On 3/13/2024 1:16 PM, Ross Finlayson wrote:
On 03/12/2024 09:00 PM, olcott wrote:
On 3/12/2024 10:49 PM, Ross Finlayson wrote:
On 03/12/2024 08:23 PM, Ross Finlayson wrote:
On 03/12/2024 07:52 PM, olcott wrote:
On 3/12/2024 9:28 PM, Richard Damon wrote:
On 3/12/24 4:31 PM, olcott wrote:
On 3/12/2024 6:11 PM, Richard Damon wrote:
On 3/12/24 3:53 PM, olcott wrote:
On 3/12/2024 5:30 PM, Richard Damon wrote:
On 3/12/24 2:34 PM, olcott wrote:∀ H ∈ Turing_Machines_Returning_Boolean
On 3/12/2024 4:23 PM, Richard Damon wrote:
On 3/12/24 1:11 PM, olcott wrote:
On 3/12/2024 2:40 PM, Richard Damon wrote:
On 3/12/24 12:02 PM, olcott wrote:
On 3/12/2024 1:31 PM, immibis wrote:
On 12/03/24 19:12, olcott wrote:
∀ H ∈ Turing_Machine_DecidersAnd it can be a different TMD to each H.
∃ TMD ∈ Turing_Machine_Descriptions | >>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD) >>>>>>>>>>>>>>>>>>
There is some input TMD to every H such that >>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD) >>>>>>>>>>>>>>>>>
When we disallow decider/input pairs that are incorrect >>>>>>>>>>>>>>>>>> questions where both YES and NO are the wrong answer >>>>>>>>>>>>>>>>>Once we understand that either YES or NO is the right >>>>>>>>>>>>>>>>> answer, the whole rebuttal is tossed out as invalid and >>>>>>>>>>>>>>>>> incorrect.
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩
halts
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn // Ĥ applied to ⟨Ĥ⟩
does
not halt
BOTH YES AND NO ARE THE WRONG ANSWER FOR EVERY Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩
No, because a given H will only go to one of the answers. >>>>>>>>>>>>>>> THAT
will be wrong, and the other one right.
∀ H ∈ Turing_Machine_Deciders
∃ TMD ∈ Turing_Machine_Descriptions |
Predicted_Behavior(H, TMD) != Actual_Behavior(TMD) >>>>>>>>>>>>>>
Not exactly. A pair of otherwise identical machines that >>>>>>>>>>>>>> (that are contained within the above specified set) >>>>>>>>>>>>>> only differ by return value will both be wrong on the >>>>>>>>>>>>>> same pathological input.
You mean a pair of DIFFERENT machines. Any difference is >>>>>>>>>>>>> different.
Every decider/input pair (referenced in the above set) has a >>>>>>>>>>>> corresponding decider/input pair that only differs by the >>>>>>>>>>>> return
value of its decider.
Nope.
∃ TMD ∈ Turing_Machine_Descriptions |
Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
Every H/TMD pair (referenced in the above set) has a
corresponding H/TMD pair that only differs by the return
value of its Boolean_TM.
That isn't in the set above.
Nope, since both aren't in the set selected.
That both of these H/TMD pairs get the wrong answer proves that >>>>>>>>>> their question was incorrect because the opposite answer to the >>>>>>>>>> same question is also proven to be incorrect.
When they are deciders that must get the correct answer both
of them are not in the set.
*IF* they are correct decider.
WHen we select from all Turing Machine Deciders, there is no
requirement that any of them get any particular answer right.
So, ALL deciders are in the set that we cycle through and apply the >>>>>>> following logic to ALL of them.
Each is them paired with an input that it will get wrong, and the >>>>>>> existance of the input was what as just proven, the ^ template
When they are Turing_Machines_Returning_Boolean the this
set inherently includes identical pairs that only differ
by return value.
But in the step of select and input that they will get wrong, they >>>>>>> will be givne DIFFERENT inputs.
You just don't understand what that statement is saying.No the problem is that you are not paying attention.
I've expalined it, but it seems over you head.
No, you keep on making STUPID mistakes, like thinking that select a >>>>>>> input that the machine will get wrong needs to be the same for two >>>>>>> differnt machines.
For Every H, we show we can find at least one input (chosenWhen we use machine templates then we can see instances of
just for
that machine) that it will get wrong.
the same machine that only differs by return value where both
get the wrong answer on the same input. By same input I mean
the same finite string of numerical values.
But if they returned differnt values, they will have different
descriptions.
