What are the Intuitive Examples of the Liar Paradox?

[Charlie-Boo]

What are the Intuitive Examples (a.k.a. IEs) of the Liar Paradox? What do we want to formalize and evaluate? For starters:

1. This is false.
2. ???

(That is ALWAYS the first step in formalizing. It gives us a gauge as to the success of any proposal, something missing from virtually all accounts. Someone asked on FOM if anyone had ever described a gauge.)

C-B

[Peter Percival]

Charlie-Boo wrote:
>
> What are the Intuitive Examples (a.k.a. IEs) of the Liar Paradox? What do we want to formalize and evaluate? For starters:
>
> 1. This is false.
> 2. ???

One is enough. Elsewhere I referred to Monk's text. He gives two ways
of interpreting The Liar. Interested parties may like to look at Thm
15.20 and the remark following it, and Ex 17.14. That's pages 275,276,308.

> (That is ALWAYS the first step in formalizing. It gives us a gauge as to the success of any proposal, something missing from virtually all accounts.

What accounts are those?

> Someone asked on FOM if anyone had ever described a gauge.)
>
> C-B
>

--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan

[peterolcott]

https://en.wikipedia.org/wiki/No_Silver_Bullet#Summary
The Truth Teller Paradox, eliminates inessential complexity from the analysis of the Liar Paradox.

s = "This sentence is true."

When we formalize this using Tarski
https://plato.stanford.edu/entries/tarski-truth/#ForCor
"This sentence" becomes a variable representing the entire sentence: x.

"true" becomes True(x)

"is" can be represented by many things, some are more accurately descriptive than others.

https://plato.stanford.edu/entries/model-theory/#Cons
https://en.wikipedia.org/wiki/Logical_consequence#Semantic_consequence
https://en.wikipedia.org/wiki/Satisfiability#Satisfiability_in_model_theory

x is True(x)
x ⊨ True(x)
x ⊢ True(x)

[X.Y. Newberry]

I would start here:
http://www.columbia.edu/~hg17/gaifman6.pdf

[Charlie-Boo]

That's the same mistake that leads to the Liar Paradox in the first place. The Liar Paradox occurred because man developed a notion of truth and proof (propositional calculus then predicate calculus) without reference to "This is false." If you consider other instances of self reference, you will see many more results (e.g. proofs) and additional constructions needed to represent them.

It also gives us a much bigger set of examples to look through for patterns. And patterns there are!

Look up Uniform Resolution to the Liar Paradox.

Considering siblings is an excellent way to shore up ones understanding of any concept!

See the 7 entries under Possible Resolutions at https://en.wikipedia.org/wiki/Liar_paradox

C-B

[Charlie-Boo]

No, not at all. He gives only one example:

Line 1 The sentence on line 1 is not true.
Line 2 The sentence on line 1 is not true.

Actually, his not considering things like,

This is this.

(which nobody considers - try to find a single Google result with it.)

If he did, he would see that being groundless does NOT make a sentence unevaluatable. He is truly throwing the baby out with the bath water.

There are many more. (See the dozen I refer to in my posts.)

Don't be fooled by the length of the paper or its apparent complexity. In the final analysis, it's BS like all the rest. They don't know the answer - AT ALL!!

C-B

[Peter Percival]

Charlie-Boo wrote:
> On Wednesday, February 8, 2017 at 11:28:58 AM UTC-5, Newberry wrote:
>> Charlie-Boo wrote:
>>>
>>> What are the Intuitive Examples (a.k.a. IEs) of the Liar Paradox? What do we want to formalize and evaluate? For starters:
>>>
>>> 1. This is false.
>>> 2. ???
>>>
>>> (That is ALWAYS the first step in formalizing. It gives us a gauge as to the success of any proposal, something missing from virtually all accounts. Someone asked on FOM if anyone had ever described a gauge.)
>>>
>>> C-B
>>>
>>
>> I would start here:
>> http://www.columbia.edu/~hg17/gaifman6.pdf
>
> No, not at all. He gives only one example:
>
> Line 1 The sentence on line 1 is not true.
> Line 2 The sentence on line 1 is not true.
>
> Actually, his not considering things like,
>
> This is this.
>
> (which nobody considers - try to find a single Google result with it.

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

> )
>
> If he did, he would see that being groundless does NOT make a sentence unevaluatable. He is truly throwing the baby out with the bath water.
>
> There are many more. (See the dozen I refer to in my posts.)
>
> Don't be fooled by the length of the paper or its apparent complexity. In the final analysis, it's BS like all the rest. They don't know the answer - AT ALL!!
>
> C-B
>

[Charlie-Boo]

What does https://en.wikipedia.org/wiki/Law_of_identity have to do with any of this? Looks like a totally BS reference.

C-B

[Peter Percival]

Charlie-Boo wrote:

>
> What does https://en.wikipedia.org/wiki/Law_of_identity have to do with any of this? Looks like a totally BS reference.

You mentioned "this is this" which is an instance of the law of identity.

[Charlie-Boo]

I should be glad I didn't have to buy a BS book from Amazon to prove it!

This illustrates the fallacy of requiring someone to obtain an elusive reference instead of simply writing out the result. Obfuscation!

C-B

[Charlie-Boo]

Here's the context:

ME: "Actually, his not considering things like, This is this. (which nobody considers - try to find a single Google result with it.

YOU: "https://en.wikipedia.org/wiki/Law_of_identity"

It sounds like you're claiming that you found a Google result with it, doesn't it?

And who said we have a problem with the law of identity?

You know, when they can't use logic or obfuscation, and they're one of those who would never admit the truth, this is what you get: irrelevant (read: BS) references.

C-B

[X.Y. Newberry]

Have you found anything wrong with his argument?

[Charlie-Boo]

Yes. He says that if a sentence refers to a sentence then you must go to the sentence referred to and evaluate it, and that if you get into a closed loop then it is not true. However, “This is this.” has a closed loop but it is in fact true.

C-B

[Ross A. Finlayson]

OT: Does FOM have a portal these days? It's moved several times.

(FOM is a moderated academic "Foundations of Mathematics" e-mail reflector.)

[X.Y. Newberry]

"Had the instruction been non-semantic, say to count the number of words
in the sentence (‘The sentence on line 1 has an odd number of words’) or
to perform an orthographic check (‘The sentence on line 1 contains no
misspellings’), there would have been no loop and no paradox." (page 3)

>
> C-B
>

[Charlie-Boo]

Of course. So?

He will never see that "This is this." is true, or "This is true." is false etc.

