these words
>>> denumerable + sequences + elements
>>>
>>> Then we get the set of all tuples of D,
>>
>> No "we" don't. You see? Even the simplest thing about Mendelson's
>> presentation confuses you. How can you have the nerve to suggest that
>> "he doesn't get to true/false" when you have no idea what he's saying?
>
> Go back and see my quote that you got wrong.
> The compositional meaning of denumerable + sequences + elements
>
> (a) https://en.wikipedia.org/wiki/Countable_set
> (b) https://www.lexico.com/en/definition/sequence
> (c) The objects used to form a set are called its element or its
> members.

The set of denumerable sequences of D is not the set of all tuples of D.
I won't explain further because, currently, your mind is closed to the
idea of your being wrong.

>>> yet this seems a little crazy because a 5-tuple cannot possibly
>>> satisfy a 6-tuple relation.
>>
>> How long have you been trying to understand what an interpretation is?
>
> It is really hard to understand these things when they diverge from
> their common meaning.

Yes, I noticed that you struggle with that. I thought at first is was
laziness -- that you hoped to wing it using the common meaning rather
than having to the learn the jargon -- but maybe it goes deeper than
that. If so, mathematics will be hard for because it is full of
re-purposed terms: theory, proof, model, function, group, satisfy, real,
rational, complex... The list goes on and on.

>>> When you bake a cake the ingredients must be food items. If you
>>> consider 5-tuples for a 6-tuple relation that is like considering a
>>> box of rocks as a possible ingredient to the cake.
>>>
>>> -- Definitions
>>> -- 1. A wf B is true for the interpretation M (written ⊨M B)
>>> -- if and only if every sequence in Σ satisfies B.
>>
>> Give an actual B and M and I'll explain this.
>>
>>> So if Let Σ is the set of all denumerable sequences of elements of D
>>> Then unless every sequence of humans is not the father of Bill then
>>> Bill has no father.
>>
>> Oh dear. So far from anything Mendelson is saying that you need to go
>> back and start again. The first step would be to stop waffling about
>> fathers and start writing formulas. Interpretations with domains apply
>> to formulas in formal languages. You have never once written a valid
>> formula in a formal language that you have specified.
>
> I need to have each term of the art fully defined using words that are
> not terms of the art at all.

I don't know any textbook that will do that for you. But you haven't
even given Mendelson a chance. You should read it from the beginning.

--
Ben.

[olcott]

These diverge and they can't both be right:

every n-place predicate symbol is mapped to a function from Dⁿ to the
set comprised of two values true and false.
https://mathworld.wolfram.com/Interpretation.html

b. For each predicate letter Anj of L, an assignment of an n-place
relation (Anj)ᴹ in D.

>>>>> And I suppose you never read page 61 where there is a list of
properties
>>>>> of true and false sentences.
>>>>>
>>>>
>>>> Definitions 1. A wf B is true for the interpretation M (written ⊨M B)
>>>> if and only if every sequence in Σ satisfies B.
>>>>
>>>> We still don't know what Σ is.
>>>
>>> You mean you don't. I do. Maths students who read this text do.
>>>
>>>> Let Σ be the set of all denumerable sequences of elements of D.
>>>>
>>>> If we take the precise compositional meaning of these words
>>>> denumerable + sequences + elements
>>>>
>>>> Then we get the set of all tuples of D,
>>>
>>> No "we" don't. You see? Even the simplest thing about Mendelson's
>>> presentation confuses you. How can you have the nerve to suggest that
>>> "he doesn't get to true/false" when you have no idea what he's saying?
>>
>> Go back and see my quote that you got wrong.
>> The compositional meaning of denumerable + sequences + elements
>>
>> (a) https://en.wikipedia.org/wiki/Countable_set
>> (b) https://www.lexico.com/en/definition/sequence
>> (c) The objects used to form a set are called its element or its
>> members.
>
> The set of denumerable sequences of D is not the set of all tuples of D.
> I won't explain further because, currently, your mind is closed to the
> idea of your being wrong.
>

It utterly freaks me out that terms of the art make incompatible changes
to the meaning of common words and don't even explain that they changed
the meaning.

I will try that a little.

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> On 8/6/2020 5:48 AM, Ben Bacarisse wrote:
>> olcott <No...@NoWhere.com> writes:

<cut>

>>> WHERE DOES IT SAY ANYTHING ABOUT TRUE AND FALSE IN THERE?
>>> OTHER AUTHORS SIMPLY HAVE B MAP TO BOOLEAN VALUES.
>>
>> You have a choice to make. You can keep think you know better and
>> therefore keep making a fool of yourself, or you change your way of
>> thinking to that of a student. Search for what you might have got
>> wrong. Ask people who might know to help.
>>
>> By the way, you have also misunderstood the "other" ways of doing it.
>> They don't map B (a formula) to Boolean values at this point either.
>> True and false come later, once satisfaction is explained.
>
> These diverge and they can't both be right:

Of course they can. They are provably equivalent ways to arrive at the
same result. Why would you not immediately suspect your lack of
understanding?

>>>>> Then we get the set of all tuples of D,
>>>>
>>>> No "we" don't. You see? Even the simplest thing about Mendelson's
>>>> presentation confuses you. How can you have the nerve to suggest that
>>>> "he doesn't get to true/false" when you have no idea what he's saying?
>>>
>>> Go back and see my quote that you got wrong.
>>> The compositional meaning of denumerable + sequences + elements
>>>
>>> (a) https://en.wikipedia.org/wiki/Countable_set
>>> (b) https://www.lexico.com/en/definition/sequence
>>> (c) The objects used to form a set are called its element or its
>>> members.
>>
>> The set of denumerable sequences of D is not the set of all tuples of D.
>> I won't explain further because, currently, your mind is closed to the
>> idea of your being wrong.
>
> It utterly freaks me out that terms of the art make incompatible
> changes to the meaning of common words and don't even explain that
> they changed the meaning.

What meaning do you think has been changed? I am not aware of any
common or previous meaning for "the set of denumerable sequences of D".

And how can you have the nerve to say that the meaning is not explained?
You haven't read the book, have you?

>>> I need to have each term of the art fully defined using words that are
>>> not terms of the art at all.
>>
>> I don't know any textbook that will do that for you. But you haven't
>> even given Mendelson a chance. You should read it from the beginning.
>
> I will try that a little.

You will find out what a denumerable sequence is after only eight pages.
In fact, you can infer what you've misunderstood about it just by
reading the contents page.

--
Ben.

[olcott]

On 8/6/2020 2:04 PM, Ben Bacarisse wrote:
> olcott <No...@NoWhere.com> writes:
>
>> On 8/6/2020 5:48 AM, Ben Bacarisse wrote:
>>> olcott <No...@NoWhere.com> writes:
> <cut>
>>>> WHERE DOES IT SAY ANYTHING ABOUT TRUE AND FALSE IN THERE?
>>>> OTHER AUTHORS SIMPLY HAVE B MAP TO BOOLEAN VALUES.
>>>
>>> You have a choice to make. You can keep think you know better and
>>> therefore keep making a fool of yourself, or you change your way of
>>> thinking to that of a student. Search for what you might have got
>>> wrong. Ask people who might know to help.
>>>
>>> By the way, you have also misunderstood the "other" ways of doing it.
>>> They don't map B (a formula) to Boolean values at this point either.
>>> True and false come later, once satisfaction is explained.
>>
>> These diverge and they can't both be right:
>
> Of course they can. They are provably equivalent ways to arrive at the
> same result. Why would you not immediately suspect your lack of
> understanding?

I wish that you would not cut out these details it really
seems like a dishonest play to attempt to win an argument:

On 8/6/2020 10:47 AM, olcott wrote:
> every n-place predicate symbol is mapped to a function from Dⁿ to the
> set comprised of two values true and false.
> https://mathworld.wolfram.com/Interpretation.html
>
> b. For each predicate letter Anj of L, an assignment of an n-place
> relation (Anj)ᴹ in D.

One says that it maps to Boolean and the other says that it does not map
to Boolean it maps n-tuples.

>>>>>> Then we get the set of all tuples of D,
>>>>>
>>>>> No "we" don't. You see? Even the simplest thing about Mendelson's
>>>>> presentation confuses you. How can you have the nerve to suggest that
>>>>> "he doesn't get to true/false" when you have no idea what he's saying?
>>>>
>>>> Go back and see my quote that you got wrong.
>>>> The compositional meaning of denumerable + sequences + elements
>>>>
>>>> (a) https://en.wikipedia.org/wiki/Countable_set
>>>> (b) https://www.lexico.com/en/definition/sequence
>>>> (c) The objects used to form a set are called its element or its
>>>> members.
>>>
>>> The set of denumerable sequences of D is not the set of all tuples of D.
>>> I won't explain further because, currently, your mind is closed to the
>>> idea of your being wrong.
>>
>> It utterly freaks me out that terms of the art make incompatible
>> changes to the meaning of common words and don't even explain that
>> they changed the meaning.
>
> What meaning do you think has been changed? I am not aware of any
> common or previous meaning for "the set of denumerable sequences of D".

On 8/6/2020 5:48 AM, Ben Bacarisse wrote:
> The set of denumerable sequences of D is not the set of all tuples
> of D.

On 8/5/2020 8:09 PM, olcott wrote: // Mendelson quote

> Let Σ be the set of all denumerable sequences of elements of D.

Denumerable means countable.
A sequence is an ordered list.

The objects used to form a set are called its element or its members.

thus we have countable ordered lists of elements AKA tuples of elements.

https://en.wikipedia.org/wiki/Tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements.
I see no plausible way to conclude that you are not lying.

A {tuple} is a sequence of elements.
Mendelson refers to "denumerable sequences of elements of D."

Tuples are countable thus denumerable thus a tuple is a
"denumerable sequence of elements".

AND YOU LIED (or less probably) were mistaken.

On 8/6/2020 5:48 AM, Ben Bacarisse wrote:
> The set of denumerable sequences of D is not the set of all tuples
> of D.

>

> And how can you have the nerve to say that the meaning is not explained?
> You haven't read the book, have you?
>
>>>> I need to have each term of the art fully defined using words that are
>>>> not terms of the art at all.
>>>
>>> I don't know any textbook that will do that for you. But you haven't
>>> even given Mendelson a chance. You should read it from the beginning.
>>
>> I will try that a little.
>
> You will find out what a denumerable sequence is after only eight pages.
> In fact, you can infer what you've misunderstood about it just by
> reading the contents page.
>


--

Copyright 2020 Pete Olcott

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> On 8/6/2020 2:04 PM, Ben Bacarisse wrote:
>> olcott <No...@NoWhere.com> writes:
>>
>>> On 8/6/2020 5:48 AM, Ben Bacarisse wrote:
>>>> olcott <No...@NoWhere.com> writes:
>> <cut>
>>>>> WHERE DOES IT SAY ANYTHING ABOUT TRUE AND FALSE IN THERE?
>>>>> OTHER AUTHORS SIMPLY HAVE B MAP TO BOOLEAN VALUES.
>>>>
>>>> You have a choice to make. You can keep think you know better and
>>>> therefore keep making a fool of yourself, or you change your way of
>>>> thinking to that of a student. Search for what you might have got
>>>> wrong. Ask people who might know to help.
>>>>
>>>> By the way, you have also misunderstood the "other" ways of doing it.
>>>> They don't map B (a formula) to Boolean values at this point either.
>>>> True and false come later, once satisfaction is explained.
>>>
>>> These diverge and they can't both be right:
>>
>> Of course they can. They are provably equivalent ways to arrive at the
>> same result. Why would you not immediately suspect your lack of
>> understanding?
>
> I wish that you would not cut out these details it really
> seems like a dishonest play to attempt to win an argument:

The details are not important. I know they are different and I know
exactly how they are different and I know that both approaches give the
same results. You don't yet know that. In fact, up until a few moments
ago you thought Mendelson's approach did not give true/false answers for
sentences.

There are lots of ways of getting the same result but I don't think you
are up to following two different presentations of the same ideas right
now so you should pick one and stick with it.

Here's another hint to aid learning: I am one of the more honest people
you are likely to encounter. If you think I am being dishonest, it is
more likely you don't know what is going on.

> On 8/6/2020 10:47 AM, olcott wrote:
>> every n-place predicate symbol is mapped to a function from Dⁿ to the
>> set comprised of two values true and false.
>> https://mathworld.wolfram.com/Interpretation.html
>>
>> b. For each predicate letter Anj of L, an assignment of an n-place
>> relation (Anj)ᴹ in D.
>
> One says that it maps to Boolean and the other says that it does not
> map to Boolean it maps n-tuples.

You don't understand the details. To find out how both approaches give
exactly the same results, you would have to understand Mendelson.

>>>> The set of denumerable sequences of D is not the set of all tuples of D.
>>>> I won't explain further because, currently, your mind is closed to the
>>>> idea of your being wrong.
>>>
>>> It utterly freaks me out that terms of the art make incompatible
>>> changes to the meaning of common words and don't even explain that
>>> they changed the meaning.
>>
>> What meaning do you think has been changed? I am not aware of any
>> common or previous meaning for "the set of denumerable sequences of
>> D".
>
> On 8/6/2020 5:48 AM, Ben Bacarisse wrote:
>> The set of denumerable sequences of D is not the set of all tuples
>> of D.
>
> On 8/5/2020 8:09 PM, olcott wrote: // Mendelson quote
>> Let Σ be the set of all denumerable sequences of elements of D.
>
> Denumerable means countable.