Otherwise, how could a UTM get the right answer, since it only gets >>>>>>> the description.
We can get around all of this stuff by simply using this criteria: >>>>>> Date 10/13/2022 11:29:23 AM
*MIT Professor Michael Sipser agreed this verbatim paragraph is
correct*
(He has neither reviewed nor agreed to anything else in this paper) >>>>>> (a) If simulating halt decider H correctly simulates its input D
until H
correctly determines that its simulated D would never stop running >>>>>> unless aborted then
(b) H can abort its simulation of D and correctly report that D
specifies a non-halting sequence of configurations.
*When we apply this criteria* (elaborated above)
Will you halt if you never abort your simulation?
*Then the halting problem is conquered*
When two different machines implementing this criteria
get different results from identical inputs then we
know that Pathological Self-Reference has been detected.
We don't even need to know that for:
*denial-of-service-attack detection*
*NO always means reject as unsafe*
But, Halting theorem never said "there's an input that halts
all machines", it just says "for any machine, there's an input
that halts it".
Where "halt the machine" means "put it in an infinite loop".
So, rather, Halting theorem never said "there's an input that
exhausts all machines", it just says, "for any machine, there's
an input that exhausts it".
I still don't see how that would be with infinite tapes though,
without a means of checking all the way right the tape in one
step, i.e. that it's 1's or 0's or any pattern at all, any
input that unbounded with respect to the machine basically
exhausts it or where the machine would run in the unbounded.
Of course any finite tape, has a static analysis that is
not infinite, that decides whether or not it halts
(or, loops, or grows, the state space of the decider).
Static analysis has to either enumerate or _infer_ the
state-space, where equal values in what's determined
the idempotent can detect loops, while inequalities
or proven divergence, ..., can detect unbounded growth.
Now, proving convergence or divergence is its own kind
of thing. For example, there are series that converge
very slowly, and series that diverge very slowly. These
are subtly intractable to analysis.
Then the usual idea of progressions that rapidly grow
yet return to what's detectable, are pathological to
analysis, only in resources not correctness or completion,
vis-a-vis the subtle intractability of the convergence or
divergence what either halts, or loops, or grows unboundedly.
Separately "not-halts" into "loops or grows unboundedly",
has that really for most matters of interest of understanding
the state-space of programs, is "halts or enters loops"
and not "grows unboundedly".
I.e. if the Halting problem is basically subject the
subtle intractability of slow convergence, that otherwise
it can just infer divergence and decide, practically
it's sort of more relevant what the machine would be
doing on the input on the tape, then with respect to
beyond the Turing theory, of the state of the read-head,
what happens when somebody modifies the tape, or events,
the write-head.
Anyways though for bounded inputs, besides slow divergence,
it's to be made clear that _most all_ and _almost all_
programs _are_ decided their behavior by static analysis.
Though, "most all" and "almost all" might be a bit strong,
but pretty much all that don't involve "the subtle intractability
of slow divergence".
Giving the idea that an existence result
is in any way the expected result here
seems sort of the root of this dilem-na.
(Though that the real numbers in ZFC have a well-ordering
and if they had a normal ordering that was a well-ordering,
that would be a thing, because ZFC has a well-ordering of
[0,1], but can't give one.)
What I'm saying is that you'd need to solve all
cases of slow convergence and slow divergence
to have a correct and complete halt decider,
for the unbounded, and finite, if not the infinite.
Most though setups of convergence though
would effectively exhaust according to the
existence of criteria of convergence as modeling
the computation, though, so, if you exclude
slow convergence and slow divergence,
then a subset of very usual machines is of
course quite all decidable. (Or decide-able,
or as I put it once, "not deciddable".)
Well-ordering the reals in ZFC, that's it own
sort issue - that a well-ordering implies the
existence of a bijection from ordinals to a
set, then as that as tuples, subsets of those
are well-ordered by their ordinal part, then
that if ever uncountably many of those are
in normal order their real part and irrationals,
then between those are rationals. I.e.,
ZFC doesn't give an example of well-ordering
the reals, but there is one.
So, Peter, I think you're kind of misreading that
quote you authoritated. It just says "if static
analysis detects a loop or unboundedly growing
then it's not-halts".
Of course, for any bounded resource, there
are arbitrarily larger bounded resources
required by what's called pathological,
that an arbitrarily larger resource can though solve.