C-B

> >
> > C-B
> >

[Ross A. Finlayson]

( http://www.cs.nyu.edu/pipermail/fom/ -- seems it's returned )

[Dan Christensen]

On Wednesday, February 8, 2017 at 2:03:44 PM UTC-5, Charlie-Boo wrote:
> On Wednesday, February 8, 2017 at 10:26:29 AM UTC-5, Peter Percival wrote:
> > Charlie-Boo wrote:
> > >
> > > What are the Intuitive Examples (a.k.a. IEs) of the Liar Paradox? What do we want to formalize and evaluate? For starters:
> > >
> > > 1. This is false.
> > > 2. ???
> >
> > One is enough. Elsewhere I referred to Monk's text. He gives two ways
> > of interpreting The Liar. Interested parties may like to look at Thm
> > 15.20 and the remark following it, and Ex 17.14. That's pages 275,276,308.
> >
> > > (That is ALWAYS the first step in formalizing. It gives us a gauge as to the success of any proposal, something missing from virtually all accounts.
> >
> > What accounts are those?
> >
> > > Someone asked on FOM if anyone had ever described a gauge.)
> > >
> > > C-B
> > >
> >
> >
> > --
> > Do, as a concession to my poor wits, Lord Darlington, just explain
> > to me what you really mean.
> > I think I had better not, Duchess. Nowadays to be intelligible is
> > to be found out. -- Oscar Wilde, Lady Windermere's Fan
>
> That's the same mistake that leads to the Liar Paradox in the first place. The Liar Paradox occurred because man developed a notion of truth and proof (propositional calculus then predicate calculus) without reference to "This is false." If you consider other instances of self reference, you will see many more results (e.g. proofs) and additional constructions needed to represent them.
>

The original liar paradox, supposedly from the Cretan poet Epimenides ("Cretans always lie"), is easily resolved: His self-referential, anti-Cretan rant was a lie and, at the very least, one Cretan once told the truth. (See "The original liar paradox" including formal proof at my math blog.)

A more politically correct version, "This sentence is false" has no straightforward resolution other than to simply dismiss it as nonsense. Likewise "This sentence is true." For better or worse, it's an issue that just doesn't seem to concern the vast majority of mathematicians. After thousands of years, however, it still seems to whip philosophers into a frenzy.


Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

[X.Y. Newberry]

"This sentence is not true" goes into infinite loop. "This sentence has
five words" does not. "This is this" is an atypical sentence but I would
put it in the same category as the latter.

My guess is that Gaifman would see it the same way. But the bottom line
is that he does not need to know anything about "This is this" in order
to analyze the liar paradox. Nor do I think that analyzing "This is
this" is very illuminating.

>
> C-B
>
>>>
>>> C-B
>>>
>

[peterolcott]

On Wednesday, February 8, 2017 at 9:09:12 AM UTC-6, Charlie-Boo wrote:

This sentence is false.
This sentence is not true.

Any other examples add unnecessary complexity to the analysis and this thwarts clear understanding.

When I formalize this using Tarski:True(x) I have to use ~True(x) because Tarski never defined False(x).

x.hasProperty(~True(x)) // This sentence is not true. (in OOP)

Here it is in quantified second order logic:
∃x ∈ Predicates ∃P ∈ Properties | x ↔ ( P(x) & ~True(x) )

Since the language of predicate logic is far more clumsy and far less self-descriptive I am still not sure whether or not it accurately specifies the OOP notational conventions that I specified above.

My original way of saying it is more semantically correct yet incomprehensible within typical logical notational conventions:

x ⊨ ~True(x)
x.Logically-Entails(~True(x))

This is a new way of saying it.
True(x) ↦ Boolean.True
True(x) <mathematically maps to> Boolean.True

[Peter Percival]

Charlie-Boo wrote:

> Look up Uniform Resolution to the Liar Paradox.

I gave Google "Uniform Resolution to the Liar Paradox" (sans quote
marks) and got 1,440,000 results. What should I do with them? I tried
again with the quote marks and was told "No results found".

> Considering siblings is an excellent way to shore up ones understanding of any concept!
>
> See the 7 entries under Possible Resolutions at https://en.wikipedia.org/wiki/Liar_paradox

I searched for the word "uniform" in that article and it wasn't found.
Mind you, it's not all negative, I came upon mention of Bradley
Armour-Garb. I've no idea who he is, but I like the surname!

[Charlie-Boo]

He's presenting a general algorithm to evaluate sentence, that fails. In fact, if he's only interested in one sentence, what's the use of a general algorithm?

So he develops this big messy algorithm (with lots of kludges and exceptions) but then says that it's not guaranteed to work? Hmmm ...

C-B

[Charlie-Boo]

If "This is this." doesn't go into an infinite loop then where does it go - what would be the result?

The problem is he (and many others) say that if it doesn't ultimately refer to something other than itself, then it is "ungrounded" and meaningless. Long ago I considered how a statement about only itself could be know to be true or to be false, and I came up with "This is this." and "This is not this." Judging from the lack of any Google results of a paradoxical "THIS IS THIS" or PARADOX "THIS IS THIS", he is not alone in missing these points.

If you really consider the possibilities when you propose a grand scheme to resolve paradoxes surrounding natural language statements, you very quickly go from "This is false." or "This is not true." to "True is true." (TRUE), "This is this." (TRUE) and "This is not true." (FALSE)

But the real answer to all of this is to use ordinary mathematics to evaluate the problem, and there are at least a dozen places where Mathematicians have already done that.

I can only wonder what people would think when confronted with that fact, that, after all these years, the truth was staring us in the face the whole time?

C-B

[Charlie-Boo]

Oh my goodness. What is the Scientific Method all about?

Isn't that being a wee bit like an ostrich?

If you wanted to determine a formula for an infinite series of numbers, would you rather know the first 3 or the first 10?

C-B

[Charlie-Boo]

The first relevant result I found:

https://antimeta.wordpress.com/2007/05/22/five-days-of-formal-philosophy-and-uniform-solutions/

C-B

[peterolcott]

That is great that people are referencing the exact verbiage of my original paper. That is showing that my ideas are not actually falling on deaf ears.