Have you read the first eight pages of Mendelson? If not, why not? How
long could it take?

> A sequence is an ordered list.
> The objects used to form a set are called its element or its members.
> thus we have countable ordered lists of elements AKA tuples of elements.
>
> https://en.wikipedia.org/wiki/Tuple
> In mathematics, a tuple is a finite ordered list (sequence) of elements.
> I see no plausible way to conclude that you are not lying.

How about reading page eight of Mendelson?

But here's another tip: to learn, you need to switch off that reflex.
If you ever feel forced into thinking I am lying, you have either
misunderstood something or I have made a mistake. The best way to
resolve the matter would be to ask me. It's interesting that in all
these posts, you have not once asked me what it is that you have got
wrong and "the set of denumerable sequences of D".

> A {tuple} is a sequence of elements.
> Mendelson refers to "denumerable sequences of elements of D."

How about reading page eight of Mendelson?

> Tuples are countable thus denumerable thus a tuple is a
> "denumerable sequence of elements".
>
> AND YOU LIED (or less probably) were mistaken.

You have that backwards. I am often mistaken, but I don't lie.
Normally I'd be deeply offended by that accusation, but I don't think
you can help it. Textbooks are wrong, teachers lie, everyone has got
the theorems wrong -- anything but consider that you don't know what you
are talking about. Learning will be very hard for you.

> On 8/6/2020 5:48 AM, Ben Bacarisse wrote:
>> The set of denumerable sequences of D is not the set of all tuples
>> of D.

A righteous and true statement. Have you read the first eight pages of
Mendelson yet?

>> And how can you have the nerve to say that the meaning is not explained?
>> You haven't read the book, have you?

You haven't read page eight for sure.

>>>>> I need to have each term of the art fully defined using words that are
>>>>> not terms of the art at all.
>>>>
>>>> I don't know any textbook that will do that for you. But you haven't
>>>> even given Mendelson a chance. You should read it from the beginning.
>>>
>>> I will try that a little.

How long does making these posts defending your misunderstandings take?
How long can reading the eight pages of a book take? Your time is not
being well spent.

--
Ben.

[olcott]

The can't be the same ideas, they can at most be similar ideas.

> Here's another hint to aid learning: I am one of the more honest people
> you are likely to encounter. If you think I am being dishonest, it is
> more likely you don't know what is going on.
>

When I looked up the nutty crap the denumerable sequences are I am
seeing this.

A denumerable
sequence is a function s whose domain is the set of positive integers;

I need to know how to apply formal semantics to billions of different
things besides natural numbers.

>> On 8/6/2020 10:47 AM, olcott wrote:
>>> every n-place predicate symbol is mapped to a function from Dⁿ to the
>>> set comprised of two values true and false.
>>> https://mathworld.wolfram.com/Interpretation.html
>>>
>>> b. For each predicate letter Anj of L, an assignment of an n-place
>>> relation (Anj)ᴹ in D.
>>
>> One says that it maps to Boolean and the other says that it does not
>> map to Boolean it maps n-tuples.
>
> You don't understand the details.

There is no way to reconcile X is a dog with X is not a dog.

> To find out how both approaches give
> exactly the same results, you would have to understand Mendelson.
>

If one says X and the other says not X then one is wrong. If one says X
leads to Y and Y leads to Z and the other one only mentions that X leads
to Y they should be clear that they are leaving a step out.

>>>>> The set of denumerable sequences of D is not the set of all tuples of D.
>>>>> I won't explain further because, currently, your mind is closed to the
>>>>> idea of your being wrong.
>>>>
>>>> It utterly freaks me out that terms of the art make incompatible
>>>> changes to the meaning of common words and don't even explain that
>>>> they changed the meaning.
>>>
>>> What meaning do you think has been changed? I am not aware of any
>>> common or previous meaning for "the set of denumerable sequences of
>>> D".
>>
>> On 8/6/2020 5:48 AM, Ben Bacarisse wrote:
>>> The set of denumerable sequences of D is not the set of all tuples
>>> of D.
>>
>> On 8/5/2020 8:09 PM, olcott wrote: // Mendelson quote
>>> Let Σ be the set of all denumerable sequences of elements of D.
>>
>> Denumerable means countable.
>
> Have you read the first eight pages of Mendelson? If not, why not? How
> long could it take?

I spend many hours on the one exercise.

>> A sequence is an ordered list.
>> The objects used to form a set are called its element or its members.
>> thus we have countable ordered lists of elements AKA tuples of elements.
>>
>> https://en.wikipedia.org/wiki/Tuple
>> In mathematics, a tuple is a finite ordered list (sequence) of elements.
>> I see no plausible way to conclude that you are not lying.
>
> How about reading page eight of Mendelson?

A denumerable
sequence is a function s whose domain is the set of positive integers;

This does not follow from the defintions of "denumerable" and "sequence".

It is like saying that a "birthday" + "cake" is a vanilla cake with
vanilla frosting served on one's birthday. A cholcolate cake served on
one's birthday is not a "Birthday cake".

> But here's another tip: to learn, you need to switch off that reflex.
> If you ever feel forced into thinking I am lying, you have either
> misunderstood something or I have made a mistake. The best way to
> resolve the matter would be to ask me. It's interesting that in all
> these posts, you have not once asked me what it is that you have got
> wrong and "the set of denumerable sequences of D".
>
>> A {tuple} is a sequence of elements.
>> Mendelson refers to "denumerable sequences of elements of D."
>
> How about reading page eight of Mendelson?
>
>> Tuples are countable thus denumerable thus a tuple is a
>> "denumerable sequence of elements".
>>
>> AND YOU LIED (or less probably) were mistaken.
>
> You have that backwards. I am often mistaken, but I don't lie.

Well there is increasing evidence that you don't ever lie.
I am very glad about that.
Every single time that I accused you of lying I was incorrect.

> Normally I'd be deeply offended by that accusation, but I don't think
> you can help it. Textbooks are wrong, teachers lie, everyone has got
> the theorems wrong -- anything but consider that you don't know what you
> are talking about. Learning will be very hard for you.
>
>> On 8/6/2020 5:48 AM, Ben Bacarisse wrote:
>>> The set of denumerable sequences of D is not the set of all tuples
>>> of D.
>
> A righteous and true statement. Have you read the first eight pages of
> Mendelson yet?
>

A set X is denumerable if it is equinumerous with the set of positive
integers

>>> And how can you have the nerve to say that the meaning is not explained?
>>> You haven't read the book, have you?
>
> You haven't read page eight for sure.
>
>>>>>> I need to have each term of the art fully defined using words that are
>>>>>> not terms of the art at all.
>>>>>
>>>>> I don't know any textbook that will do that for you. But you haven't
>>>>> even given Mendelson a chance. You should read it from the beginning.
>>>>
>>>> I will try that a little.
>
> How long does making these posts defending your misunderstandings take?
> How long can reading the eight pages of a book take? Your time is not
> being well spent.
>

A set X is denumerable if it is equinumerous with the set of positive
integers.

In mathematics, two sets or classes A and B are equinumerous if there
exists a one-to-one correspondence (a bijection) between them.
https://en.wikipedia.org/wiki/Equinumerosity

A denumerable sequence is a function s whose domain is the set of
positive integers;

The second sentence does not logically follow for the first sentence.
Here is the version that does follow from the first sequence

A denumerable sequence is a function s of domain D that maps each
element of domain D to an element of the set of positive integers;

[olcott]

A set X is denumerable if it is equinumerous with the set of positive
integers.

A denumerable sequence is a function s whose domain is the set of
positive integers;

Let Σ be the set of all denumerable sequences of elements of D.

So when D is the set of humans the denumerable sequences of elements of
D is the empty set, so to satisfy the "father of" relation we must find
all of the biological human clones that had the empty set for a father.

[olcott]

Because the set of integers in a denumerable sequence is disjoint with
the set of humans. There are no humans that are integers.

[David Kleinecke]

"when D is the set of humans the SET OF denumerable sequences of
elements of D is the empty set" is comprehensible but false. It
is only true if D is empty.

[olcott]

The problem is that Mendelson begins talking about the generic concept
of interpretation of all WFF of FOL and then suddenly limits the domain
to natural numbers as if the only possible domain for FOL is natural
numbers.

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> A set X is denumerable if it is equinumerous with the set of positive
> integers.

Denumerable means countably infinite.

> A denumerable sequence is a function s whose domain is the set of
> positive integers;

... and whose range is some other set.

Can you write out the start of a few denumerable sequences? For example
using the set {true, false} and then set {apple, grape, orange} as the
range of the function?

> Let Σ be the set of all denumerable sequences of elements of D.
>
> So when D is the set of humans the denumerable sequences of elements
> of D is the empty set,

I can't see how you get that. A denumerable sequence of humans is an
endless sequence of humans. The set of all such sequences is vast.

> so to satisfy the "father of" relation we must
> find all of the biological human clones that had the empty set for a
> father.

Sounds daft, doesn't it? So presumably you concluded that you've
misunderstood something. Why did you not follow on with some question
about what you might have misunderstood?

--
Ben.

[olcott]

On 8/6/2020 8:26 PM, Ben Bacarisse wrote:
> olcott <No...@NoWhere.com> writes:
>
>> A set X is denumerable if it is equinumerous with the set of positive
>> integers.
>
> Denumerable means countably infinite.
>
>> A denumerable sequence is a function s whose domain is the set of
>> positive integers;
>
> ... and whose range is some other set.
>

Oh great, it finally makes sense.

> Can you write out the start of a few denumerable sequences? For example
> using the set {true, false} and then set {apple, grape, orange} as the
> range of the function?
>

Yet those those sets are not equinumerous with the set of positive
integers so we quickly run out of mappings.

>> Let Σ be the set of all denumerable sequences of elements of D.
>>
>> So when D is the set of humans the denumerable sequences of elements
>> of D is the empty set,
>
> I can't see how you get that. A denumerable sequence of humans is an
> endless sequence of humans.

I thought that a domain in set theory was an entirey different word than
the domain from algebra.

There is no such endless sequence of humans in the whole physical universe.

> The set of all such sequences is vast.
>
>> so to satisfy the "father of" relation we must
>> find all of the biological human clones that had the empty set for a
>> father.
>
> Sounds daft, doesn't it? So presumably you concluded that you've
> misunderstood something. Why did you not follow on with some question
> about what you might have misunderstood?

That domain was paired with range cleared up a lot.

>> Let Σ be the set of all denumerable sequences of elements of D.

Still makes no sense at all when D is finite.

[Ben Bacarisse]

The same idea (how an interpretation gives a truth value to a sentence)
can be described (formalised in fact) in different ways.

>> Here's another hint to aid learning: I am one of the more honest people
>> you are likely to encounter. If you think I am being dishonest, it is
>> more likely you don't know what is going on.
>
> When I looked up the nutty crap the denumerable sequences are I am
> seeing this.

That nutty crap is called mathematics.

> A denumerable
> sequence is a function s whose domain is the set of positive integers;
>
> I need to know how to apply formal semantics to billions of different
> things besides natural numbers.

And when you read Mendelson you will find out how to do that. The
domain of a function is the set it applies to. The domain of an
interpretation is the set it applies to. They are not, usually, the
same set.

>>> On 8/6/2020 10:47 AM, olcott wrote:
>>>> every n-place predicate symbol is mapped to a function from Dⁿ to the
>>>> set comprised of two values true and false.
>>>> https://mathworld.wolfram.com/Interpretation.html
>>>>
>>>> b. For each predicate letter Anj of L, an assignment of an n-place
>>>> relation (Anj)ᴹ in D.
>>>
>>> One says that it maps to Boolean and the other says that it does not
>>> map to Boolean it maps n-tuples.
>>
>> You don't understand the details.
>
> There is no way to reconcile X is a dog with X is not a dog.

So what do you conclude? That you have got the wrong end of the stick?

>> To find out how both approaches give
>> exactly the same results, you would have to understand Mendelson.
>
> If one says X and the other says not X then one is wrong.

Indeed. If different explanations of what an interpretation is give
different truth values to the same sentences, then something is
seriously wrong. So what do you conclude? Mathematicians are all
bonkers, or you have misunderstood something?

> If one says
> X leads to Y and Y leads to Z and the other one only mentions that X
> leads to Y they should be clear that they are leaving a step out.

If... So many ifs. Maybe, just maybe, that's not what's happening.
Maybe you have misunderstood the steps involved?

>> How about reading page eight of Mendelson?
>
> A denumerable
> sequence is a function s whose domain is the set of positive integers;
>
> This does not follow from the defintions of "denumerable" and
> "sequence".

Finite sequences are function from {1, 2, ... n} (for some n) and
denumerable sequences are functions from {1, 2, ...}. Makes sense to
me, but at least you know what they are now, even if it seems odd to
you.

>>> Tuples are countable thus denumerable thus a tuple is a
>>> "denumerable sequence of elements".
>>>
>>> AND YOU LIED (or less probably) were mistaken.
>>
>> You have that backwards. I am often mistaken, but I don't lie.
>
> Well there is increasing evidence that you don't ever lie.
> I am very glad about that.
> Every single time that I accused you of lying I was incorrect.