So anyways "slow convergence" and "slow divergence",
have that in mathematics, it's unknown for some series
whether they converge or diverge, though it's usually
accepted that the inverse powers of 2 converge and
that the sum of integers diverges, and that periodic
functions do neither.
I.e. the criteria of convergence and existence of a limit,
and the special case "diverges as going to infinity",
are not yet closed in today's mathematics.
It's sort of a conjecture or independent whether
they ever could be, closed, and complete, the criteria.
I have spent 20 years on The conventional halting problem proofs,
Gödel 1931 Incompleteness and the Liar Paradox. I have only spent
about seven years on Tarski Undefinability.
The Halting problem is the only one of these that has a formal system
that cannot have hidden gaps its its reasoning. If a machine can do it
then it can be done.
"Algorithmics" is a study of algorithms. The most usual
consideration is "asymptotics", "asymptotics of complexity",
with the idea that computing does work, that there are
closures and completions, and there are approximations and
attenuations, as it were, in the full and the final and
the initial and the partial.
The most usual resources are time, and space, that actions
occur in time, and according to a structure, in space.
Of course any CS grad pretty much knows that.
The objects of mathematics, have it so, that the finite
and bounded resources of any collection of digital and
binary machines, are finite and bounded, while, the unbounded
models of mathematics that represent them, are unbounded,
it's bad that anybody ever said a Turing tape was "infinite",
except that it's unbounded as an object of mathematics, and
the model of any finite, digital, binary machine.
One might aver "digital" has a complement in computing,
"analog", which represents the continuous instead of
the discrete, and one might aver "binary" has a complement,
either "real-valued" or "fuzzy" or variously about the
"multi-valent", in logic, the models, according to the
mechanisms, the models, their mechanics, machines.
The Turing machine though is a model of unbounded machine
as what any thusly simply mechanical model can implement,
up to its bounds.
So, the objects of mathematics, then involve analytical
results. I can't draw a circle, though a compass may
portray a diagram arbitrarily close, and no different.
It's the analytical results, that make so for why
mathematics is: "sensible, fungible, and tractable",
and for "the ubiquitous success of mathematics" in
all such matters of physics, which is a continuum mechanics,
and as well discrete mechanics.
Of course, results exist in mathematical objects,
after the unbounded the unbounded and complete,
which is why closures and completions are so
relevant to all such matters, vis-a-vis approximations
and attenuations, which in bounded terms result
indistinguishably non-different results,
up to precision, say, or tolerances.
So, "machine" might be discrete, binary, and digital,
or it might be continuous, multivalent, and analog.
The Turing machine in a concrete form, is bounded.
Similar models, here mostly for the stack models,
stack machines, among finite-state-machines then
for example concepts like "unbounded nondeterminism",
here is pretty much about determinism, about the
defined behavior of a machine according to a
mathematical model that given defined behavior
of the mechanisms results a mechanical model of
a mathematical model, such a theory as this.
So, the theory of computing, with mechanical or
computational models, models of computing, is
pretty much exactly like the theory of physics,
with the attachment of physical models to the
mathematical models, and rather, "interpreted as",
the models, the models, with respect to each other.
I.e. the extensionality that so follows "they're
equal, or same, or not different, they represent
each other, they're equi-interpretable, they model
each other", the models, of logical and mathematical
and computational and mechanical and physical models,
help represent that over all the entire thing there
is a usual theory for the entire thing, it's a theory
where model theory models extensionality, and in identity
intensionality, about equality, tautology, identity,
and sameness/difference, and nearness/farness, and all
these usual aspects of the mathematical models'
arithmetic, algebra, and geometry. (And function
theory and topology, with respect to categories and
types, sets and parts, relations and predicates,
then all the model-building among that which is
all the propositional or the stipulated or axiomatic.)
The idea is that this sort of platonistic universe of
mathematical objects has always existed, and it's just
discovered, then that axiomatics for example, just
sort of results a model theory of it, with regards
to models, the modular, modulation, modality, and the mode.
So, a machine, with respect to computation, establishing
the validity of interpretations of models, is subject to
filling out all the above, vis-a-vis what it can do,
and it can-do.
Then, "slowly converging" and "slowly diverging"
are examples that get get into that there are more
law(s) of large numbers, than the usual classical
law of large numbers.
Some people variously do or don't have a mathematical
model of larger numbers, or the infinite, at all.
It's a totally ancient and dogmatic tradition that
no, we finite peoples, persons, or agents of limited
and finite means, no we do not have discrete, digital,
binary mechanical models of the infinite.