[This sentence has five words]
https://philpapers.org/archive/PETFDS-3.pdf

[Peter Percival]

Charlie-Boo wrote:
> On Friday, February 10, 2017 at 10:55:19 AM UTC-5, Peter Percival wrote:
>> Charlie-Boo wrote:
>>
>>> Look up Uniform Resolution to the Liar Paradox.
>>
>> I gave Google "Uniform Resolution to the Liar Paradox" (sans quote
>> marks) and got 1,440,000 results. What should I do with them? I tried
>> again with the quote marks and was told "No results found".
>>
>>> Considering siblings is an excellent way to shore up ones understanding of any concept!
>>>
>>> See the 7 entries under Possible Resolutions at https://en.wikipedia.org/wiki/Liar_paradox
>>
>> I searched for the word "uniform" in that article and it wasn't found.
>> Mind you, it's not all negative, I came upon mention of Bradley
>> Armour-Garb. I've no idea who he is, but I like the surname!

> The first relevant result I found:
>
> https://antimeta.wordpress.com/2007/05/22/five-days-of-formal-philosophy-and-uniform-solutions/

Thank you.

[Peter Percival]

X.Y. Newberry wrote:

> "This sentence is not true" goes into infinite loop. "This sentence has
> five words" does not. "This is this" is an atypical sentence but I would
> put it in the same category as the latter.

For me, "This is this" is an instance of the law of identity and is
therefore true.

[X.Y. Newberry]

When I suggested that since the recursion theorem does not hold for

The sentence "This sentence is not true" is not true <--> This
sentence is not true

then it ought not to hold for

~(Ex)Prf(x,<#G#>) <--> ~(Ex)(Ey)(Prf(x,y) & This(y))

or

T(<R>,w) halts <--> R(w) halts

you objected that they were all different.

>
> C-B
>

[peterolcott]

On Friday, February 10, 2017 at 4:20:32 PM UTC-6, Peter Percival wrote:
> X.Y. Newberry wrote:
>
> > "This sentence is not true" goes into infinite loop. "This sentence has
> > five words" does not. "This is this" is an atypical sentence but I would
> > put it in the same category as the latter.
>
> For me, "This is this" is an instance of the law of identity and is
> therefore true.
>
>

I would agree 100%.
"This sentence is itself."

[peterolcott]

I am guessing that you mean this one:
https://en.wikipedia.org/wiki/Kleene%27s_recursion_theorem

I would agree that they are all essentially the same thus the correct resolution to anyone of them resolves all three.

https://antimeta.wordpress.com/2007/05/22/five-days-of-formal-philosophy-and-uniform-solutions/

What I am proposing is the "principle of uniform solution"
in that every paradox is semantically ill-formed in one way or another.

The test for semantic well-formedness of an atomic proposition is (informally) two steps:
(1) Proposition P ∈ Relations // Finally a clear way to say it !!!

(2) Proposition P ↔ Relation R and (informally) all of the arguments to R are semantically compatible with R. // I will figure out a formal way to say this.

The comments indicate several "solutions" to the Liar Paradox, yet I have never encountered any. If my understanding is correct Kripke (1975) cited Tarski's infinite hierarchy of languages as an approach.

I have never seen any solution the unravels the Liar Paradox like my own:
x ↔ HasProperty( x, Tarski:True(x) )

when evaluated using Tarski:True(x) becomes:

Tarski:True( HasProperty( x, Tarski:True(x) ) )

[Ross A. Finlayson]

It seems there's just a difference: constants, terms, and expressions then
as of the references and self-contained references, there are many available
tools usually of the classical (as the classical as the perfect must be accommodated), about then what purpose there is about extending the extra,
or, supporting the extra, here that the "extra" is the non-classical and
the purpose is completion.

(It's established there is the non-classical, then about there to be established all the extra and the completion/completed.)

The Liar as a template, in constants, terms, and expressions, as above
has where in terms of expressions it's a term and in terms of terms it's
a constant. And, its value is false. This is where, as above, in the
affirmatory of the truth-value as the constants and truisms as terms,
that the Liar is just part of the extra. In this way it has a very
simple, direct, and central placement for what it is: a template for
the evaluation of higher-order terms (and expressions) as non-contradictory
(or rather, that the Liar is contradictory).

[Charlie-Boo]

And if,
1. He says that the way to handle the Liar is to disallow it because evaluating it leads to a loop,
2. "This is this." is easily seen to be simply true, and
3. "This is this." also gets into a loop when evaluating it.

C-B

[Charlie-Boo]

On Friday, February 10, 2017 at 5:20:32 PM UTC-5, Peter Percival wrote:
> X.Y. Newberry wrote:
>
> > "This sentence is not true" goes into infinite loop. "This sentence has
> > five words" does not. "This is this" is an atypical sentence but I would
> > put it in the same category as the latter.
>
> For me, "This is this" is an instance of the law of identity and is
> therefore true.
>

Yes, but the evaluation algorithms get in a loop at the 2nd "this" instead of realizing it is an instance of x=x.

C-B

[peterolcott]

On Friday, February 10, 2017 at 4:20:32 PM UTC-6, Peter Percival wrote:

> X.Y. Newberry wrote:
>
> > "This sentence is not true" goes into infinite loop. "This sentence has
> > five words" does not. "This is this" is an atypical sentence but I would
> > put it in the same category as the latter.
>
> For me, "This is this" is an instance of the law of identity and is
> therefore true.
>

"This is this." seems to be the law of identity if it only refers to itself.
"
This is that." provides a very simple case of semantic ungroundedness, no loops or self-reference required.

[peterolcott]

The way to disallow paradoxes is to disallow propositions that are not relations between compatible types.

I am still working on how to formalize this as predicate logic:
∀x ∈ Propositions ∃R ∈ Relations ∃a ∈ Argument-to-Relation |
x ↔ R & ∀Ra Compatible-Types(Ra)

[Peter Percival]

Charlie-Boo wrote:
> On Friday, February 10, 2017 at 5:20:32 PM UTC-5, Peter Percival wrote:
>> X.Y. Newberry wrote:
>>
>>> "This sentence is not true" goes into infinite loop. "This sentence has
>>> five words" does not. "This is this" is an atypical sentence but I would
>>> put it in the same category as the latter.
>>
>> For me, "This is this" is an instance of the law of identity and is
>> therefore true.
>>
>
> Yes, but the evaluation algorithms get in a loop at the 2nd "this" instead of realizing it is an instance of x=x.

Then the evaluation algorithm is no good. Not that I know what the
evaluation algorithm is.

[Peter Percival]

peterolcott wrote:

> The way to disallow paradoxes is to disallow propositions that are not relations between compatible types.

So Russell did your job for you back in 1908.

[Charlie-Boo]

Paradoxes have produced a number of branches of mathematics/logic/computer science e.g. Incompleteness in Logic (Godel), Theory of Computation (Turing) and Recursion Theory (Kleene.)