Thank you.

> A set X is denumerable if it is equinumerous with the set of positive integers.
>
> In mathematics, two sets or classes A and B are equinumerous if there
> exists a one-to-one correspondence (a bijection) between them.
> https://en.wikipedia.org/wiki/Equinumerosity
>
> A denumerable sequence is a function s whose domain is the set of
> positive integers;
>
> The second sentence does not logically follow for the first sentence.

Yes it does. Here is (the start of) a denumerable sequence of ASCII
digits. It's elements are clearly in a one-to-one correspondance with
the positive integers:

('3', '1', '4', '1', '5', '9', '2', '6', '5', ...)
| | | | | | | | |
1 2 3 4 5 6 7 8 9 ...

--
Ben.

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> On 8/6/2020 8:26 PM, Ben Bacarisse wrote:
>> olcott <No...@NoWhere.com> writes:
>>
>>> A set X is denumerable if it is equinumerous with the set of positive
>>> integers.
>>
>> Denumerable means countably infinite.
>>
>>> A denumerable sequence is a function s whose domain is the set of
>>> positive integers;
>>
>> ... and whose range is some other set.
>>
>
> Oh great, it finally makes sense.
>
>> Can you write out the start of a few denumerable sequences? For example
>> using the set {true, false} and then set {apple, grape, orange} as the
>> range of the function?
>
> Yet those those sets are not equinumerous with the set of positive
> integers so we quickly run out of mappings.

So... Have I asked you to do something impossible or have you
misunderstood something? Is there something you can do to find out
which is the case?

>>> Let Σ be the set of all denumerable sequences of elements of D.
>>>
>>> So when D is the set of humans the denumerable sequences of elements
>>> of D is the empty set,
>>
>> I can't see how you get that. A denumerable sequence of humans is an
>> endless sequence of humans.
>
> I thought that a domain in set theory was an entirey different word
> than the domain from algebra.

It's a very widely used word. Functions have domains. Interpretations
have domains. There is even a highly technical meaning that is used in
denotational semantics of programming languages.

> There is no such endless sequence of humans in the whole physical
> universe.

You can have an endless sequence even if the range has one member.

>> The set of all such sequences is vast.
>>
>>> so to satisfy the "father of" relation we must
>>> find all of the biological human clones that had the empty set for a
>>> father.
>>
>> Sounds daft, doesn't it? So presumably you concluded that you've
>> misunderstood something. Why did you not follow on with some question
>> about what you might have misunderstood?
>
> That domain was paired with range cleared up a lot.
>
>>> Let Σ be the set of all denumerable sequences of elements of D.
> Still makes no sense at all when D is finite.

Can you image a function from ℕ to {true, false}? Maybe the function
p(n) that is true when n is prime? That is a denumerable sequence of
truth values. It starts

p_n = (false, true, true, false, true, false, true, false, false, ....)

(We tend to write sequences with a subscript rather than brackets: p_n
rather than p(n) but I would always write p(n) were it not for centuries
people writing p_n.)

--
Ben.

[André G. Isaak]

On 2020-08-05 21:31, olcott wrote:

> I need to have each term of the art fully defined using words that are
> not terms of the art at all.

All terms of art are defined in terms of other terms of art.

A good textbook will attempt to start out by defining some small number
of terms in ways that don't require you to know many technical
definitions and will then build upon those terms until you are in a
position to understand more advanced concepts.

But the important thing is that you have to start at the *beginning*.
That means starting with Mendelson page 1 and working forward from
there, stopping at each set of exercises to ensure that you are able to
complete them before proceeding further.

Your strategy is to try to learn things by starting in the middle of a
textbook or simply from random wikipedia pages, but that isn't going to
work since wikipedia, unlike a good textbook, doesn't start at the
beginning and generally assumes you know a fair amount of other material
in each of its articles.

Start at the beginning of Mendelson. When Mendelson presents a
definition, read that definition and *do not* attempt to import anything
into that definition from colloquial meanings or from words as they are
used in other fields.

[olcott]

It seems like the non Mendelson articles define the terms of the art in
term of the common meaning of common words.

Moreover, instead of talking about the n-tuples of objects that satisfy
a wf that has n free variables, it is much more convenient from a
technical standpoint to deal uniformly with denumerable sequences.

When we need to work with finite sets denumerable sequences are
inapplicable. Sets of n-tuples of objects can be either finite or infinite.

[olcott]

He did not say it that way. He disgarded finite sequences:

Moreover, instead of talking about the n-tuples of objects
that satisfy a wf that has n free variables, it is much more convenient
from a technical standpoint to deal uniformly with denumerable sequences.

>>>> Tuples are countable thus denumerable thus a tuple is a
>>>> "denumerable sequence of elements".
>>>>
>>>> AND YOU LIED (or less probably) were mistaken.
>>>
>>> You have that backwards. I am often mistaken, but I don't lie.
>>
>> Well there is increasing evidence that you don't ever lie.
>> I am very glad about that.
>> Every single time that I accused you of lying I was incorrect.
>
> Thank you.

I am really very sincerely devoted to the truth. This requires me to
acknowledge my repeated mistake of calling you a liar.

>
>> A set X is denumerable if it is equinumerous with the set of positive integers.
>>
>> In mathematics, two sets or classes A and B are equinumerous if there
>> exists a one-to-one correspondence (a bijection) between them.
>> https://en.wikipedia.org/wiki/Equinumerosity
>>
>> A denumerable sequence is a function s whose domain is the set of
>> positive integers;
>>
>> The second sentence does not logically follow for the first sentence.
>
> Yes it does.

When we add the domain/range dichotomy to this then it makes sense.

When he says this:

Moreover, instead of talking about the n-tuples of objects that satisfy
a wf that has n free variables, it is much more convenient from a
technical standpoint to deal uniformly with denumerable sequences.

He is meaning that he is referring to elements of the set of n-tuple
objects using a natural number as short-hand to identify each n-tuple in
the set. It may make his subsequent notation simpler yet these extra
layers of unecessary complexity make what he is saying much more
difficult to understand. They way that he said it was much more
convoluted than necessary.

A specific 5-tuple of D could be referred to as <n7, n22, n3, n66, n27>
and encoded in shorthand as m3. Giving n-tuples subscripted names is
consistent with his other notational conventions. Giving n-tuples bare
natural numbers with no alpha part is inconsistent with his other
notational conventions and inconsistent with common naming conventions
across all of mathematics and computer science.

> Here is (the start of) a denumerable sequence of ASCII
> digits. It's elements are clearly in a one-to-one correspondance with
> the positive integers:
>
> ('3', '1', '4', '1', '5', '9', '2', '6', '5', ...)
> | | | | | | | | |
> 1 2 3 4 5 6 7 8 9 ...
>


--

Copyright 2020 Pete Olcott

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> It seems like the non Mendelson articles define the terms of the art
> in term of the common meaning of common words.

Sounds unlikely but without citations, I can't check.

> Moreover, instead of talking about the n-tuples of objects that
> satisfy a wf that has n free variables, it is much more convenient
> from a technical standpoint to deal uniformly with denumerable
> sequences.

This is a quote. The author deserves credit. Don't quote without
attribution.

> When we need to work with finite sets denumerable sequences are
> inapplicable. Sets of n-tuples of objects can be either finite or
> infinite.

Since this is wrong, I'm guessing it's your words. Mendelson uses
denumerable sequences, so you seem to be suggesting he's wrong. The way
to learn is to stop making assertions like this and to start asking
questions: "how can we use denumerable sequences when working with
finite sets?".

--
Ben.

[olcott]

On 8/6/2020 9:05 PM, Ben Bacarisse wrote:
> olcott <No...@NoWhere.com> writes:
>
>> On 8/6/2020 8:26 PM, Ben Bacarisse wrote:
>>> olcott <No...@NoWhere.com> writes:
>>>
>>>> A set X is denumerable if it is equinumerous with the set of positive
>>>> integers.
>>>
>>> Denumerable means countably infinite.
>>>
>>>> A denumerable sequence is a function s whose domain is the set of
>>>> positive integers;
>>>
>>> ... and whose range is some other set.
>>>
>>
>> Oh great, it finally makes sense.
>>
>>> Can you write out the start of a few denumerable sequences? For example
>>> using the set {true, false} and then set {apple, grape, orange} as the
>>> range of the function?
>>
>> Yet those those sets are not equinumerous with the set of positive
>> integers so we quickly run out of mappings.
>
> So... Have I asked you to do something impossible or have you
> misunderstood something? Is there something you can do to find out
> which is the case?
>

He simply said this incorrectly.

Moreover, instead of talking about the n-tuples of objects that satisfy
a wf that has n free variables, it is much more convenient from a
technical standpoint to deal uniformly with denumerable sequences.

it is much more convenient from a technical standpoint to deal uniformly

with denumerable sequences [or finite sequences].

He could have been much more clear

Moreover, instead of talking about the n-tuples of objects that satisfy
a wf that has n free variables, it is much more convenient from a

technical standpoint to deal uniformly with denumerable sequences [or
finite sequences] such that 5-tuple <x27, x3, x99, x39, x3> would be
referred to by a short-hand name such as y3.

>>>> Let Σ be the set of all denumerable sequences of elements of D.
>>>>
>>>> So when D is the set of humans the denumerable sequences of elements
>>>> of D is the empty set,
>>>
>>> I can't see how you get that. A denumerable sequence of humans is an
>>> endless sequence of humans.
>>
>> I thought that a domain in set theory was an entirey different word
>> than the domain from algebra.
>
> It's a very widely used word. Functions have domains. Interpretations
> have domains. There is even a highly technical meaning that is used in
> denotational semantics of programming languages.
>
>> There is no such endless sequence of humans in the whole physical
>> universe.
>
> You can have an endless sequence even if the range has one member.

The bijection between the natural numbers and the elements of a finite
set breaks when one reaches the size-of-set element of the set.

To say that there is a complete bijection between the natural numbers
and the elements of a finite set is psychotic.

>>> The set of all such sequences is vast.
>>>
>>>> so to satisfy the "father of" relation we must
>>>> find all of the biological human clones that had the empty set for a
>>>> father.
>>>
>>> Sounds daft, doesn't it? So presumably you concluded that you've
>>> misunderstood something. Why did you not follow on with some question
>>> about what you might have misunderstood?
>>
>> That domain was paired with range cleared up a lot.
>>
>>>> Let Σ be the set of all denumerable sequences of elements of D.
>> Still makes no sense at all when D is finite.
>
> Can you image a function from ℕ to {true, false}? Maybe the function
> p(n) that is true when n is prime? That is a denumerable sequence of
> truth values. It starts
>
> p_n = (false, true, true, false, true, false, true, false, false, ....)

No I can't imagine this. p(n) operates on individual constants.

> (We tend to write sequences with a subscript rather than brackets: p_n
> rather than p(n) but I would always write p(n) were it not for centuries
> people writing p_n.)
>

There are Unicode subscripts: pₙ

[olcott]

On 8/7/2020 10:38 AM, Ben Bacarisse wrote:
> olcott <No...@NoWhere.com> writes:
>
>> It seems like the non Mendelson articles define the terms of the art
>> in term of the common meaning of common words.
>
> Sounds unlikely but without citations, I can't check.

I included all of them in the first post of this thread.

>> Moreover, instead of talking about the n-tuples of objects that
>> satisfy a wf that has n free variables, it is much more convenient
>> from a technical standpoint to deal uniformly with denumerable
>> sequences.
>
> This is a quote. The author deserves credit. Don't quote without
> attribution.
>
>> When we need to work with finite sets denumerable sequences are
>> inapplicable. Sets of n-tuples of objects can be either finite or
>> infinite.
>
> Since this is wrong, I'm guessing it's your words. Mendelson uses
> denumerable sequences, so you seem to be suggesting he's wrong. The way
> to learn is to stop making assertions like this and to start asking
> questions: "how can we use denumerable sequences when working with
> finite sets?".

How does an infinite set of nautral numbers form a bijection with a
finite set of elements? It does not Mendelson goofed.

[Ben Bacarisse]

Yes he did:

"A /denumerable sequence/ is a function s whose domain is the set of
positive integers; one usually writes s_n instead of s(n). A /finite
sequence/ is a function whose domain is the empty set or { 1, 2, ...,
n} for some positive integer n."

/.../ indicates italics. This is Mendelson defining the terms. What
you quote immediately below is just a use of the term.

> He disgarded finite sequences:
>
> Moreover, instead of talking about the n-tuples of objects
> that satisfy a wf that has n free variables, it is much more
> convenient from a technical standpoint to deal uniformly with
> denumerable sequences.

If using something more convenient is "disregarding" the other, then so
be it.

>>> Every single time that I accused you of lying I was incorrect.
>>
>> Thank you.
>
> I am really very sincerely devoted to the truth. This requires me to
> acknowledge my repeated mistake of calling you a liar.

This is not consistent with your behaviour elsewhere. You claimed to
have had an "actual Turing machine" "fully encoded" when you did not.
And you had previously claimed to have a TM that refuted Rice's theorem
when you did not -- or do you still claim to have such a thing? In the
new spirit of generosity, I will accept that your were mistaken about
what having an actual Turing machine, fully encoded means to the rest of
the world, provided you conceded that it was indeed a misunderstanding
and you never had such a thing.