Yet, it's also about the first thing that deduction
arrives at just from the plain contemplation of the
beyond the unbounded, like the theories of Democritus
and Atomism or the theories of Zeno and Continuity,
that make it so we at least have something to arrive
at for models of the unbounded, as "methods of exhaustion",
the approximation and attenuative what results
the asymptotics of algorithmics.
So, we do have mental models of the continuous and infinite.
And, where physics is a continuum mechanics,
there are physical models of it as well, then
with regards to the philosophy of science and
the scientific method and the Big Science and
from the Primary Science, lots to establish
that the continuous and analog has mathematical
results, as whethe, "sensible, fungible, and tractable",
to machines at all.
Numerical methods then, and, approximation and
attenuation, result from analysis, why they exist,
here for the criteria of convergence and existence
of limits, in the closed, in the complete.
So, a metonymy, unless it's "the Strong Metonymy",
which is utter intensionality of "the ground model",
is yet a metaphor and a metaphor yet a weaker metonymy
joint among weaker metonymys, that, ontology, has
that there are or may be a completion, of ontology,
as a "strong ontology" and "strong metonymy",
but otherwise it's just a bag of facts.
Here then there's certainly a perceived analytical
development Cantor space, and, Square Cantor space,
and, Signal Cantor space as it were, about that among
the law(s) of large numbers, there are definitions of
continuity, at least including field-reals, line-reals,
and signal-reals, three sorts continuous domains.
Mathematics owes physics this to better begin to
explain, to disclose, to find truths of, continuum mechanics.
Then that analog, fuzzy, multi-value, "quantum" computing
models (parts) of that, that discrete, binary, digital
computing does not, is a thing.
The convergence of terms in series is an example
of a model of things, that, sometimes has clear
and direct and perfectly accurate representations,
models, in stack machine, and sometimes doesn't.
Of course, for any bounded input, there is a sufficiently
larger bounded analyzer, that does solve it. So, why not
just put those all together? It would be infinite,
yet most things are not. (Or, you know, are.)
It's called "foundations" the study of all these things
as a fundamental coherency, a thing together, while of
its parts.
So, students should well have the concepts of "abacus"
and "slide rule" and other "tabulators and computers"
of the usual manual mechanical variety down, according
to the principles of the mathematical models behind them,
then introducing the law(s) of large numbers, then
introducing the idea of a standard model of an infinite,
about rates, related rates, asymptotics, and bounds.
"Foundations" then the idea is that it's been around
forever, it's our dogma and from our canon and it develops
over time, and the current edition is called "modern".
Did I make any mistakes?
Tarski anchors his proof in the Liar Paradox https://liarparadox.org/Tarski_247_248.pdf
How Tarski encodes the Liar Paradox
x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x
This is transformed into line 01 of his proof
by replacing True with Provable
*Here is the Tarski Undefinability Theorem proof*
(1) x ∉ Provable if and only if p // assumption
(2) x ∈ True if and only if p // assumption
(3) x ∉ Provable if and only if x ∈ True.
(4) either x ∉ True or x̄ ∉ True; // axiom: ~True(x) ∨ ~True(~x) (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
(6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: Provable(~x) → True(~x)
(7) x ∈ True
(8) x ∉ Provable
(9) x̄ ∉ Provable
Tarski's Full proof
https://liarparadox.org/Tarski_275_276.pdf
On 3/14/2024 11:58 AM, Ross Finlayson wrote:
On 03/13/2024 10:20 PM, olcott wrote:
On 3/13/2024 1:16 PM, Ross Finlayson wrote:
On 03/12/2024 09:00 PM, olcott wrote:
On 3/12/2024 10:49 PM, Ross Finlayson wrote:
On 03/12/2024 08:23 PM, Ross Finlayson wrote:
On 03/12/2024 07:52 PM, olcott wrote:
On 3/12/2024 9:28 PM, Richard Damon wrote:
On 3/12/24 4:31 PM, olcott wrote:
On 3/12/2024 6:11 PM, Richard Damon wrote:
On 3/12/24 3:53 PM, olcott wrote:
On 3/12/2024 5:30 PM, Richard Damon wrote:
On 3/12/24 2:34 PM, olcott wrote:∀ H ∈ Turing_Machines_Returning_Boolean
On 3/12/2024 4:23 PM, Richard Damon wrote:
On 3/12/24 1:11 PM, olcott wrote:
On 3/12/2024 2:40 PM, Richard Damon wrote:
On 3/12/24 12:02 PM, olcott wrote:
On 3/12/2024 1:31 PM, immibis wrote:
On 12/03/24 19:12, olcott wrote:
∀ H ∈ Turing_Machine_DecidersAnd it can be a different TMD to each H. >>>>>>>>>>>>>>>>>>>
∃ TMD ∈ Turing_Machine_Descriptions | >>>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD) >>>>>>>>>>>>>>>>>>>>
There is some input TMD to every H such that >>>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD) >>>>>>>>>>>>>>>>>>>
When we disallow decider/input pairs that are incorrect >>>>>>>>>>>>>>>>>>>> questions where both YES and NO are the wrong answer >>>>>>>>>>>>>>>>>>>Once we understand that either YES or NO is the right >>>>>>>>>>>>>>>>>>> answer, the whole rebuttal is tossed out as invalid and >>>>>>>>>>>>>>>>>>> incorrect.