Russell's logic disallows the Natural Numbers (any recursive definitions.) The job isn't to find self reference and somehow stamp it out. That is throwing the baby out with the bathwater.

C-B

[Peter Percival]

Charlie-Boo wrote:
>

> The job isn't to find self reference and somehow stamp it out. That is throwing the baby out with the bathwater.
>

I agree.

[Charlie-Boo]

"This is that." is a wff with a free variable (input.) (I personally prefer "This is it.", using "it", "that" and "the other" as the first 3 input variables, as needed.) It is not a statement (no free variables) and can't be treated as such.

C-B

[Charlie-Boo]

Then how do you (we - fearless leader) tell the difference between a vicious circle and a valid recursive definition?

I give "This is this." and "This is not this." as examples of unproblematical self reference, as opposed to "This is false." or "This is not true."

C-B

[Jim Burns]

On 2/11/2017 9:28 AM, Peter Percival wrote:
> peterolcott wrote:

>> The way to disallow paradoxes is to disallow propositions
>> that are not relations between compatible types.
>
> So Russell did your job for you back in 1908.

The problem with that plan is that Goedel did his job
back in 1931. Maybe Kurt Goedel wasn't aware of Russell's
earlier work, and thus re-creating it might possibly answer
Goedel. But that wouldn't be the way to bet.

----
I would say that the problem that Goedel described was _not_
a formal version of the Liar
"This sentence is false"
I would say Goedel described a formal version of
Quine's paradox
"yields falsehood when preceded by its quotation"
yields falsehood when preceded by its quotation.

The problem I see with type theory (a problem which maybe has
been resolved somewhere) is that, if a lower type, such
as numbers, can represent a higher type, such as arithmetic
expressions, the order that the types are intended to
establish does not get (completely) established.

That's what Goedel accomplished. He used numbers to
represent arithmetic expressions, and arithmetic
expressions to represent _us_ when we are manipulating
arithmetic expressions. Once the type-order has been
disrupted that way, the sort of non-well-founded
references that lead to paradoxes become inevitable.

How can we ban _all_ such type-boosting representations?
Once our language has the expressive-capability to do
that, the range of ways to represent higher types runs
well beyond our poor mortal powers of comprehension. Surely,
we need to be able to _perceive_ a rule-breaking if we
plan to enforce a rule. Suppose we rule out type-boosting
representations. And suppose you show me an enormously complex
expression ECX. Has ECX broken the rule? Elephino.

It doesn't work in both directions. (Oh, if only!)
I _look hard_ at ECX ( _really, really_ hard) and I don't
see a type-boosting representation. I pass ECX through
my type-boosting-representation-checker. Then you go
<Simpsons' brat voice> "HAH HAH!" You show me a way to
represent a higher type that I didn't check for, and I am
forced to go "D'Oh!"

----
This is why I like the Revision Theory of Truth.[1]
Non-well-founded references gonna happen. Get used to it.
RTT is about describing what we actually see, instead of
claiming, for example, that we know all the ways of
representing higher types by lower types. Which we don't.

[1]
https://plato.stanford.edu/entries/truth-revision/

[peterolcott]

On Saturday, February 11, 2017 at 8:28:23 AM UTC-6, Peter Percival wrote:
> peterolcott wrote:
>
> > The way to disallow paradoxes is to disallow propositions that are not relations between compatible types.
>
> So Russell did your job for you back in 1908.
>
>

So Russell solved the (1936) Halting Problem and correctly refuted (1936) Tarski Undefinability way back in 1908? If not then he did not do what I have done.

[peterolcott]

If it is pointing to something outside of itself then it has a free variable. If it is pointing to itself then it has no free variable, and is thus incoherent. Semantically it seems to be a direct refutation of the mathematical identity principle.

[peterolcott]

On Saturday, February 11, 2017 at 11:19:01 AM UTC-6, Jim Burns wrote:
> On 2/11/2017 9:28 AM, Peter Percival wrote:
> > peterolcott wrote:
>
> >> The way to disallow paradoxes is to disallow propositions
> >> that are not relations between compatible types.
> >
> > So Russell did your job for you back in 1908.
>
> The problem with that plan is that Goedel did his job
> back in 1931. Maybe Kurt Goedel wasn't aware of Russell's
> earlier work, and thus re-creating it might possibly answer
> Goedel. But that wouldn't be the way to bet.
>
> ----
> I would say that the problem that Goedel described was _not_
> a formal version of the Liar
> "This sentence is false"
> I would say Goedel described a formal version of
> Quine's paradox
> "yields falsehood when preceded by its quotation"
> yields falsehood when preceded by its quotation.

That only seems to be pure gibberish to me.

>
> The problem I see with type theory (a problem which maybe has
> been resolved somewhere) is that, if a lower type, such
> as numbers, can represent a higher type, such as arithmetic
> expressions, the order that the types are intended to
> establish does not get (completely) established.
>
> That's what Goedel accomplished. He used numbers to
> represent arithmetic expressions, and arithmetic
> expressions to represent _us_ when we are manipulating
> arithmetic expressions. Once the type-order has been
> disrupted that way, the sort of non-well-founded
> references that lead to paradoxes become inevitable.

Yes you and I are on the same track now.

>
> How can we ban _all_ such type-boosting representations?

We simply boost one of them high enough to explicitly formalize the error of the lower ones.

[peterolcott]

I am with Peter Percival on this (1) Identity Principle (2) Contradiction neither one has an infinite loop in its evaluation.

[Peter Percival]

Those aren't paradoxes. Let me remind you what you wrote:

The way to disallow paradoxes is to disallow propositions
that are not relations between compatible types.

It was Russell who actually invented a theory of types to do that.

[X.Y. Newberry]

>> "This sentence is not true" goes into infinite loop. "This sentence
>> has five words" does not. "This is this" is an atypical sentence
>> but I would put it in the same category as the latter.
>>

>> My guess is that Gaifman would see it the same way. But the bottom
>> line is that he does not need to know anything about "This is this"
>> in order to analyze the liar paradox. Nor do I think that analyzing
>> "This is this" is very illuminating.
>>
>>>
>>> C-B
>>>
>>>>>
>>>>> C-B
>>>>>
>>>
>

> If "This is this." doesn't go into an infinite loop then where does
> it go - what would be the result?

What does the "this" in your sentence point to?

> The problem is he (and many others) say that if it doesn't ultimately
> refer to something other than itself, then it is "ungrounded" and
> meaningless. Long ago I considered how a statement about only itself
> could be know to be true or to be false, and I came up with "This is
> this." and "This is not this." Judging from the lack of any Google
> results of a paradoxical "THIS IS THIS" or PARADOX "THIS IS THIS", he
> is not alone in missing these points.