>>> A set X is denumerable if it is equinumerous with the set of positive integers.
>>>
>>> In mathematics, two sets or classes A and B are equinumerous if there
>>> exists a one-to-one correspondence (a bijection) between them.
>>> https://en.wikipedia.org/wiki/Equinumerosity
>>>
>>> A denumerable sequence is a function s whose domain is the set of
>>> positive integers;
>>>
>>> The second sentence does not logically follow for the first sentence.
>>
>> Yes it does.
>
> When we add the domain/range dichotomy to this then it makes sense.
>
> When he says this:
> Moreover, instead of talking about the n-tuples of objects that
> satisfy a wf that has n free variables, it is much more convenient
> from a technical standpoint to deal uniformly with denumerable
> sequences.
>
> He is meaning that he is referring to elements of the set of n-tuple
> objects using a natural number as short-hand to identify each n-tuple
> in the set.

Nope.

> It may make his subsequent notation simpler yet these extra layers of
> unecessary complexity make what he is saying much more difficult to
> understand. They way that he said it was much more convoluted than
> necessary.

How would you know? I've seen (and understood) it done both ways and
you haven't. You are assuming that the author is wrong to say it is
more convenient without, I suspect, having understood how the
satisfaction of formulas is formally defined.

> A specific 5-tuple of D could be referred to as <n7, n22, n3, n66, n27>
> and encoded in shorthand as m3.

This makes it clear to me that you don't know how Mendeson goes on to
use Σ. The whole point is to describe a general process applicable to
any formula.

> Giving n-tuples subscripted names is consistent with his other
> notational conventions. Giving n-tuples bare natural numbers with no
> alpha part is inconsistent with his other notational conventions and
> inconsistent with common naming conventions across all of mathematics
> and computer science.

You have no standing to make such claims. You don't yet know how to
write even simple formulas. All I can be bothered to say is that
Mendelson's notation is not inconsistent. Stop pontificating and start
learning from an author who know vastly more about writing mathematics
than you do.

--
Ben.

[olcott]

My actual Turing Machine was more fully encoded than any Turing machine
that Turing himself ever wrote.

> And you had previously claimed to have a TM that refuted Rice's theorem
> when you did not -- or do you still claim to have such a thing? In the

Refuting the halting problem counter-example includes refuting Rice. I
didn't know that I would ever be able to actually refute the halting
problem proofs so my initial claim was smaller.

> new spirit of generosity, I will accept that your were mistaken about
> what having an actual Turing machine, fully encoded means to the rest of
> the world, provided you conceded that it was indeed a misunderstanding
> and you never had such a thing.
>

My actual Turing Machine was more fully encoded than any Turing machine
that Turing himself ever wrote.

The above is a general process that could be applied to any formula and
it provides a concrete example of how this process is applied to a
single instance of a 5-tuple.

M_1...M_n could refer to the set of n-tuples of D.

If one only speaks in generalities and never shows all of the exact
details how these generalities are specifically applied then one
communicates much less effectively.

>> Giving n-tuples subscripted names is consistent with his other
>> notational conventions. Giving n-tuples bare natural numbers with no
>> alpha part is inconsistent with his other notational conventions and
>> inconsistent with common naming conventions across all of mathematics
>> and computer science.
>
> You have no standing to make such claims. You don't yet know how to
> write even simple formulas. All I can be bothered to say is that
> Mendelson's notation is not inconsistent. Stop pontificating and start
> learning from an author who know vastly more about writing mathematics
> than you do.
>


--

Copyright 2020 Pete Olcott

[André G. Isaak]

Another possibility which you need to consider is that they have defined
them using the technical vocabulary of mathematics, and you have tried
to interpret those definition using common meanings of the terms which
means you will not have actually understood those definitions.

> Moreover, instead of talking about the n-tuples of objects that satisfy
> a wf that has n free variables, it is much more convenient from a
> technical standpoint to deal uniformly with denumerable sequences.
>
> When we need to work with finite sets denumerable sequences are
> inapplicable. Sets of n-tuples of objects can be either finite or infinite.

Neither of the above two paragraphs appear related to my post.

[Kaz Kylheku]

On 2020-08-07, olcott <No...@NoWhere.com> wrote:
> My actual Turing Machine was more fully encoded than any Turing machine
> that Turing himself ever wrote.

I'm sorry, but all the evidence so far points toward the conclusion that
you couldn't code your way out of a brown paper bag, even if it were
soaked wet to ease the difficulty level.

[André G. Isaak]

On 2020-08-07 10:48, olcott wrote:

> My actual Turing Machine was more fully encoded than any Turing machine
> that Turing himself ever wrote.

What does that even mean? You obviously have some new and entirely
idiosyncratic meaning for the word 'encoded'. How does encode a Turing
Machine more filly than Turing did?

[olcott]

That statement is based on an assessment that is far too superficial.

[olcott]

On 8/7/2020 12:13 PM, André G. Isaak wrote:
> On 2020-08-07 10:48, olcott wrote:
>
>> My actual Turing Machine was more fully encoded than any Turing
>> machine that Turing himself ever wrote.
>
> What does that even mean? You obviously have some new and entirely
> idiosyncratic meaning for the word 'encoded'. How does encode a Turing
> Machine more filly than Turing did?
>
> André

If I write pseudo-code that can be directly translated into fully
operational software this pseudo-code has all of the details of the
required algorithm fully specified, thus is fully encoded.

To the best of my understanding Turing never wrote any details of any
algorithm using his concept of a Turing machine. He merely specified the
underlying infrastructure that could be used for specifying algorithms.

[Keith Thompson]

See, this is the problem. I haven't read Mendelson, but I'd be
willing to bet (not any large amount) that he didn't goof, and
in particular that he didn't say or imply that an infinite set of
natural numbers forms a bijection with a finite set of elements.

If you think that some work that's been read by experts for decades
in multiple editions has an error of this magnitude, your first
thought should not be that the author forgot about the distinction
between finite and infinite sets and that you're the first person
to notice. It should be that you've misunderstood something.
And if you can't figure it out, you can *ask*.

It's (barely) possible that you've found some fundamental error
that everyone else has missed, but that's not the way to bet.

--
Keith Thompson (The_Other_Keith) Keith.S.T...@gmail.com
Working, but not speaking, for Philips Healthcare
void Void(void) { Void(); } /* The recursive call of the void */

[Keith Thompson]

olcott <No...@NoWhere.com> writes:
[...]

> The bijection between the natural numbers and the elements of a finite
> set breaks when one reaches the size-of-set element of the set.

Yes, though it's simpler to say (correctly) that there is no such
bijection.

> To say that there is a complete bijection between the natural numbers
> and the elements of a finite set is psychotic.

Please calm down. Such a claim would not be "psychotic" (I presume
you're not qualified to make such a psychiatric diagnosis). It would
simply be factually incorrect.

But I don't believe that anyone has made such a claim. Please cite
the exact wording (from Mendelson, if I've followed the context
correctly) that you've interpreted that way. I expect (with less
than 100% certainty) that someone here will be able to explain how
you've misunderstood it.

[...]

[André G. Isaak]

On 2020-08-07 11:40, olcott wrote:
> On 8/7/2020 12:13 PM, André G. Isaak wrote:
>> On 2020-08-07 10:48, olcott wrote:
>>
>>> My actual Turing Machine was more fully encoded than any Turing
>>> machine that Turing himself ever wrote.
>>
>> What does that even mean? You obviously have some new and entirely
>> idiosyncratic meaning for the word 'encoded'. How does encode a Turing
>> Machine more filly than Turing did?
>>
>> André
>
> If I write pseudo-code that can be directly translated into fully
> operational software this pseudo-code has all of the details of the
> required algorithm fully specified, thus is fully encoded.

What is any of the above supposed to mean? A Turing Machine is an
abstract mathematical model of computation. It is not software. Writing
something in pseudo-code is not a Turing Machine.

A Turing Machine is specified by a 7-tuple <Q, Γ, b, Σ, δ, q₀, F>. If
what you have is anything other than such a 7-tuple, it is not a Turing
Machine, 'fully encoded' or otherwise.

> To the best of my understanding Turing never wrote any details of any
> algorithm using his concept of a Turing machine. He merely specified the
> underlying infrastructure that could be used for specifying algorithms.

Of course Turing dealt with actual Turing Machines.

[Keith Thompson]

olcott <No...@NoWhere.com> writes:
> On 8/7/2020 12:13 PM, André G. Isaak wrote:
>> On 2020-08-07 10:48, olcott wrote:
>>
>>> My actual Turing Machine was more fully encoded than any Turing
>>> machine that Turing himself ever wrote.
>>
>> What does that even mean? You obviously have some new and entirely
>> idiosyncratic meaning for the word 'encoded'. How does encode a
>> Turing Machine more filly than Turing did?
>>
>> André
>
> If I write pseudo-code that can be directly translated into fully
> operational software this pseudo-code has all of the details of the
> required algorithm fully specified, thus is fully encoded.

To be clear, are you still asserting that you have (had?) such a
fully encoded Turing machine? If so, I presume you could easily
determine how many states it has.

Do you know how many states it has? If so, how many states does it
have? (That's two distinct questions.)

> To the best of my understanding Turing never wrote any details of any
> algorithm using his concept of a Turing machine. He merely specified
> the underlying infrastructure that could be used for specifying
> algorithms.

I don't know whether that's correct or not. If it is, then comparing
your Turing machine to Turing's seems like meaningless bragging.

[olcott]

Apparently the authors of math textbooks are not the best communicators.
When I say that Mendelson goofed I mean that what he said is not 100%
perfectly precisely what he meant. Other people can infer what he meant
thus it does not seem to be an error for them.

It might not actually be an error at all. It may be that his cumbersome
way of saying things can possibly be decoded.

[olcott]

On 8/7/2020 1:24 PM, André G. Isaak wrote:
> On 2020-08-07 11:40, olcott wrote:
>> On 8/7/2020 12:13 PM, André G. Isaak wrote:
>>> On 2020-08-07 10:48, olcott wrote:
>>>
>>>> My actual Turing Machine was more fully encoded than any Turing
>>>> machine that Turing himself ever wrote.
>>>
>>> What does that even mean? You obviously have some new and entirely
>>> idiosyncratic meaning for the word 'encoded'. How does encode a
>>> Turing Machine more filly than Turing did?
>>>
>>> André
>>
>> If I write pseudo-code that can be directly translated into fully
>> operational software this pseudo-code has all of the details of the
>> required algorithm fully specified, thus is fully encoded.
>
> What is any of the above supposed to mean? A Turing Machine is an
> abstract mathematical model of computation. It is not software. Writing
> something in pseudo-code is not a Turing Machine.
>

Turing machines have been fully implemented in software as software.
http://www.lns.mit.edu/~dsw/turing/turing.html

Thus the abstraction has been empirically proven to be able to be made
perfectly concrete.

> A Turing Machine is specified by a 7-tuple <Q, Γ, b, Σ, δ, q₀, F>. If
> what you have is anything other than such a 7-tuple, it is not a Turing
> Machine, 'fully encoded' or otherwise.
>

It is equivalent.

>> To the best of my understanding Turing never wrote any details of any
>> algorithm using his concept of a Turing machine. He merely specified
>> the underlying infrastructure that could be used for specifying
>> algorithms.
>
> Of course Turing dealt with actual Turing Machines.
>

How many programs did he write in the Turing Machine description language?

> André

[olcott]

On 8/7/2020 1:29 PM, Keith Thompson wrote:
> olcott <No...@NoWhere.com> writes:
>> On 8/7/2020 12:13 PM, André G. Isaak wrote:
>>> On 2020-08-07 10:48, olcott wrote:
>>>
>>>> My actual Turing Machine was more fully encoded than any Turing
>>>> machine that Turing himself ever wrote.
>>>
>>> What does that even mean? You obviously have some new and entirely
>>> idiosyncratic meaning for the word 'encoded'. How does encode a
>>> Turing Machine more filly than Turing did?
>>>
>>> André
>>
>> If I write pseudo-code that can be directly translated into fully
>> operational software this pseudo-code has all of the details of the
>> required algorithm fully specified, thus is fully encoded.
>
> To be clear, are you still asserting that you have (had?) such a
> fully encoded Turing machine? If so, I presume you could easily
> determine how many states it has.
>

Sure just as easily as anyone can count the number of states of any
other example of pseudocode that does not yet even map to a specific
source code much less and specific machine code.

I am just now writing the context switch aspect of my x86 based UTM. The
number of states will be construed as the number of x86 instructions.

> Do you know how many states it has? If so, how many states does it
> have? (That's two distinct questions.)
>

It has the same number of states as any other program that has not been
written or compiled. It is looking like the entire system will easily
fit inside of 64K. x86 instructions vary in length. We can probably have
very reliable certainty that the number of states will be far less than
64K.