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩
halts
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn // Ĥ applied to ⟨Ĥ⟩
does
not halt
BOTH YES AND NO ARE THE WRONG ANSWER FOR EVERY Ĥ.H ⟨Ĥ⟩ >>>>>>>>>>>>>>>>>> ⟨Ĥ⟩
No, because a given H will only go to one of the answers. >>>>>>>>>>>>>>>>> THAT
will be wrong, and the other one right.
∀ H ∈ Turing_Machine_Deciders
∃ TMD ∈ Turing_Machine_Descriptions |
Predicted_Behavior(H, TMD) != Actual_Behavior(TMD) >>>>>>>>>>>>>>>>
Not exactly. A pair of otherwise identical machines that >>>>>>>>>>>>>>>> (that are contained within the above specified set) >>>>>>>>>>>>>>>> only differ by return value will both be wrong on the >>>>>>>>>>>>>>>> same pathological input.
You mean a pair of DIFFERENT machines. Any difference is >>>>>>>>>>>>>>> different.
Every decider/input pair (referenced in the above set) has a >>>>>>>>>>>>>> corresponding decider/input pair that only differs by the >>>>>>>>>>>>>> return
value of its decider.
Nope.
∃ TMD ∈ Turing_Machine_Descriptions |
Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
Every H/TMD pair (referenced in the above set) has a
corresponding H/TMD pair that only differs by the return >>>>>>>>>>>> value of its Boolean_TM.
That isn't in the set above.
Nope, since both aren't in the set selected.
That both of these H/TMD pairs get the wrong answer proves that >>>>>>>>>>>> their question was incorrect because the opposite answer to the >>>>>>>>>>>> same question is also proven to be incorrect.
When they are deciders that must get the correct answer both >>>>>>>>>> of them are not in the set.
*IF* they are correct decider.
WHen we select from all Turing Machine Deciders, there is no >>>>>>>>> requirement that any of them get any particular answer right. >>>>>>>>>
So, ALL deciders are in the set that we cycle through and apply >>>>>>>>> the
following logic to ALL of them.
Each is them paired with an input that it will get wrong, and the >>>>>>>>> existance of the input was what as just proven, the ^ template >>>>>>>>>
When they are Turing_Machines_Returning_Boolean the this
set inherently includes identical pairs that only differ
by return value.
But in the step of select and input that they will get wrong, they >>>>>>>>> will be givne DIFFERENT inputs.
You just don't understand what that statement is saying. >>>>>>>>>>>No the problem is that you are not paying attention.
I've expalined it, but it seems over you head.
No, you keep on making STUPID mistakes, like thinking that
select a
input that the machine will get wrong needs to be the same for two >>>>>>>>> differnt machines.
For Every H, we show we can find at least one input (chosen >>>>>>>>>>> just forWhen we use machine templates then we can see instances of >>>>>>>>>> the same machine that only differs by return value where both >>>>>>>>>> get the wrong answer on the same input. By same input I mean >>>>>>>>>> the same finite string of numerical values.
that machine) that it will get wrong.
But if they returned differnt values, they will have different >>>>>>>>> descriptions.
Otherwise, how could a UTM get the right answer, since it only >>>>>>>>> gets
the description.
We can get around all of this stuff by simply using this criteria: >>>>>>>> Date 10/13/2022 11:29:23 AM
*MIT Professor Michael Sipser agreed this verbatim paragraph is >>>>>>>> correct*
(He has neither reviewed nor agreed to anything else in this paper) >>>>>>>> (a) If simulating halt decider H correctly simulates its input D >>>>>>>> until H
correctly determines that its simulated D would never stop running >>>>>>>> unless aborted then
(b) H can abort its simulation of D and correctly report that D >>>>>>>> specifies a non-halting sequence of configurations.