"This sentence has five words" is also known to be true. I do not see
any problem here.

> If you really consider the possibilities when you propose a grand
> scheme to resolve paradoxes surrounding natural language statements,
> you very quickly go from "This is false." or "This is not true." to
> "True is true." (TRUE), "This is this." (TRUE) and "This is not
> true." (FALSE)
>
> But the real answer to all of this is to use ordinary mathematics to
> evaluate the problem, and there are at least a dozen places where
> Mathematicians have already done that.
>
> I can only wonder what people would think when confronted with that
> fact, that, after all these years, the truth was staring us in the
> face the whole time?

What might that be?

>
> C-B
>

[X.Y. Newberry]

Peter Percival wrote:
> Charlie-Boo wrote:
>> On Friday, February 10, 2017 at 5:20:32 PM UTC-5, Peter Percival wrote:
>>> X.Y. Newberry wrote:
>>>
>>>> "This sentence is not true" goes into infinite loop. "This sentence has
>>>> five words" does not. "This is this" is an atypical sentence but I
>>>> would
>>>> put it in the same category as the latter.
>>>
>>> For me, "This is this" is an instance of the law of identity and is
>>> therefore true.
>>>
>>
>> Yes, but the evaluation algorithms get in a loop at the 2nd "this"
>> instead of realizing it is an instance of x=x.
>
> Then the evaluation algorithm is no good. Not that I know what the
> evaluation algorithm is.

I don't either.

[peterolcott]

On Saturday, February 11, 2017 at 1:51:11 PM UTC-6, Peter Percival wrote:
> peterolcott wrote:
> > On Saturday, February 11, 2017 at 8:28:23 AM UTC-6, Peter Percival wrote:
> >> peterolcott wrote:
> >>
> >>> The way to disallow paradoxes is to disallow propositions that are not relations between compatible types.
> >>
> >> So Russell did your job for you back in 1908.
> >>
> >>
> >
> > So Russell solved the (1936) Halting Problem and correctly refuted (1936) Tarski Undefinability way back in 1908? If not then he did not do what I have done.
> >
> Those aren't paradoxes. Let me remind you what you wrote:
>
> The way to disallow paradoxes is to disallow propositions
> that are not relations between compatible types.
>
> It was Russell who actually invented a theory of types to do that.
>

Yes but (a really big (and fat) butt) it was never correctly applied to solve either the Halting Problem or Tarski undefinability.

[Peter Percival]

There are no paradoxes there.

[X.Y. Newberry]

He did not say anything about "disallowing". You seem to be missing the
point that the meaning, more precisely its sense of the predicate
"true()" requires it to evaluate another sentence. If the other sentence
is itself then it goes into an infinite loop, and therefore it is unable
to evaluate anything. Thus it lacks reference.

> 2. "This is this." is easily seen to be simply true, and

> 3. "This is this." also gets into a loop when evaluating it.

If you say "this" do you mean "this sentence" or do you mean something
inferred from the context or circumstances. Suppose I point to a lamp on
my desk and say "This is this". By substituting the lamp for "this" we
get "The lamp is the lamp." And indeed that's what the lamp is. But it
still has nothing to do with the liar paradox.



>
> C-B
>

[Charlie-Boo]

On Wednesday, February 8, 2017 at 10:26:29 AM UTC-5, Peter Percival wrote:
> Charlie-Boo wrote:
> >
> > What are the Intuitive Examples (a.k.a. IEs) of the Liar Paradox? What do we want to formalize and evaluate? For starters:
> >
> > 1. This is false.
> > 2. ???
>

> One is enough. Elsewhere I referred to Monk's text. He gives two ways
> of interpreting The Liar. Interested parties may like to look at Thm
> 15.20 and the remark following it, and Ex 17.14. That's pages 275,276,308.

>
> > (That is ALWAYS the first step in formalizing. It gives us a gauge as to the success of any proposal, something missing from virtually all accounts.
>

> What accounts are those?

>
> > Someone asked on FOM if anyone had ever described a gauge.)
> >
> > C-B
> >
>
>

> --
> Do, as a concession to my poor wits, Lord Darlington, just explain
> to me what you really mean.
> I think I had better not, Duchess. Nowadays to be intelligible is
> to be found out. -- Oscar Wilde, Lady Windermere's Fan

“One is enough.”

On one of my visits to SRI in California, someone showed me his program to calculate factorial using an axiomatic system. He entered in 3 and, sure enough, it displayed 6. So I asked him, “How do you know it isn’t calculating the Nth triangular number?”

C-B

[Charlie-Boo]

"Yes but (a really big (and fat) butt) it was never correctly applied to solve the Halting Problem"

Why am I not surprised that he didn't solve the Halting Problem? (Aren't they still working on that?)

Yes, we agree that you have a big fat butt.

C-B

[Charlie-Boo]

What is the most general definition of a "paradox"? Note that it is subjective! In fact, some supposed paradoxes are (or were) resolved by changing one's opinion.

C-B

[Charlie-Boo]

Then we darn well better check both the baby and the bathwater, shouldn't we?

C-B

[Jim Burns]

On 2/11/2017 2:33 PM, peterolcott wrote:
> On Saturday, February 11, 2017 at 11:19:01 AM UTC-6,
> Jim Burns wrote:

>> I would say that the problem that Goedel described
>> was _not_ a formal version of the Liar
>> "This sentence is false"
>> I would say Goedel described a formal version of
>> Quine's paradox
>> "yields falsehood when preceded by its quotation"
>> yields falsehood when preceded by its quotation.
>
> That only seems to be pure gibberish to me.

Can I call it "sentence Q"?
Sentence Q has the quote

"yields falsehood when preceded by its quotation"

as its subject.

It might be easier to read if something less visually
odd-looking were the subject. What if, instead I wrote

The phrase I had tattooed on my left shoulder

yields falsehood when preceded by its quotation.

Would that sentence be somewhat less pure gibberish?

What if the phrase I had tattooed were

yields falsehood when preceded by its quotation.

Would that then be less pure or more pure, gibberish-wise?

My point (WV Quine's point) is that a perfectly fine
description of sentence Q is

"yields falsehood when preceded by its quotation"

preceded by its quotation

So, what sentence Q says is that sentence Q is false.
Paradox ensues.

The difference between sentence Q and sentence L
"This sentence is false"
is that sentence Q _describes_ itself and sentence L
_points to_ itself.