>> To the best of my understanding Turing never wrote any details of any
>> algorithm using his concept of a Turing machine. He merely specified
>> the underlying infrastructure that could be used for specifying
>> algorithms.
>
> I don't know whether that's correct or not. If it is, then comparing
> your Turing machine to Turing's seems like meaningless bragging.
>

My whole point was that any pseudo code that can be directly translated
into working software is fully encoded. Since my Turing Machine
description language is x86 machine language my fully encoded pseudocode
will be translated into x86 machine language via "c".

[Keith Thompson]

Oh, is that what you meant? I wonder why that meaning was not
communicated clearly.

If other people are able to infer what he meant and you aren't,
could it be that Mendelson isn't the problem?

> It might not actually be an error at all. It may be that his
> cumbersome way of saying things can possibly be decoded.

--

[Keith Thompson]

olcott <No...@NoWhere.com> writes:
> On 8/7/2020 1:29 PM, Keith Thompson wrote:
>> olcott <No...@NoWhere.com> writes:
>>> On 8/7/2020 12:13 PM, André G. Isaak wrote:
>>>> On 2020-08-07 10:48, olcott wrote:
>>>>
>>>>> My actual Turing Machine was more fully encoded than any Turing
>>>>> machine that Turing himself ever wrote.
>>>>
>>>> What does that even mean? You obviously have some new and entirely
>>>> idiosyncratic meaning for the word 'encoded'. How does encode a
>>>> Turing Machine more filly than Turing did?
>>>

>>> If I write pseudo-code that can be directly translated into fully
>>> operational software this pseudo-code has all of the details of the
>>> required algorithm fully specified, thus is fully encoded.
>>
>> To be clear, are you still asserting that you have (had?) such a
>> fully encoded Turing machine? If so, I presume you could easily
>> determine how many states it has.
>
> Sure just as easily as anyone can count the number of states of any
> other example of pseudocode that does not yet even map to a specific
> source code much less and specific machine code.
>
> I am just now writing the context switch aspect of my x86 based
> UTM. The number of states will be construed as the number of x86
> instructions.

I didn't ask about any x86 based whatever-it-is that you're still
working on. I asked about the fully encoded actual Turing machine that
you claimed to have several years ago. Any actual Turing machine has a
well defined number of states. If yours doesn't, then it's not a Turing
machine.

>> Do you know how many states it has? If so, how many states does it
>> have? (That's two distinct questions.)
>
> It has the same number of states as any other program that has not
> been written or compiled. It is looking like the entire system will
> easily fit inside of 64K. x86 instructions vary in length. We can
> probably have very reliable certainty that the number of states will
> be far less than 64K.

So you don't know how many states it has (or you could have said "Yes"
followed by a specific number) and all you've ever had was a vague idea
for a Turing machine, not an actual one. And in fact you don't even
seem to understand what "states" are.

A system that fits in 64K (that's 65536 bytes of 8 bits each) can have
*far* more than 64K states. It could have as many as 2**524288 states,
one for each configuration of those 524288 bits (using "**" to denote
exponentiation).

A system with 16 bits of storage can have as many as 65536 distinct
states.

>>> To the best of my understanding Turing never wrote any details of any
>>> algorithm using his concept of a Turing machine. He merely specified
>>> the underlying infrastructure that could be used for specifying
>>> algorithms.
>>
>> I don't know whether that's correct or not. If it is, then comparing
>> your Turing machine to Turing's seems like meaningless bragging.
>
> My whole point was that any pseudo code that can be directly
> translated into working software is fully encoded. Since my Turing
> Machine description language is x86 machine language my fully encoded
> pseudocode will be translated into x86 machine language via "c".

You have said nothing that provides any evidence that what you have
is a "Turing Machine".

As André G. Isaak wrote:

A Turing Machine is specified by a 7-tuple <Q, Γ, b, Σ, δ, q₀, F>.
If what you have is anything other than such a 7-tuple, it is not a
Turing Machine, 'fully encoded' or otherwise.

I don't believe you have a Turing Machine. (I do believe that you
*think* you do.)

[olcott]

It is a proven biological fact that people that are very analytical are
proportionally less good at communicating and the reverse. This is based
on the size of the brain tissue that connects to the two brain
hemispheres: corpus callosum.

>> It might not actually be an error at all. It may be that his
>> cumbersome way of saying things can possibly be decoded.
>


--

Copyright 2020 Pete Olcott

[olcott]

Do you understand the difference between source-code and machine code?
Do you understand the difference between microcode and machine code?

If you do then you can understand that you cannot count the number of
microcode states of a c++ program.

>>> Do you know how many states it has? If so, how many states does it
>>> have? (That's two distinct questions.)
>>
>> It has the same number of states as any other program that has not
>> been written or compiled. It is looking like the entire system will
>> easily fit inside of 64K. x86 instructions vary in length. We can
>> probably have very reliable certainty that the number of states will
>> be far less than 64K.
>
> So you don't know how many states it has (or you could have said "Yes"
> followed by a specific number) and all you've ever had was a vague idea
> for a Turing machine, not an actual one. And in fact you don't even
> seem to understand what "states" are.
>
> A system that fits in 64K (that's 65536 bytes of 8 bits each) can have
> *far* more than 64K states. It could have as many as 2**524288 states,
> one for each configuration of those 524288 bits (using "**" to denote
> exponentiation).
>
> A system with 16 bits of storage can have as many as 65536 distinct
> states.

It has 32-bit storage. I won't need more than 64K of it for code.

>>>> To the best of my understanding Turing never wrote any details of any
>>>> algorithm using his concept of a Turing machine. He merely specified
>>>> the underlying infrastructure that could be used for specifying
>>>> algorithms.
>>>
>>> I don't know whether that's correct or not. If it is, then comparing
>>> your Turing machine to Turing's seems like meaningless bragging.
>>
>> My whole point was that any pseudo code that can be directly
>> translated into working software is fully encoded. Since my Turing
>> Machine description language is x86 machine language my fully encoded
>> pseudocode will be translated into x86 machine language via "c".
>
> You have said nothing that provides any evidence that what you have
> is a "Turing Machine".
>

You might not be able to understand the Turing equivalence of the x86
language.

> As André G. Isaak wrote:
>
> A Turing Machine is specified by a 7-tuple <Q, Γ, b, Σ, δ, q₀, F>.
> If what you have is anything other than such a 7-tuple, it is not a
> Turing Machine, 'fully encoded' or otherwise.
>
> I don't believe you have a Turing Machine. (I do believe that you
> *think* you do.)
>

Turing equivalence

[olcott]

No one in the universe has an actual hardware Turing Machine because
they require unlimited memory. On the other hand unless we are counting
to infinity and storing the result hardly any actual computations
require unlimited memory.

This model of computation is equivalent to a Turing machine for the set
of all computations that do not require unlimited memory.
https://en.wikipedia.org/wiki/Von_Neumann_architecture

[olcott]

https://en.wikipedia.org/wiki/Turing_completeness
Of course, no physical system can have infinite memory; but if the
limitation of finite memory is ignored, most programming languages are
otherwise Turing-complete.

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> On 8/7/2020 11:12 AM, Ben Bacarisse wrote:
>> olcott <No...@NoWhere.com> writes:

<cut>

>>> I am really very sincerely devoted to the truth. This requires me to
>>> acknowledge my repeated mistake of calling you a liar.
>>
>> This is not consistent with your behaviour elsewhere. You claimed to
>> have had an "actual Turing machine" "fully encoded" when you did not.
>
> My actual Turing Machine was more fully encoded than any Turing
> machine that Turing himself ever wrote.

Ah, you are back to lying about it. Shame.

>> And you had previously claimed to have a TM that refuted Rice's theorem
>> when you did not -- or do you still claim to have such a thing? In the
>
> Refuting the halting problem counter-example includes refuting Rice.

Still ducking and diving I see. Do you still claim to have had such a
machine when you first make that preposterous claim?

--
Ben.

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> On 8/6/2020 9:05 PM, Ben Bacarisse wrote:
>> olcott <No...@NoWhere.com> writes:
>>
>>> On 8/6/2020 8:26 PM, Ben Bacarisse wrote:
>>>> olcott <No...@NoWhere.com> writes:
>>>>
>>>>> A set X is denumerable if it is equinumerous with the set of positive
>>>>> integers.
>>>>
>>>> Denumerable means countably infinite.
>>>>
>>>>> A denumerable sequence is a function s whose domain is the set of
>>>>> positive integers;
>>>>
>>>> ... and whose range is some other set.
>>>>
>>>
>>> Oh great, it finally makes sense.
>>>
>>>> Can you write out the start of a few denumerable sequences? For example
>>>> using the set {true, false} and then set {apple, grape, orange} as the
>>>> range of the function?
>>>
>>> Yet those those sets are not equinumerous with the set of positive
>>> integers so we quickly run out of mappings.
>>
>> So... Have I asked you to do something impossible or have you
>> misunderstood something? Is there something you can do to find out
>> which is the case?
>
> He simply said this incorrectly.

No, he got it right.

> Moreover, instead of talking about the n-tuples of objects that
> satisfy a wf that has n free variables, it is much more convenient
> from a technical standpoint to deal uniformly with denumerable
> sequences.
>
> it is much more convenient from a technical standpoint to deal
> uniformly with denumerable sequences [or finite sequences].

What is that "[or finite sequences]"? Mendelson didn't say it because
it's wrong.

> He could have been much more clear

You are not the target audience, so while I agree that he could be
clearer if writing for someone like you, it's not a fair criticism of
the book because he isn't.

> Moreover, instead of talking about the n-tuples of objects that
> satisfy a wf that has n free variables, it is much more convenient
> from a technical standpoint to deal uniformly with denumerable
> sequences [or finite sequences] such that 5-tuple <x27, x3, x99, x39,
> x3> would be referred to by a short-hand name such as y3.

This, however, is junk. You have no idea how Σ is used so you should
not pretend to know how to "be clearer".

>>>>> Let Σ be the set of all denumerable sequences of elements of D.
>>>>>
>>>>> So when D is the set of humans the denumerable sequences of elements
>>>>> of D is the empty set,
>>>>
>>>> I can't see how you get that. A denumerable sequence of humans is an
>>>> endless sequence of humans.
>>>
>>> I thought that a domain in set theory was an entirey different word
>>> than the domain from algebra.
>>
>> It's a very widely used word. Functions have domains. Interpretations
>> have domains. There is even a highly technical meaning that is used in
>> denotational semantics of programming languages.
>>
>>> There is no such endless sequence of humans in the whole physical
>>> universe.
>>
>> You can have an endless sequence even if the range has one member.
>
> The bijection between the natural numbers and the elements of a finite
> set breaks when one reaches the size-of-set element of the set.

There are no bijections between ℕ and any finite set. Why are you
stating this rather obvious fact? You must think it relates to my
remark "you can have an endless sequence even if the range has one
member" but I can't image how.

> To say that there is a complete bijection between the natural numbers
> and the elements of a finite set is psychotic.

Yes, that would be daft, wouldn't it? So what should your first thought
be? Remember, you want to know what's going on.

>>>>> Let Σ be the set of all denumerable sequences of elements of D.
>>> Still makes no sense at all when D is finite.
>>
>> Can you image a function from ℕ to {true, false}? Maybe the function
>> p(n) that is true when n is prime? That is a denumerable sequence of
>> truth values. It starts
>>
>> p_n = (false, true, true, false, true, false, true, false, false, ....)
>
> No I can't imagine this. p(n) operates on individual constants.

If you can't imagine a function from ℕ to {true, false}, you need
another hobby. Goodness knows what "constants" have to do with it.

Remember, constants form a syntactic category in the language of some
theory. You are using the word incorrectly for this context, but I
don't think you can change.

--
Ben.

[Keith Thompson]

olcott <No...@NoWhere.com> writes:
> On 8/7/2020 3:24 PM, Keith Thompson wrote:

[...]

>> You have said nothing that provides any evidence that what you have
>> is a "Turing Machine".
>>
>> As André G. Isaak wrote:
>>
>> A Turing Machine is specified by a 7-tuple <Q, Γ, b, Σ, δ, q₀, F>.
>> If what you have is anything other than such a 7-tuple, it is not a
>> Turing Machine, 'fully encoded' or otherwise.
>>
>> I don't believe you have a Turing Machine. (I do believe that you
>> *think* you do.)
>
> No one in the universe has an actual hardware Turing Machine because
> they require unlimited memory. On the other hand unless we are
> counting to infinity and storing the result hardly any actual
> computations require unlimited memory.

Of course a Turing Machine is an abstraction. A specific TM can be
specified as a 7-tuple of the kind André G. Isaak mentioned above.

A program written in some programming language could in principle be
specified in that form. Enumerating the states of, say, a 100-line C
program is theoretically possible but impractical. If its data consists
entirely of a single 32-bit integer that can take on any value, it has
billions of states. I'm skeptical of your claim that your so-called
Turing Machine has as few as 64,000 states.

I don't believe you have anything that you can express in the form of a
Turing Machine.

> This model of computation is equivalent to a Turing machine for the
> set of all computations that do not require unlimited memory.
> https://en.wikipedia.org/wiki/Von_Neumann_architecture

So what you have is some pseudocode that you might eventually turn
into a working program. You don't have anything that you could
express as a 7-tuple of the kind André G. Isaak referred to.

By calling it a "fully encoded Turing Machine", you gave the
impression that you could express it as the 7-tuple that defines
the term. The way you described it was grossly misleading. I think
you had a more or less vague idea for a computer program that you
still haven't fully implemented, and you thought that calling it a
"Turing Machine" would make it sound more impressive.