*When we apply this criteria* (elaborated above)
Will you halt if you never abort your simulation?
*Then the halting problem is conquered*
When two different machines implementing this criteria
get different results from identical inputs then we
know that Pathological Self-Reference has been detected.
We don't even need to know that for:
*denial-of-service-attack detection*
*NO always means reject as unsafe*
But, Halting theorem never said "there's an input that halts
all machines", it just says "for any machine, there's an input
that halts it".
Where "halt the machine" means "put it in an infinite loop".
So, rather, Halting theorem never said "there's an input that
exhausts all machines", it just says, "for any machine, there's
an input that exhausts it".
I still don't see how that would be with infinite tapes though,
without a means of checking all the way right the tape in one
step, i.e. that it's 1's or 0's or any pattern at all, any
input that unbounded with respect to the machine basically
exhausts it or where the machine would run in the unbounded.
Of course any finite tape, has a static analysis that is
not infinite, that decides whether or not it halts
(or, loops, or grows, the state space of the decider).
Static analysis has to either enumerate or _infer_ the
state-space, where equal values in what's determined
the idempotent can detect loops, while inequalities
or proven divergence, ..., can detect unbounded growth.
Now, proving convergence or divergence is its own kind
of thing. For example, there are series that converge
very slowly, and series that diverge very slowly. These
are subtly intractable to analysis.
Then the usual idea of progressions that rapidly grow
yet return to what's detectable, are pathological to
analysis, only in resources not correctness or completion,
vis-a-vis the subtle intractability of the convergence or
divergence what either halts, or loops, or grows unboundedly.
Separately "not-halts" into "loops or grows unboundedly",
has that really for most matters of interest of understanding
the state-space of programs, is "halts or enters loops"
and not "grows unboundedly".
I.e. if the Halting problem is basically subject the
subtle intractability of slow convergence, that otherwise
it can just infer divergence and decide, practically
it's sort of more relevant what the machine would be
doing on the input on the tape, then with respect to
beyond the Turing theory, of the state of the read-head,
what happens when somebody modifies the tape, or events,
the write-head.
Anyways though for bounded inputs, besides slow divergence,
it's to be made clear that _most all_ and _almost all_
programs _are_ decided their behavior by static analysis.
Though, "most all" and "almost all" might be a bit strong,
but pretty much all that don't involve "the subtle intractability >>>>>>> of slow divergence".
Giving the idea that an existence result
is in any way the expected result here
seems sort of the root of this dilem-na.
(Though that the real numbers in ZFC have a well-ordering
and if they had a normal ordering that was a well-ordering,
that would be a thing, because ZFC has a well-ordering of
[0,1], but can't give one.)
What I'm saying is that you'd need to solve all
cases of slow convergence and slow divergence
to have a correct and complete halt decider,
for the unbounded, and finite, if not the infinite.
Most though setups of convergence though
would effectively exhaust according to the
existence of criteria of convergence as modeling
the computation, though, so, if you exclude
slow convergence and slow divergence,
then a subset of very usual machines is of
course quite all decidable. (Or decide-able,
or as I put it once, "not deciddable".)
Well-ordering the reals in ZFC, that's it own
sort issue - that a well-ordering implies the
existence of a bijection from ordinals to a
set, then as that as tuples, subsets of those
are well-ordered by their ordinal part, then
that if ever uncountably many of those are
in normal order their real part and irrationals,
then between those are rationals. I.e.,
ZFC doesn't give an example of well-ordering
the reals, but there is one.
So, Peter, I think you're kind of misreading that
quote you authoritated. It just says "if static
analysis detects a loop or unboundedly growing
then it's not-halts".
Of course, for any bounded resource, there
are arbitrarily larger bounded resources
required by what's called pathological,
that an arbitrarily larger resource can though solve.
So anyways "slow convergence" and "slow divergence",
have that in mathematics, it's unknown for some series
whether they converge or diverge, though it's usually
accepted that the inverse powers of 2 converge and
that the sum of integers diverges, and that periodic
functions do neither.
I.e. the criteria of convergence and existence of a limit,
and the special case "diverges as going to infinity",
are not yet closed in today's mathematics.