Let's suppose we have a formal theory of pointing.
Then maybe we could ban non-well-founded references
(what I am currently blaming for these paradoxes).
The details matter whether we could, but we'd at
least be a step closer.

On the other hand, _can_ we have a formal theory of describing?
I don't mean a theory of describing one _particular_ way. I
mean a theory that covers _all possible ways_ of describing
something -- describing a sentence, let's say.

I don't think we can have such a theory. I could be
wrong about that, and I'd be willing to be shown that
I'm wrong. But whether I'm right or whether I'm wrong,
sentence Q presents a different and more difficult
problem for paradox-banning. Even if you could get
rid of sentence L and all its variants, you would still
have a big job ahead.

[Jim Burns]

On 2/12/2017 9:11 AM, Charlie-Boo wrote:
> On Saturday, February 11, 2017 at 3:08:10 PM UTC-5,
> Peter Percival wrote:

>> There are no paradoxes there.

> "There are no paradoxes there."
>
> What is the most general definition of a "paradox"?
> Note that it is subjective! In fact, some supposed
> paradoxes are (or were) resolved by changing one's opinion.

What about this as a starting point?
<SEP>

Epistemic paradoxes are riddles that turn on the concept
of knowledge ( _episteme_ is Greek for knowledge).
Typically, there are conflicting, well-credentialed answers
to these questions (or pseudo-questions). Thus the riddle
immediately informs us of an inconsistency. In the long run,
the riddle goads and guides us into correcting at least one
deep error -- if not directly about knowledge, then about
its kindred concepts such as justification, rational belief,
and evidence.

</SEP>[1]

Discuss.

<https://plato.stanford.edu/entries/epistemic-paradoxes/>

[Ross A. Finlayson]

Riddle, or puzzle?

I applaud the general direction, the idea that
paradoxes are logical challenges, not barriers.

The transfer principle and bridge results about
what is between the finite and infinite where
they're each of the parts and about the whole,
it's almost as if there is some purely logical
paradox, not as contradiction but alternation.

With comprehension, that comprehending nothing
is not nothing, thus something, there are no
paradoxes because that would be a paradox.

This is again about the simple conservation and
symmetry and diversity and variety, that the
abstract reasoning expands comprehension.

It's what it was for what it is,
that's not a paradox.

The universe is infinite /
infinite sets are equivalent.

[khongdo...@gmail.com]

It's interesting that the knowledge operator K() isn't a private notion of a
sci.logic poster that a couple of others seem to have alluded to.

Anyway in the link, they formalize ‘p but p is not known’ as K(p & ~Kp): I think this
is wrong. In fact (K(p) => TRUE(p)), but not necessarily the other way around.

[khongdo...@gmail.com]

They should have written: (TRUE(p) and neg(K(p))).

[khongdo...@gmail.com]

To be more precise: (K(TRUE(p)) => TRUE(p)).

[Charlie-Boo]

Charlie-Boo's Paradox uses standard English:

"It is not true of itself." is true of "It is not true of itself."

tinyurl.com/thousandparadoxes

C-B

[Jim Burns]

On 2/12/2017 6:58 PM,

khongdo...@gmail.com wrote:
> On Sunday, 12 February 2017 07:29:42 UTC-7,
> Jim Burns wrote:

>> <https://plato.stanford.edu/entries/epistemic-paradoxes/>
>
> It's interesting that the knowledge operator K() isn't
> a private notion of a sci.logic poster that a couple of
> others seem to have alluded to.

How odd. I remember those conversations very differently.

The privateness of the definition of K() and other
definitions was something insisted upon by that sci.logic
poster (you, Nam Nguyen) in the face of incessant calls
for clarification from essentially everyone who took any
notice of you at all, myself most definitely included.

You have been successful for years now in keeping whatever
notion you may have of K() private -- not merely in the
sense of that notion not being anyone else's, but also in
the sense of you declining every opportunity to share that
notion with sci.logic. It's probably too much of a stretch
to say that it is a settled question as to whether you
actually have a notion of K(), as opposed to the ability to
type "K()", but the evidence to date is that you have no
idea what you are talking about in that respect.

I will grant you an uncanny ability to _sound as though_
you are talking about K() and mathematics and logic --
a variety of glossolalia, perhaps. Congratulations for
that, anyway.

> Anyway in the link, they formalize ‘p but p is not known’
> as K(p & ~Kp): I think this is wrong. In fact
> (K(p) => TRUE(p)), but not necessarily the other way around.

Nam, when you say "impossible to know", what do you
mean by that?

You've apparently just looked at a page discussing what
kinds of things someone might mean. Maybe now you believe
that this is a question that needs an answer, if you
plan on being understood.

[khongdo...@gmail.com]

Idiotic ranting from Jim Burns, as usual.

>
> > Anyway in the link, they formalize ‘p but p is not known’
> > as K(p & ~Kp): I think this is wrong. In fact
> > (K(p) => TRUE(p)), but not necessarily the other way around.
>
> Nam, when you say "impossible to know", what do you
> mean by that?

See. While Nam has posted the technical observation above, in response Jim has
nothing to counter (offer), except idiotic ranting, character assassination, ad hominem attack.

>
> You've apparently just looked at a page discussing what
> kinds of things someone might mean. Maybe now you believe
> that this is a question that needs an answer, if you
> plan on being understood.

Again: typical idiotic ad hominem attack from Jim Burns.

I've observed:

Anyway in the link, they formalize ‘p but p is not known’
as K(p & ~Kp): I think this is wrong. In fact
(K(p) => TRUE(p)), but not necessarily the other way around.

Any technical comment from you on this observation?

[khongdo...@gmail.com]

On Sunday, 12 February 2017 17:36:10 UTC-7, Jim Burns wrote:

By "impossible to know" p in this context, it'd mean:

neg(K(TRUE(p))) and neg(K(TRUE(neg(p)))) and (K(p') => TRUE(p')).

That's a short and very technical description. Care to let sci.logic know if you'd be
able to understand that?

> You've apparently just looked at a page discussing what
> kinds of things someone might mean. Maybe now you believe
> that this is a question that needs an answer, if you
> plan on being understood.

Again, are you able to comprehend the above?

[khongdo...@gmail.com]

Now, I've technically answered your question. Hope you'd answer my question whether
or not you'd technically understand my above answering your question "what do you mean by that?".

[Ross A. Finlayson]

"Knowledge of the conditional is conditional knowledge
(that is, conditional upon learning the antecedent and
applying the inference rule modus ponens: If P then Q,
P, therefore Q). But the next section is devoted to some
known conditionals that are repudiated when we learn their
antecedents."