(It's been a long time since I studied this stuff. I invite others
to correct anything I've described unclearly or incorrectly.)

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> On 8/7/2020 10:38 AM, Ben Bacarisse wrote:
>> olcott <No...@NoWhere.com> writes:
>>
>>> It seems like the non Mendelson articles define the terms of the art
>>> in term of the common meaning of common words.
>>
>> Sounds unlikely but without citations, I can't check.
>
> I included all of them in the first post of this thread.

None of them formally define how to determine the truth value of a
sentence. They are all overviews. Mendelson is going into the details
and it in those details that you've come unstuck.

And if you consider those sources to be using "the common meaning of
common words" then I am flabbergasted. But I won't press the point.
Use these other overview sources. You are not coping with Mendelson.

>>> Moreover, instead of talking about the n-tuples of objects that
>>> satisfy a wf that has n free variables, it is much more convenient
>>> from a technical standpoint to deal uniformly with denumerable
>>> sequences.
>>
>> This is a quote. The author deserves credit. Don't quote without
>> attribution.
>>
>>> When we need to work with finite sets denumerable sequences are
>>> inapplicable. Sets of n-tuples of objects can be either finite or
>>> infinite.
>>
>> Since this is wrong, I'm guessing it's your words. Mendelson uses
>> denumerable sequences, so you seem to be suggesting he's wrong. The way
>> to learn is to stop making assertions like this and to start asking
>> questions: "how can we use denumerable sequences when working with
>> finite sets?".
>
> How does an infinite set of nautral numbers form a bijection with a
> finite set of elements? It does not Mendelson goofed.

It doesn't. You've asked the wrong question. And of course Mendelson
did not goof.

--
Ben.

[olcott]

On 8/7/2020 5:41 PM, Ben Bacarisse wrote:
> olcott <No...@NoWhere.com> writes:
>
>> On 8/7/2020 11:12 AM, Ben Bacarisse wrote:
>>> olcott <No...@NoWhere.com> writes:
> <cut>
>>>> I am really very sincerely devoted to the truth. This requires me to
>>>> acknowledge my repeated mistake of calling you a liar.
>>>
>>> This is not consistent with your behaviour elsewhere. You claimed to
>>> have had an "actual Turing machine" "fully encoded" when you did not.
>>
>> My actual Turing Machine was more fully encoded than any Turing
>> machine that Turing himself ever wrote.
>
> Ah, you are back to lying about it. Shame.
>

I am not aware of Turing ever writing any actual code, therefore
pseudo-code that directly maps to "c" code would be more fully encoded
than anything Turing ever wrote.

>>> And you had previously claimed to have a TM that refuted Rice's theorem
>>> when you did not -- or do you still claim to have such a thing? In the
>>
>> Refuting the halting problem counter-example includes refuting Rice.
>
> Still ducking and diving I see. Do you still claim to have had such a
> machine when you first make that preposterous claim?




--

Copyright 2020 Pete Olcott

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> On 8/7/2020 5:41 PM, Ben Bacarisse wrote:
>> olcott <No...@NoWhere.com> writes:
>>
>>> On 8/7/2020 11:12 AM, Ben Bacarisse wrote:
>>>> olcott <No...@NoWhere.com> writes:
>> <cut>
>>>>> I am really very sincerely devoted to the truth. This requires me to
>>>>> acknowledge my repeated mistake of calling you a liar.
>>>>
>>>> This is not consistent with your behaviour elsewhere. You claimed to
>>>> have had an "actual Turing machine" "fully encoded" when you did not.
>>>
>>> My actual Turing Machine was more fully encoded than any Turing
>>> machine that Turing himself ever wrote.
>>
>> Ah, you are back to lying about it. Shame.
>
> I am not aware of Turing ever writing any actual code, therefore
> pseudo-code that directly maps to "c" code would be more fully encoded
> than anything Turing ever wrote.

You don't have, and never have had, anything that could reasonably be
called a Turing machine. I note that all your most recent replies are
just dodges -- you don't repeat the lie explicitly, you just say
something unrelated that you think sounds vaguely exculpatory.

The really sad part is the you never needed to lie in the first place.
Having a logically impossible algorithm in pseudo-code would have been
just as impressive a boast. In fact, everyone would have preferred it
because Turing machines are so very tedious.

>>>> And you had previously claimed to have a TM that refuted Rice's theorem
>>>> when you did not -- or do you still claim to have such a thing? In the
>>>
>>> Refuting the halting problem counter-example includes refuting Rice.
>>
>> Still ducking and diving I see. Do you still claim to have had such a
>> machine when you first make that preposterous claim?

Still dodging this question I see. Do you still claim to have had it?

--
Ben.

[olcott]

How the Hell do you form a bijection between the set of natural numbers
and the elements of a finite set?

> You are not the target audience, so while I agree that he could be
> clearer if writing for someone like you, it's not a fair criticism of
> the book because he isn't.
>
>> Moreover, instead of talking about the n-tuples of objects that
>> satisfy a wf that has n free variables, it is much more convenient
>> from a technical standpoint to deal uniformly with denumerable
>> sequences [or finite sequences] such that 5-tuple <x27, x3, x99, x39,
>> x3> would be referred to by a short-hand name such as y3.
>
> This, however, is junk. You have no idea how Σ is used so you should
> not pretend to know how to "be clearer".

I am proposing one very easy way of encoding a sort-hand notation for
encoding sets of n-tuples.

>>>>>> Let Σ be the set of all denumerable sequences of elements of D.
>>>>>>
>>>>>> So when D is the set of humans the denumerable sequences of elements
>>>>>> of D is the empty set,
>>>>>
>>>>> I can't see how you get that. A denumerable sequence of humans is an
>>>>> endless sequence of humans.
>>>>
>>>> I thought that a domain in set theory was an entirey different word
>>>> than the domain from algebra.
>>>
>>> It's a very widely used word. Functions have domains. Interpretations
>>> have domains. There is even a highly technical meaning that is used in
>>> denotational semantics of programming languages.
>>>
>>>> There is no such endless sequence of humans in the whole physical
>>>> universe.
>>>
>>> You can have an endless sequence even if the range has one member.
>>
>> The bijection between the natural numbers and the elements of a finite
>> set breaks when one reaches the size-of-set element of the set.
>
> There are no bijections between ℕ and any finite set.

A set X is denumerable if it is equinumerous with the set of positive
integers.

A denumerable sequence is a function s whose domain is the set of
positive integers;

instead of talking about the n-tuples of objects that satisfy a wf that
has n free variables, it is much more convenient from a technical
standpoint to deal uniformly with denumerable sequences.

When we satify the "father of" relation we do not have a denumerable
sequence we have a finite sequence therefore his generalization of

...talking about the n-tuples of objects that satisfy a wf that has n
free variables...
using denumerable sequences is incorrect. In the case of the "father of"
relation we have a finite sequence of ordered pairs where the first
element of this pair comes from the set of every father that ever lived.

> Why are you
> stating this rather obvious fact? You must think it relates to my
> remark "you can have an endless sequence even if the range has one
> member" but I can't image how.
>

The bijection does not endlessly continue it stops at one.

>> To say that there is a complete bijection between the natural numbers
>> and the elements of a finite set is psychotic.
>
> Yes, that would be daft, wouldn't it? So what should your first thought
> be? Remember, you want to know what's going on.
>

Yet another case where the terms of the art make sure to totally screw
up the common meaning of these same terms. It is like saying that the
medical term "dead" means {could not be more healthy}.

>>>>>> Let Σ be the set of all denumerable sequences of elements of D.

>>>> Still makes no sense at all when D is finite.
>>>
>>> Can you image a function from ℕ to {true, false}? Maybe the function
>>> p(n) that is true when n is prime? That is a denumerable sequence of
>>> truth values. It starts
>>>
>>> p_n = (false, true, true, false, true, false, true, false, false, ....)
>>
>> No I can't imagine this. p(n) operates on individual constants.
>
> If you can't imagine a function from ℕ to {true, false}, you need
> another hobby. Goodness knows what "constants" have to do with it.

I can imagine a function from two elements of ℕ to each of the elements
of True and False. I cannot imagine the entire set of ℕ maps to the two
elements of true and false. I would map false to zero and true to 1.

> Remember, constants form a syntactic category in the language of some
> theory. You are using the word incorrectly for this context, but I
> don't think you can change.
>

A function is a relation that uniquely associates members of one set
with members of another set. More formally, a function from A to B is an
object f such that every a ∈ A is uniquely associated with an object
f(a) ∈ B. https://mathworld.wolfram.com/Function.html

every a ∈ ℕ is uniquely associated with an object f(a) ∈ {true, false}.

[olcott]

On 8/7/2020 6:20 PM, Ben Bacarisse wrote:
> olcott <No...@NoWhere.com> writes:
>
>> On 8/7/2020 5:41 PM, Ben Bacarisse wrote:
>>> olcott <No...@NoWhere.com> writes:
>>>
>>>> On 8/7/2020 11:12 AM, Ben Bacarisse wrote:
>>>>> olcott <No...@NoWhere.com> writes:
>>> <cut>
>>>>>> I am really very sincerely devoted to the truth. This requires me to
>>>>>> acknowledge my repeated mistake of calling you a liar.
>>>>>
>>>>> This is not consistent with your behaviour elsewhere. You claimed to
>>>>> have had an "actual Turing machine" "fully encoded" when you did not.
>>>>
>>>> My actual Turing Machine was more fully encoded than any Turing
>>>> machine that Turing himself ever wrote.
>>>
>>> Ah, you are back to lying about it. Shame.
>>
>> I am not aware of Turing ever writing any actual code, therefore
>> pseudo-code that directly maps to "c" code would be more fully encoded
>> than anything Turing ever wrote.
>
> You don't have, and never have had, anything that could reasonably be
> called a Turing machine.

The x86 architecture is as close as any physical machine gets to Turing
complete. https://en.wikipedia.org/wiki/Von_Neumann_architecture

https://en.wikipedia.org/wiki/Turing_completeness
Of course, no physical system can have infinite memory; but if the
limitation of finite memory is ignored, most programming languages are
otherwise Turing-complete.

> I note that all your most recent replies are
> just dodges -- you don't repeat the lie explicitly, you just say
> something unrelated that you think sounds vaguely exculpatory.
>

Perhaps you don't understand what Turing complete means?

> The really sad part is the you never needed to lie in the first place.
> Having a logically impossible algorithm in pseudo-code would have been
> just as impressive a boast. In fact, everyone would have preferred it
> because Turing machines are so very tedious.
>

The only plausible way that I can see that you could still assert than I
lied is that you are clueless about what Turing Complete means.

>>>>> And you had previously claimed to have a TM that refuted Rice's theorem
>>>>> when you did not -- or do you still claim to have such a thing? In the
>>>>
>>>> Refuting the halting problem counter-example includes refuting Rice.
>>>
>>> Still ducking and diving I see. Do you still claim to have had such a
>>> machine when you first make that preposterous claim?
>
> Still dodging this question I see. Do you still claim to have had it?
>

I have almost completed my UTM such that the Turing Machine description
language is x86 machine language. I am implementing the operating system
context switch architecture of these UTMs such that one UTM executes
another UTM in a chain of arbitrary depth.

[Ben Bacarisse]

You don't. So what do think is happening? Has Mendelson said something
monstrously silly, or have you misunderstood what he's saying? Where do
you think any such thing is said, implied or required to be the case?

You keep asking the wrong sort of question. If you think Dr A. N. Umber
"goofed" in his seminal work "How to add" because 2+2 can't possibly be
5 you should not keep asking "how can 2+2 be 5?". You should ask "On
page 1562 he seems to suggest that 2+2 is 5. What have I
misunderstood?".

>> You are not the target audience, so while I agree that he could be
>> clearer if writing for someone like you, it's not a fair criticism of
>> the book because he isn't.
>>
>>> Moreover, instead of talking about the n-tuples of objects that
>>> satisfy a wf that has n free variables, it is much more convenient
>>> from a technical standpoint to deal uniformly with denumerable
>>> sequences [or finite sequences] such that 5-tuple <x27, x3, x99, x39,
>>> x3> would be referred to by a short-hand name such as y3.
>>
>> This, however, is junk. You have no idea how Σ is used so you should
>> not pretend to know how to "be clearer".
>
> I am proposing one very easy way of encoding a sort-hand notation for
> encoding sets of n-tuples.

You should be try to learn how to do it properly. The chances that you
can improve on Mendelson are slim. At any reate, I will ignore your
suggestions until you clearly know what Mendeson is doing. Then you can
tell the world how to do it more clearly from the point of view of
someone who at least knows what "it" is.