It's sort of a conjecture or independent whether
they ever could be, closed, and complete, the criteria.
I have spent 20 years on The conventional halting problem proofs,
Gödel 1931 Incompleteness and the Liar Paradox. I have only spent
about seven years on Tarski Undefinability.
The Halting problem is the only one of these that has a formal system >>>>> that cannot have hidden gaps its its reasoning. If a machine can do it >>>>> then it can be done.
"Algorithmics" is a study of algorithms. The most usual
consideration is "asymptotics", "asymptotics of complexity",
with the idea that computing does work, that there are
closures and completions, and there are approximations and
attenuations, as it were, in the full and the final and
the initial and the partial.
The most usual resources are time, and space, that actions
occur in time, and according to a structure, in space.
Of course any CS grad pretty much knows that.
The objects of mathematics, have it so, that the finite
and bounded resources of any collection of digital and
binary machines, are finite and bounded, while, the unbounded
models of mathematics that represent them, are unbounded,
it's bad that anybody ever said a Turing tape was "infinite",
except that it's unbounded as an object of mathematics, and
the model of any finite, digital, binary machine.
One might aver "digital" has a complement in computing,
"analog", which represents the continuous instead of
the discrete, and one might aver "binary" has a complement,
either "real-valued" or "fuzzy" or variously about the
"multi-valent", in logic, the models, according to the
mechanisms, the models, their mechanics, machines.
The Turing machine though is a model of unbounded machine
as what any thusly simply mechanical model can implement,
up to its bounds.
So, the objects of mathematics, then involve analytical
results. I can't draw a circle, though a compass may
portray a diagram arbitrarily close, and no different.
It's the analytical results, that make so for why
mathematics is: "sensible, fungible, and tractable",
and for "the ubiquitous success of mathematics" in
all such matters of physics, which is a continuum mechanics,
and as well discrete mechanics.
Of course, results exist in mathematical objects,
after the unbounded the unbounded and complete,
which is why closures and completions are so
relevant to all such matters, vis-a-vis approximations
and attenuations, which in bounded terms result
indistinguishably non-different results,
up to precision, say, or tolerances.
So, "machine" might be discrete, binary, and digital,
or it might be continuous, multivalent, and analog.
The Turing machine in a concrete form, is bounded.
Similar models, here mostly for the stack models,
stack machines, among finite-state-machines then
for example concepts like "unbounded nondeterminism",
here is pretty much about determinism, about the
defined behavior of a machine according to a
mathematical model that given defined behavior
of the mechanisms results a mechanical model of
a mathematical model, such a theory as this.
So, the theory of computing, with mechanical or
computational models, models of computing, is
pretty much exactly like the theory of physics,
with the attachment of physical models to the
mathematical models, and rather, "interpreted as",
the models, the models, with respect to each other.
I.e. the extensionality that so follows "they're
equal, or same, or not different, they represent
each other, they're equi-interpretable, they model
each other", the models, of logical and mathematical
and computational and mechanical and physical models,
help represent that over all the entire thing there
is a usual theory for the entire thing, it's a theory
where model theory models extensionality, and in identity
intensionality, about equality, tautology, identity,
and sameness/difference, and nearness/farness, and all
these usual aspects of the mathematical models'
arithmetic, algebra, and geometry. (And function
theory and topology, with respect to categories and
types, sets and parts, relations and predicates,
then all the model-building among that which is
all the propositional or the stipulated or axiomatic.)
The idea is that this sort of platonistic universe of
mathematical objects has always existed, and it's just
discovered, then that axiomatics for example, just
sort of results a model theory of it, with regards
to models, the modular, modulation, modality, and the mode.
So, a machine, with respect to computation, establishing
the validity of interpretations of models, is subject to
filling out all the above, vis-a-vis what it can do,
and it can-do.
Then, "slowly converging" and "slowly diverging"
are examples that get get into that there are more
law(s) of large numbers, than the usual classical
law of large numbers.
Some people variously do or don't have a mathematical
model of larger numbers, or the infinite, at all.
It's a totally ancient and dogmatic tradition that
no, we finite peoples, persons, or agents of limited
and finite means, no we do not have discrete, digital,
binary mechanical models of the infinite.
Yet, it's also about the first thing that deduction
arrives at just from the plain contemplation of the
beyond the unbounded, like the theories of Democritus
and Atomism or the theories of Zeno and Continuity,
that make it so we at least have something to arrive
at for models of the unbounded, as "methods of exhaustion",
the approximation and attenuative what results
the asymptotics of algorithmics.