"The student can know that the announcement is true
after it becomes true – but not before."

That said, not all teachers are liars. And, here they have
the discretion to make surprise tests. And, there's no
reason that it's not on Friday, which is still before the
end of the week. The surprise test is always ever now.
The student's failure was that the week isn't over until
it's over. (Surprise!)

Otherwise, the probability that the test has occurred
goes to one, and when the bell tolls, it will have occurred.
It's a continuous time domain (and everybody and everything
has only one present state, in it: now). This is a classical time
domain and such notions as non-classical progress in time
are not material to it.

The "Surprise Test" (or "Hangman") paradox is just Zeno's
motion in disguise, for the probability of the quiz, or, result
(before) compared to the probability of an opposite result
(after or "not"). At some point, the event, they exchange,
the running probabilities for and against. In a sense that
is the event. It's a matter for Zeno's relay runners.

This is a simple extension of Zeno's runners with a round track:
each going opposite ways until they meet and met.

There just aren't that many paradoxes (or none).
Expectation and knowledge are simple different things.

The values of expectations are always conditional,
the state of expectations are always opinion,
events of knowledge are matters of fact.

"You just can't be amazed /
Even if you pull the pin /
from your hand grenade."

"How soon is now?"


By now the student should have learned
that matters of belief aren't matters of fact,
except as matters of the fact of belief, not
the belief of fact.

Mathematics establishes facts.

[Charlie-Boo]

Typical paradoxes are apparent proofs that Mathematics (including Logic) is inconsistent.

A more general definition is ??

C-B

[Jim Burns]

On 2/12/2017 7:48 PM,

One comment. Let's see what you can make of it.

You've missed the point of
K(p & ~Kp)
That's the premise to a paradox.[1]
Your observation adds nothing to the discussion,
like "observing" that 1 + 1 = 2 in the middle of a
proof a Fermat's Last Theorem. So it is. So what?

In case you didn't catch it earlier, the paradox is
that we can make very reasonable assumptions about
the nature of knowledge and still derive a contradiction.

Suppose that
1. If we _know_ a thing p, then p is true.
Kp -> p
2. If we _know_ a conjunction p & q is true, then
we _know_ p is true and we _know_ q is true.
K(p & q) -> Kp & Kq
3. If a statement p implies a contradiction q & ~q,
then we conclude ~P.
( p -> q & ~q ) -> ~p
4. We know that we don't know everything. That is,
we know that there is some true things p which
we do not know, ~Kp .
K( p & ~Kp )

The paradox is that K(p & ~Kp) implies a contradiction
-- which seems to imply that we _don't_ know that
we don't know everything.

5. K( p & ~Kp ) 4, assumption
6. Kp & K(~Kp) 5,2
7. Kp & ~Kp 6,1
8. ~K( p & ~Kp ) 5,7

[1]
Note that the topic of the page you found that on
is paradoxes.
<https://plato.stanford.edu/entries/epistemic-paradoxes/>
5.2 The “Knowability Paradox”

[Jim Burns]

On 2/12/2017 8:07 PM,

What you've done is give another of your unexplained
formalisms to explain your earlier unexplained formalism.
So, no, you haven't technically answered my question.

> Hope you'd answer my question whether or not
> you'd technically understand my above answering
> your question "what do you mean by that?".

Your response was vacuous technical-sounding gibberish.
So, no, I don't understand your vacuous technical-sounding
gibberish. Do you?

[Ross A. Finlayson]

Paradox _would be_ contradiction arising from
otherwise mathematical collections of related
inferences, that they're not then is the extra-
ordinary of the logic.

Contradictions are just contradictions.

The very form of the Liar directly informs the
reader its self-contradiction, for its exclusion
when it exists in vacuo except as an example, and
for the discovery of what otherwise would be
contradiction when otherwise collected inferences
contradict each other.

Given a single contradiction there are many ways
to re-write a mathematical collection and that
starting from within it would lead to contradiction.

The point is then that there is a distinguished pair
of collections of the otherwise consistent statements
then joined with the cut-point of the contradiction.
This is the Liar as template and constant among
expressions.

(As above I establish an opinion with an otherwise
affirmatory logic to be able to establish truth as
not containing the Liar, that the Liar exists as an
extra-ordinary constant not as just the collection
of the words.)

Paradox otherwise is basically the advice as to
either the inconsistency of the theory or that
there are implicit features of the theory as so
found the resolution of what would have otherwise
been the paradox.

Either way: the foundation has no paradox. This
is carefully qualified so that where there "is" a
paradox, it is an extra-ordinary constant that
"cancels itself away", that it's so distinguished
and special (some "root probabilistic flaw"),
that it "was" a paradox as it's so resolved, and
that the resolution of the paradox is the root
of the inference.

Here it's that inference does carry.



Paradox as among the non-logical is again another
indication of over- or under-definition of the theory.
For example, atoms as particles and waves, has driven
theory into the super-classical, and not just that
particles are waves, but that the objects combine
the properties of otherwise the particles and waves,
that they are super-classical and surpass our models
in this way.

That's similar then in the logical, the Nothing and
Being advise each other that as either is the primary
or ur-element, that they're each other and immaterial.
This is that the model must accommodate all the properties,
and that it somehow does.

This isn't then that the objects are non-classical,
instead, it's that the objects are the super-classical,
then that that's neo-classical.

The Liar would be a paradox if you believed it,
but luckily, you're a thinker not just a reader.


Another way of looking at paradox is that it's what
remains when two otherwise incompatible mathematical
collections are collected together. Picking then one
or the other and not both is selective ignorance (or,
"definition", eg, restriction of comprehension),
establishing the necessary augments to both (the
relevant properties of the objects as satsify both)
then is a new mathematical collection.

Or "paradox is like a red flag to a matador".

[khongdo...@gmail.com]

If you were a math professor or even a poster having a Bachelor degree in mathematics
then that might have sounded a little less idiotic and less coward: but you were
neither ...

> So, no, I don't understand your vacuous technical-sounding gibberish. Do you?

If you don't understand such _ONE_ freaking _short_ sentence technical definition,
then just say so as a man of courage. Stop sounding like a math professor who you're nowhere near being: that would just make you sound like a lying coward than anything else.

[Peter Percival]

Roy Sorensen has used the notation common in epistemic logic.

[Peter Percival]

khongdo...@gmail.com wrote:

> I've observed:
>
> Anyway in the link, they formalize ‘p but p is not known’
> as K(p & ~Kp): I think this is wrong. In fact
> (K(p) => TRUE(p)), but not necessarily the other way around.
>
> Any technical comment from you on this observation?