>>>>>>> Let Σ be the set of all denumerable sequences of elements of D.
>>>>>>>
>>>>>>> So when D is the set of humans the denumerable sequences of elements
>>>>>>> of D is the empty set,
>>>>>>
>>>>>> I can't see how you get that. A denumerable sequence of humans is an
>>>>>> endless sequence of humans.
>>>>>
>>>>> I thought that a domain in set theory was an entirey different word
>>>>> than the domain from algebra.
>>>>
>>>> It's a very widely used word. Functions have domains. Interpretations
>>>> have domains. There is even a highly technical meaning that is used in
>>>> denotational semantics of programming languages.
>>>>
>>>>> There is no such endless sequence of humans in the whole physical
>>>>> universe.
>>>>
>>>> You can have an endless sequence even if the range has one member.
>>>
>>> The bijection between the natural numbers and the elements of a finite
>>> set breaks when one reaches the size-of-set element of the set.
>>
>> There are no bijections between ℕ and any finite set.
>
> A set X is denumerable if it is equinumerous with the set of positive
> integers.

Sigh. Yes, that's a quote from someone who knows shit.

> A denumerable sequence is a function s whose domain is the set of
> positive integers;

Yup, another quote from someone who knows.

> instead of talking about the n-tuples of objects that satisfy a wf
> that has n free variables, it is much more convenient from a technical
> standpoint to deal uniformly with denumerable sequences.

And still the keep coming... Surely you will get round to saying
something wrong soon.

> When we satify the "father of" relation we do not have a denumerable
> sequence we have a finite sequence therefore his generalization of

We can choose to have a denumerable sequence if we want. You'd have to
read the damn book to see how.

> ...talking about the n-tuples of objects that satisfy a wf that has n
> free variables...
>
> using denumerable sequences is incorrect. In the case of the "father
> of" relation we have a finite sequence of ordered pairs where the
> first element of this pair comes from the set of every father that
> ever lived.

You'd have to read the book to find out how this works. But you've
decided he's wrong (without knowing what Σ is used for) and you can't
get past this block.

>> Why are you
>> stating this rather obvious fact? You must think it relates to my
>> remark "you can have an endless sequence even if the range has one
>> member" but I can't image how.
>
> The bijection does not endlessly continue it stops at one.

That's badly worded. Functions don't "stop". But you are right, there
is no bijection between ℕ and a singleton set. You've said it many
times. I've agreed many times. Yet I still stand by what I wrote (now
cut unfortunately). It was this:

You can have an endless sequence even if the range has one member.

so what do you suppose is going on? Could it be you've just imagined a
problem where one does not exist?

>>> To say that there is a complete bijection between the natural numbers
>>> and the elements of a finite set is psychotic.
>>
>> Yes, that would be daft, wouldn't it? So what should your first thought
>> be? Remember, you want to know what's going on.
>
> Yet another case where the terms of the art make sure to totally screw
> up the common meaning of these same terms. It is like saying that the
> medical term "dead" means {could not be more healthy}.

A sequence is a function from the positive integers to some other set
(called the range of the function). Finite sequences are functions from
{1, 2, ..., n} and infinite sequences (denumerable sequences) are
functions from ℕ to some other set. That other set, the range of the
function, can be a finite set. It can even be a singleton set (though
it can't be empty). Here's an example

s(n) = 42

s(1) is 42, s(2) is 42 and so on. We might write the sequence like
this: (42, 42, 42, ...).

Now I don't really care if you consider some of these words screwy, but
they seem perfectly reasonable to me. More importantly, it's
high-school maths (at least in the UK it is). Everyday I fond you need
to start further back than I thought.

>>>>>>> Let Σ be the set of all denumerable sequences of elements of D.
>>>>> Still makes no sense at all when D is finite.
>>>>
>>>> Can you image a function from ℕ to {true, false}? Maybe the function
>>>> p(n) that is true when n is prime? That is a denumerable sequence of
>>>> truth values. It starts
>>>>
>>>> p_n = (false, true, true, false, true, false, true, false, false, ....)
>>>
>>> No I can't imagine this. p(n) operates on individual constants.
>>
>> If you can't imagine a function from ℕ to {true, false}, you need
>> another hobby. Goodness knows what "constants" have to do with it.
>
> I can imagine a function from two elements of ℕ to each of the
> elements of True and False. I cannot imagine the entire set of ℕ maps
> to the two elements of true and false. I would map false to zero and
> true to 1.

You need a book far more elementary than Mendelson. You've been using
words like "function" as if you know what they mean (heck, you've been
doing that with "set") but you don't. When challenged, you pull a
quote from somewhere, but that just fools people like me into think you
know what the quote means.

>> Remember, constants form a syntactic category in the language of some
>> theory. You are using the word incorrectly for this context, but I
>> don't think you can change.
>

> A function is a relation that...

Oh, talk of the Devil!

> ... uniquely associates members of one set

> with members of another set. More formally, a function from A to B is
> an object f such that every a ∈ A is uniquely associated with an
> object f(a) ∈ B. https://mathworld.wolfram.com/Function.html
>
> every a ∈ ℕ is uniquely associated with an object f(a) ∈ {true,
> false}.

Do you think that "uniquely associated" means that if, say, f(1) = true,
f(2) can't also be true? Do you think that sin(x) is not a function
because sin(x) = 1 for so many values of x?

--
Ben.

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> On 8/7/2020 6:20 PM, Ben Bacarisse wrote:
>> olcott <No...@NoWhere.com> writes:
>>
>>> On 8/7/2020 5:41 PM, Ben Bacarisse wrote:
>>>> olcott <No...@NoWhere.com> writes:
>>>>
>>>>> On 8/7/2020 11:12 AM, Ben Bacarisse wrote:
>>>>>> olcott <No...@NoWhere.com> writes:
>>>> <cut>
>>>>>>> I am really very sincerely devoted to the truth. This requires me to
>>>>>>> acknowledge my repeated mistake of calling you a liar.
>>>>>>
>>>>>> This is not consistent with your behaviour elsewhere. You claimed to
>>>>>> have had an "actual Turing machine" "fully encoded" when you did not.
>>>>>
>>>>> My actual Turing Machine was more fully encoded than any Turing
>>>>> machine that Turing himself ever wrote.
>>>>
>>>> Ah, you are back to lying about it. Shame.
>>>
>>> I am not aware of Turing ever writing any actual code, therefore
>>> pseudo-code that directly maps to "c" code would be more fully encoded
>>> than anything Turing ever wrote.
>>
>> You don't have, and never have had, anything that could reasonably be
>> called a Turing machine.
>
> The x86 architecture is as close as any physical machine gets to
> Turing complete. https://en.wikipedia.org/wiki/Von_Neumann_architecture

Evasion noted.

What is a Turing machine? (I predict you will ignore this question or
give a dishonest reply.)

>> I note that all your most recent replies are
>> just dodges -- you don't repeat the lie explicitly, you just say
>> something unrelated that you think sounds vaguely exculpatory.
>
> Perhaps you don't understand what Turing complete means?

Evasion noted. What is a Turing machine? How would you know you had
one rather than a Meeley machine or a Post system or lambda form or a
Game of Life configuration?

>> The really sad part is the you never needed to lie in the first place.
>> Having a logically impossible algorithm in pseudo-code would have been
>> just as impressive a boast. In fact, everyone would have preferred it
>> because Turing machines are so very tedious.
>
> The only plausible way that I can see that you could still assert than
> I lied is that you are clueless about what Turing Complete means.

You lied by saying something untrue. Whatever is was you had, or
thought you had, it was not a Turing machine.

>>>>>> And you had previously claimed to have a TM that refuted Rice's theorem
>>>>>> when you did not -- or do you still claim to have such a thing? In the
>>>>>
>>>>> Refuting the halting problem counter-example includes refuting Rice.
>>>>
>>>> Still ducking and diving I see. Do you still claim to have had such a
>>>> machine when you first make that preposterous claim?
>>
>> Still dodging this question I see. Do you still claim to have had it?
>
> I have almost completed my UTM such that the Turing Machine
> description language is x86 machine language. I am implementing the
> operating system context switch architecture of these UTMs such that
> one UTM executes another UTM in a chain of arbitrary depth.

Still dodging the question! Do you still claim to have had the TM that
"refutes Rice" when you said you did? I've asked and asked but you dare
not answer. You don't want to lie again, but you also don't want admit
that it was lie when you first claimed it.

--
Ben.

[olcott]

I am far too frustrated to do it that way.

Sure we can all we have to do is by pure magic create an infinite set of
humans.

> You'd have to
> read the damn book to see how.
>
>> ...talking about the n-tuples of objects that satisfy a wf that has n
>> free variables...
>>
>> using denumerable sequences is incorrect. In the case of the "father
>> of" relation we have a finite sequence of ordered pairs where the
>> first element of this pair comes from the set of every father that
>> ever lived.
>
> You'd have to read the book to find out how this works. But you've
> decided he's wrong (without knowing what Σ is used for) and you can't
> get past this block.
>



>>> Why are you
>>> stating this rather obvious fact? You must think it relates to my
>>> remark "you can have an endless sequence even if the range has one
>>> member" but I can't image how.
>>
>> The bijection does not endlessly continue it stops at one.
>
> That's badly worded. Functions don't "stop". But you are right, there
> is no bijection between ℕ and a singleton set. You've said it many
> times. I've agreed many times. Yet I still stand by what I wrote (now
> cut unfortunately). It was this:
>
> You can have an endless sequence even if the range has one member.

How would this work? An endless sequence of one singleton is self
contradictory.

> so what do you suppose is going on? Could it be you've just imagined a
> problem where one does not exist?
>
>>>> To say that there is a complete bijection between the natural numbers
>>>> and the elements of a finite set is psychotic.
>>>
>>> Yes, that would be daft, wouldn't it? So what should your first thought
>>> be? Remember, you want to know what's going on.
>>
>> Yet another case where the terms of the art make sure to totally screw
>> up the common meaning of these same terms. It is like saying that the
>> medical term "dead" means {could not be more healthy}.
>
> A sequence is a function from the positive integers to some other set
> (called the range of the function). Finite sequences are functions from
> {1, 2, ..., n} and infinite sequences (denumerable sequences) are
> functions from ℕ to some other set.

That part makes sense.

> That other set, the range of the
> function, can be a finite set. It can even be a singleton set (though
> it can't be empty). Here's an example
>
> s(n) = 42
>
> s(1) is 42, s(2) is 42 and so on. We might write the sequence like
> this: (42, 42, 42, ...).

in "c"

int f(int N)
{
return 42;
}

That is exactly the kind of concrete example explanation that I will
most often understand very quickly. Without a concrete example nailing
down exactly what is meant I many ne3ver understand.

> Now I don't really care if you consider some of these words screwy, but
> they seem perfectly reasonable to me. More importantly, it's
> high-school maths (at least in the UK it is). Everyday I fond you need
> to start further back than I thought.

As long as it is not incoherent in ways such as describing infinite sets
that are finite I can deal with it.

>>>>>>>> Let Σ be the set of all denumerable sequences of elements of D.
>>>>>> Still makes no sense at all when D is finite.
>>>>>
>>>>> Can you image a function from ℕ to {true, false}? Maybe the function
>>>>> p(n) that is true when n is prime? That is a denumerable sequence of
>>>>> truth values. It starts
>>>>>
>>>>> p_n = (false, true, true, false, true, false, true, false, false, ....)
>>>>
>>>> No I can't imagine this. p(n) operates on individual constants.
>>>
>>> If you can't imagine a function from ℕ to {true, false}, you need
>>> another hobby. Goodness knows what "constants" have to do with it.
>>
>> I can imagine a function from two elements of ℕ to each of the
>> elements of True and False. I cannot imagine the entire set of ℕ maps
>> to the two elements of true and false. I would map false to zero and
>> true to 1.
>
> You need a book far more elementary than Mendelson. You've been using
> words like "function" as if you know what they mean (heck, you've been
> doing that with "set") but you don't. When challenged, you pull a
> quote from somewhere, but that just fools people like me into think you
> know what the quote means.
>

Yes I need all the terms of the art fully defined using ordinary words.
I can handle a basic set of terms defined using ordinary words and then
use more ordinary words to build on these terms of the art.

>>> Remember, constants form a syntactic category in the language of some
>>> theory. You are using the word incorrectly for this context, but I
>>> don't think you can change.
>>
>> A function is a relation that...
>
> Oh, talk of the Devil!
>
>> ... uniquely associates members of one set
>> with members of another set. More formally, a function from A to B is
>> an object f such that every a ∈ A is uniquely associated with an
>> object f(a) ∈ B. https://mathworld.wolfram.com/Function.html
>>
>> every a ∈ ℕ is uniquely associated with an object f(a) ∈ {true,
>> false}.
>
> Do you think that "uniquely associated" means that if, say, f(1) = true,
> f(2) can't also be true? Do you think that sin(x) is not a function
> because sin(x) = 1 for so many values of x?
>

So a function is not a one-to-one bidirectional mapping?

a function from A to B is an object f such that every a ∈ A is uniquely
associated with an object f(a) ∈ B.

https://mathworld.wolfram.com/Function.html

[olcott]

Any machine or computer language that is Turing Complete (ignoring the
fact that no physical machine has unlimited memory) is sufficiciently a
Turing Machine.

>>> I note that all your most recent replies are
>>> just dodges -- you don't repeat the lie explicitly, you just say
>>> something unrelated that you think sounds vaguely exculpatory.
>>
>> Perhaps you don't understand what Turing complete means?
>
> Evasion noted. What is a Turing machine? How would you know you had
> one rather than a Meeley machine or a Post system or lambda form or a
> Game of Life configuration?
>

As long as a machine is computationally equivlent to a Turing Machine
(when the lack of unlimited memory is ignored) it is sufficiently a
Turing Machine.