Sorensen wishes to discuss 'p but p is not known' and he expressed it in
symbols in the usual way.

[Peter Percival]

khongdo...@gmail.com wrote:

> If you don't understand such _ONE_ freaking _short_ sentence technical definition,
> then just say so as a man of courage. Stop sounding like a math professor who you're nowhere near being: that would just make you sound like a lying coward than anything else.

Try not to be so aggressive. Here we were having a nice friendly
discussion about the liar paradox, and within hours of your arrival
you're accusing people of sounding like lying cowards. A bit of good
manners wouldn't go amiss. This corner of the thread was discussing the
epistemic paradoxes. Do you have anything to say about them or
Sorensen's account of them?

[Jim Burns]

On 2/12/2017 7:58 PM,

khongdo...@gmail.com wrote:
> On Sunday, 12 February 2017 17:36:10 UTC-7,
> Jim Burns wrote:
>> On 2/12/2017 6:58 PM,
>> khongdo...@gmail.com wrote:
>>> On Sunday, 12 February 2017 07:29:42 UTC-7,
>>> Jim Burns wrote:

>>>> <https://plato.stanford.edu/entries/epistemic-paradoxes/>
>>>
>>> It's interesting that the knowledge operator K() isn't
>>> a private notion of a sci.logic poster that a couple of
>>> others seem to have alluded to.
>>
>> How odd. I remember those conversations very differently.

[...]

>>> Anyway in the link, they formalize ‘p but p is not known’
>>> as K(p & ~Kp): I think this is wrong. In fact
>>> (K(p) => TRUE(p)), but not necessarily the other way around.
>>
>> Nam, when you say "impossible to know", what do you
>> mean by that?
>
> By "impossible to know" p in this context, it'd mean:
>
> neg(K(TRUE(p))) and neg(K(TRUE(neg(p)))) and
> (K(p') => TRUE(p')).
>
> That's a short and very technical description. Care to
> let sci.logic know if you'd be able to understand that?

I've already answered this question elsewhere.

Rather than answer it again, I'm taking a moment to point out
that your (NN's) "possible to know" operator K() which you
introduce here is not the _different_ "known" operator K()
to which you refer to one post earlier as evidence that _your_
K() is _not_ a private notion of yours.

You've actually done the opposite of what you intended.
I believe the technical term for this is "own goal".
Not only did you introduce a _private_ (yours alone)
operator K(), you introduced it while keeping its meaning
_private_ . (And then, you wanted _me_ to tell _you_
what you mean, a typical ploy for you, Nam.)

That you think that a "possible-to-know" K() operator and a
"known" K() operator _are the same operator_ suggests
very strongly that you have no notions of what it is you're
writing or reading, beyond the ability to copy text in a
roughly life-like manner. This would also explain your practice
of meeting any attempt to find out what your notions are
(what your _non-existent_ notions are) with evasion and
inflammatory, distracting rhetoric like "cowardly liar".
You've got nothing there for us to find out.

[Jim Burns]

On 2/11/2017 2:33 PM, peterolcott wrote:
> On Saturday, February 11, 2017 at 11:19:01 AM UTC-6,
> Jim Burns wrote:

>> The problem I see with type theory (a problem which maybe
>> has been resolved somewhere) is that, if a lower type, such
>> as numbers, can represent a higher type, such as arithmetic
>> expressions, the order that the types are intended to
>> establish does not get (completely) established.
>> That's what Goedel accomplished. He used numbers
>> to represent arithmetic expressions, and arithmetic
>> expressions to represent _us_ when we are manipulating
>> arithmetic expressions. Once the type-order has been
>> disrupted that way, the sort of non-well-founded
>> references that lead to paradoxes become inevitable.
>

> Yes you and I are on the same track now.

>
>> How can we ban _all_ such type-boosting representations?
>

> We simply boost one of them high enough to explicitly
> formalize the error of the lower ones.

I'll point out for future reference (as these sorts of
conversations tend to travel in circles, and we may well
re-play all of this again) that

-- You haven't done any such boosting.

-- You claimed earlier that you had solved ( _past tense_ )
these sorts of problems.[1]

-- You claimed a little earlier that your claims _were not_
bullshit, because you don't lie.[2]

Note: Not all of these things can be true at the same time.

Some things that I would like you to see from considering the
example of Quine's paradox:

-- "Simply" having a representation that boosts to a high enough
order (if there were such a thing) is a lot more technically
demanding than saying "let it be so". Goedel's numbering boosts
one level. Compare what you've done to what he's done.

-- It's not clear to me that there is such a thing as a
"high enough order". That would be an order higher than
all the other orders, right? Tell you what, why don't
you solve the problem of a number higher than all the
other numbers, as a warm-up exercise?

-- Even if I granted you everything else (which I'm not going
to do -- that's way too much to grant), your solution
would have to be banning the lower types that represent
higher types. Those representations are _there_ , the same
way a particular number is a prime or is a composite.
You can't just say "Let it be that 10^100+17 is prime".
Whether it is or not is baked into the system.

But what would that even mean to ban, for example, the _number_ ,
the Goedel number g of a true-but-unprovable statement?
This is a specific number, very large, which depends upon
the details of the Goedelization and sub(x,y) and Provable(y).
Would the Unique Prime Factorization theorem no longer be
true for all n? What would that mean for g+1 ? Or g-1?

-- And it only gets worse from there.

[1]
<https://groups.google.com/d/msg/sci.logic/_IpbSQSG-F0/o0bpScr8CgAJ>
<PO>

So Russell solved the (1936) Halting Problem and correctly
refuted (1936) Tarski Undefinability way back in 1908?
If not then he did not do what I have done.

</PO>

[2]
<https://groups.google.com/d/msg/sci.logic/P1cDgLXTjc8/Sm-jo3suCgAJ>
<PO>
Not really I count bullshitting as aligning myself
with Satan (the father of all lies) so I don't do it.
</PO>

[peterolcott]

I have created the foundation of the analytical framework to accomplish what I said above. I can see all the way through to the end extrapolating on the basis of what I currently have.

I have 1,500 hours into this work since last June This sums up a key point:
https://philpapers.org/archive/PETFSP.pdf

Here is Newberry agreeing with the verbatim words of my following paper:
[This sentence has five words]
https://groups.google.com/forum/#!msg/sci.logic/_IpbSQSG-F0/EkyEGmWhCgAJ

https://philpapers.org/archive/PETFDS-3.pdf