>>> The really sad part is the you never needed to lie in the first place.
>>> Having a logically impossible algorithm in pseudo-code would have been
>>> just as impressive a boast. In fact, everyone would have preferred it
>>> because Turing machines are so very tedious.
>>
>> The only plausible way that I can see that you could still assert than
>> I lied is that you are clueless about what Turing Complete means.
>
> You lied by saying something untrue. Whatever is was you had, or
> thought you had, it was not a Turing machine.

The problem seems to be on you side of failing to comprehend
computational equivalence.

>>>>>>> And you had previously claimed to have a TM that refuted Rice's theorem
>>>>>>> when you did not -- or do you still claim to have such a thing? In the
>>>>>>
>>>>>> Refuting the halting problem counter-example includes refuting Rice.
>>>>>
>>>>> Still ducking and diving I see. Do you still claim to have had such a
>>>>> machine when you first make that preposterous claim?
>>>
>>> Still dodging this question I see. Do you still claim to have had it?
>>
>> I have almost completed my UTM such that the Turing Machine
>> description language is x86 machine language. I am implementing the
>> operating system context switch architecture of these UTMs such that
>> one UTM executes another UTM in a chain of arbitrary depth.
>
> Still dodging the question! Do you still claim to have had the TM that
> "refutes Rice" when you said you did? I've asked and asked but you dare
> not answer. You don't want to lie again, but you also don't want admit
> that it was lie when you first claimed it.
>

Do you understand that refuting the halting problem proof semantically
includes refuting Rice because Rice depends upon the halting problem
proof as the key aspect of its whole basis?

[André G. Isaak]

On 2020-08-07 13:46, olcott wrote:
> On 8/7/2020 1:24 PM, André G. Isaak wrote:
>> On 2020-08-07 11:40, olcott wrote:
>>> On 8/7/2020 12:13 PM, André G. Isaak wrote:
>>>> On 2020-08-07 10:48, olcott wrote:
>>>>
>>>>> My actual Turing Machine was more fully encoded than any Turing
>>>>> machine that Turing himself ever wrote.
>>>>
>>>> What does that even mean? You obviously have some new and entirely
>>>> idiosyncratic meaning for the word 'encoded'. How does encode a
>>>> Turing Machine more filly than Turing did?
>>>>
>>>> André
>>>
>>> If I write pseudo-code that can be directly translated into fully
>>> operational software this pseudo-code has all of the details of the
>>> required algorithm fully specified, thus is fully encoded.
>>
>> What is any of the above supposed to mean? A Turing Machine is an
>> abstract mathematical model of computation. It is not software.
>> Writing something in pseudo-code is not a Turing Machine.
>>
>
> Turing machines have been fully implemented in software as software.
> http://www.lns.mit.edu/~dsw/turing/turing.html

That is a program which interprets Turing Machines. It is not a Turing
Machine any more than a program which graphs polynomials is a polynomial.

> Thus the abstraction has been empirically proven to be able to be made
> perfectly concrete.
>
>> A Turing Machine is specified by a 7-tuple <Q, Γ, b, Σ, δ, q₀, F>. If
>> what you have is anything other than such a 7-tuple, it is not a
>> Turing Machine, 'fully encoded' or otherwise.
>>
> It is equivalent.

Equivalent how? Unless you can express them as a 7-tuple as above, you
are not dealing with Turing Machines.

Do you have such a 7-tuple?

>>> To the best of my understanding Turing never wrote any details of any
>>> algorithm using his concept of a Turing machine. He merely specified
>>> the underlying infrastructure that could be used for specifying
>>> algorithms.
>>
>> Of course Turing dealt with actual Turing Machines.
>>
>
> How many programs did he write in the Turing Machine description language?

That question is non-sensical. A Turing Machine description language is
a language which describes Turing Machines, not a programming language.
Turing Machines aren't programs.

[Ben Bacarisse]

olcott <No...@NoWhere.com> writes:

> On 8/7/2020 7:20 PM, Ben Bacarisse wrote:
>> olcott <No...@NoWhere.com> writes:

<cut>

>>> How the Hell do you form a bijection between the set of natural
>>> numbers and the elements of a finite set?
>>
>> You don't. So what do think is happening? Has Mendelson said something
>> monstrously silly, or have you misunderstood what he's saying? Where do
>> you think any such thing is said, implied or required to be the case?
>>
>> You keep asking the wrong sort of question. If you think Dr A. N. Umber
>> "goofed" in his seminal work "How to add" because 2+2 can't possibly be
>> 5 you should not keep asking "how can 2+2 be 5?". You should ask "On
>> page 1562 he seems to suggest that 2+2 is 5. What have I
>> misunderstood?".
>
> I am far too frustrated to do it that way.

I am suggesting a change of plan because the current plan is not
working.

>>> When we satify the "father of" relation we do not have a denumerable
>>> sequence we have a finite sequence therefore his generalization of
>>
>> We can choose to have a denumerable sequence if we want.
>
> Sure we can all we have to do is by pure magic create an infinite set
> of humans.

A function isn't always bijective. Lots of functions exist that map
every member of ℕ to some member of the human race. There is even a
function that maps every member of ℕ to just me.

>>> Yet another case where the terms of the art make sure to totally screw
>>> up the common meaning of these same terms. It is like saying that the
>>> medical term "dead" means {could not be more healthy}.
>>
>> A sequence is a function from the positive integers to some other set
>> (called the range of the function). Finite sequences are functions from
>> {1, 2, ..., n} and infinite sequences (denumerable sequences) are
>> functions from ℕ to some other set.
>
> That part makes sense.
>
>> That other set, the range of the
>> function, can be a finite set. It can even be a singleton set (though
>> it can't be empty). Here's an example
>>
>> s(n) = 42
>>
>> s(1) is 42, s(2) is 42 and so on. We might write the sequence like
>> this: (42, 42, 42, ...).
>
> in "c"
>
> int f(int N)
> {
> return 42;
> }

Except that's not a function from ℕ to {42}. A better concrete example
would be Haskell:

f n = 42

The domain of this f is not defined to be finite, as is the case for the
C function.

>> Now I don't really care if you consider some of these words screwy, but
>> they seem perfectly reasonable to me. More importantly, it's
>> high-school maths (at least in the UK it is). Everyday I fond you need
>> to start further back than I thought.
>
> As long as it is not incoherent in ways such as describing infinite
> sets that are finite I can deal with it.

But you are having trouble even when no one has described infinite sets
that are finite.

>>> A function is a relation that...
>>
>> Oh, talk of the Devil!
>>
>>> ... uniquely associates members of one set
>>> with members of another set. More formally, a function from A to B is
>>> an object f such that every a ∈ A is uniquely associated with an
>>> object f(a) ∈ B. https://mathworld.wolfram.com/Function.html
>>>
>>> every a ∈ ℕ is uniquely associated with an object f(a) ∈ {true,
>>> false}.
>>
>> Do you think that "uniquely associated" means that if, say, f(1) = true,
>> f(2) can't also be true? Do you think that sin(x) is not a function
>> because sin(x) = 1 for so many values of x?
>
> So a function is not a one-to-one bidirectional mapping?

No it isn't (always assuming that you know what those other words mean).

> a function from A to B is an object f such that every a ∈ A is
> uniquely associated with an object f(a) ∈ B.
>
> https://mathworld.wolfram.com/Function.html

There's no question here so all I can say is that that's no worse than
many other ways of putting it. Formally a function f: A → B is a subset
of AxB such that (a, b1) ∈ f & (a, b2) ∈ f ⇒ b1 = b2 (and also that ∀a∃b
(a, b) ∈ f but that's not at issue.)

--
Ben.

[Keith Thompson]

olcott <No...@NoWhere.com> writes:
> On 8/7/2020 5:41 PM, Ben Bacarisse wrote:
>> olcott <No...@NoWhere.com> writes:

[...]

>>> To say that there is a complete bijection between the natural numbers
>>> and the elements of a finite set is psychotic.
>>
>> Yes, that would be daft, wouldn't it? So what should your first thought
>> be? Remember, you want to know what's going on.
>>
>
> Yet another case where the terms of the art make sure to totally screw
> up the common meaning of these same terms. It is like saying that the
> medical term "dead" means {could not be more healthy}.

No, it's not like that at all.

Nobody said that there is a complete bijection between the natural
numbers and the elements of a finite set. You can stop refuting that
claim that nobody made.

[...]

> I can imagine a function from two elements of ℕ to each of the
> elements of True and False.

Good.

> I cannot imagine the entire set of ℕ maps
> to the two elements of true and false. I would map false to zero and
> true to 1.

Sure you can. Here's an example in C++ code (think of it as pseudocode,
and think of "int" as representing the entire set ℕ rather than a finite
subrange as "int" does in C++):

bool func(int i) {
return i % 2 == 0;
}

Behold, a function that maps the entire set ℕ to the two elements true
and false.

Here's another:

bool func(int i) {
return i != 0;
}

No, it's not a bijection. Nobody said that it was, or that it should be.

[...]

> A function is a relation that uniquely associates members of one set
> with members of another set. More formally, a function from A to B is
> an object f such that every a ∈ A is uniquely associated with an
> object f(a) ∈ B. https://mathworld.wolfram.com/Function.html
>
> every a ∈ ℕ is uniquely associated with an object f(a) ∈ {true, false}.

You've misunderstood the way the word "uniquely" is being used here.
Is sin a function?

[André G. Isaak]

This is nonsense. A Turing Machine refers to a very specific
mathematical model of computation. Being computationally equivalent to a
Turing Machine is not the same as being a Turing Machine.

If I were to demonstrate that for any arbitrary Pascal program there is
an equivalent C program, that doesn't mean that a Pascal program can be
called a C program.

A Turing machine, *by definition* is a very specific type of
mathematical object consisting of a 7-tuple <Q, Γ, b, Σ, δ, q₀, F>.
Claiming something is equivalent to a Turing Machine doesn't magically
turn it into a 7-tuple of the relevant sort. It is therefore not a
Turing Machine.

[Ben Bacarisse]

Not an answer! I did not ask what is "sufficiently a Turing machine"
that you can kid yourself you didn't lie, I asked what a Turing machine
actually is. How would you define a Turing machine to some innocent
student of the theory of computation?

>> You lied by saying something untrue. Whatever is was you had, or
>> thought you had, it was not a Turing machine.
>
> The problem seems to be on you side of failing to comprehend
> computational equivalence.

The problem is you lying about having an "actual Turing machine" when
you did not. (And don't think for one moment that I believe you are
being sincere when you suggest that I might not comprehend computational
equivalence. I know that you know that I know more about it than you
do.)

>>>>>>>> And you had previously claimed to have a TM that refuted Rice's theorem
>>>>>>>> when you did not -- or do you still claim to have such a thing? In the
>>>>>>>
>>>>>>> Refuting the halting problem counter-example includes refuting Rice.
>>>>>>
>>>>>> Still ducking and diving I see. Do you still claim to have had such a
>>>>>> machine when you first make that preposterous claim?
>>>>
>>>> Still dodging this question I see. Do you still claim to have had it?
>>>
>>> I have almost completed my UTM such that the Turing Machine
>>> description language is x86 machine language. I am implementing the
>>> operating system context switch architecture of these UTMs such that
>>> one UTM executes another UTM in a chain of arbitrary depth.
>>
>> Still dodging the question! Do you still claim to have had the TM that
>> "refutes Rice" when you said you did? I've asked and asked but you dare
>> not answer. You don't want to lie again, but you also don't want admit
>> that it was lie when you first claimed it.
>
> Do you understand that refuting the halting problem proof semantically
> includes refuting Rice because Rice depends upon the halting problem
> proof as the key aspect of its whole basis?

Still dodging! You claimed to have this other Turing machine some time
before Dec 15th 2018. Do you still claim to have had this earlier one
/and/ the one you lied about having on Dec 15th 2018?

--
Ben.

[Ben Bacarisse]

André G. Isaak <agi...@gm.invalid> writes:

> On 2020-08-07 19:06, olcott wrote:

<cut>

>> As long as a machine is computationally equivlent to a Turing Machine
>> (when the lack of unlimited memory is ignored) it is sufficiently a
>> Turing Machine.
>
> This is nonsense. A Turing Machine refers to a very specific
> mathematical model of computation. Being computationally equivalent to
> a Turing Machine is not the same as being a Turing Machine.

Some context...

In December 2018 we were talking about Linz's proof of the undeciability
of TM halting. Linz gives a very specific transformation from one TM to
another. It's the usual construction that manipulates the states of one
TM, H, to get another he calls H^. It was in this context that PO
posted:

"I now have an actual H that decides actual halting for an actual (Ĥ,
Ĥ) input pair. I have to write the UTM to execute this code, that
should not take very long. The key thing is the H and Ĥ are 100%
fully encoded as actual Turing machines."

As you can see, he is as emphatic as he can possibly be. Add to that
the context, and how H and H^ are defined in Linz, and there is no
wriggle room at all. He was furiously backpedalling come January but he
never retracted the claim.

--
Ben.

[olcott]

It seems that you have never heard of this:
https://en.wikipedia.org/wiki/Turing_completeness