So, we do have mental models of the continuous and infinite.
And, where physics is a continuum mechanics,
there are physical models of it as well, then
with regards to the philosophy of science and
the scientific method and the Big Science and
from the Primary Science, lots to establish
that the continuous and analog has mathematical
results, as whethe, "sensible, fungible, and tractable",
to machines at all.
Numerical methods then, and, approximation and
attenuation, result from analysis, why they exist,
here for the criteria of convergence and existence
of limits, in the closed, in the complete.
So, a metonymy, unless it's "the Strong Metonymy",
which is utter intensionality of "the ground model",
is yet a metaphor and a metaphor yet a weaker metonymy
joint among weaker metonymys, that, ontology, has
that there are or may be a completion, of ontology,
as a "strong ontology" and "strong metonymy",
but otherwise it's just a bag of facts.
Here then there's certainly a perceived analytical
development Cantor space, and, Square Cantor space,
and, Signal Cantor space as it were, about that among
the law(s) of large numbers, there are definitions of
continuity, at least including field-reals, line-reals,
and signal-reals, three sorts continuous domains.
Mathematics owes physics this to better begin to
explain, to disclose, to find truths of, continuum mechanics.
Then that analog, fuzzy, multi-value, "quantum" computing
models (parts) of that, that discrete, binary, digital
computing does not, is a thing.
The convergence of terms in series is an example
of a model of things, that, sometimes has clear
and direct and perfectly accurate representations,
models, in stack machine, and sometimes doesn't.
Of course, for any bounded input, there is a sufficiently
larger bounded analyzer, that does solve it. So, why not
just put those all together? It would be infinite,
yet most things are not. (Or, you know, are.)
It's called "foundations" the study of all these things
as a fundamental coherency, a thing together, while of
its parts.
So, students should well have the concepts of "abacus"
and "slide rule" and other "tabulators and computers"
of the usual manual mechanical variety down, according
to the principles of the mathematical models behind them,
then introducing the law(s) of large numbers, then
introducing the idea of a standard model of an infinite,
about rates, related rates, asymptotics, and bounds.
"Foundations" then the idea is that it's been around
forever, it's our dogma and from our canon and it develops
over time, and the current edition is called "modern".
Did I make any mistakes?
Tarski anchors his proof in the Liar Paradox
https://liarparadox.org/Tarski_247_248.pdf
How Tarski encodes the Liar Paradox
x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x
This is transformed into line 01 of his proof
by replacing True with Provable
*Here is the Tarski Undefinability Theorem proof*
(1) x ∉ Provable if and only if p // assumption
(2) x ∈ True if and only if p // assumption
(3) x ∉ Provable if and only if x ∈ True.
(4) either x ∉ True or x̄ ∉ True; // axiom: ~True(x) ∨ ~True(~x)
(5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x) >>> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: Provable(~x) → True(~x)
(7) x ∈ True
(8) x ∉ Provable
(9) x̄ ∉ Provable
Tarski's Full proof
https://liarparadox.org/Tarski_275_276.pdf
Well I have some collected works of Tarski so
I'll be looking to them, the excerpt there you
reference basically talks about the model under
consideration, as a fragment, of "the meta-theory",
i.e., "Tarski's meta-theory" is "rather underdefined",
that what he does is that he says that in the meta-theory,
that there's something about the model under consideration,
that isn't true in the model but is true, and he invokes
Goedel to basically says that Goedel's incompleteness
applies to this model, which of course must thusly by
"strong enough to support arithmetic", where otherwise
of course, Goedel's incompleteness does _not_ apply
to theories that may be entirely closed categories.
It seems the issue is that you think that you have
entirely closed categories, in your meta-theory,
then are applying this meta-theory to another model
under consideration, which isn't, so it seems like
you have implicits on your model, which are underdefined,
rather as the "properly logical" or "non-logical" are
with respect to being modeled by the logical, that
you have the some sort of "un-logical" that are the
result of that the axioms are un-logical, with regards
to that your "meta-theory" is logical and your model
under consideration, its _logical_ objects, are actually
"un-logical" objects with respect to your, "meta-theory".
That's a problem overall with "meta-theory", here the
goal is to arrive at a theory that is its own meta-theory,
that's also all one language, with regards to otherwise
Yes that is exactly correct.
*True(L,x) returns TRUE when x is True otherwise returns FALSE*
*Easier to understand in English*
LP = "This sentence is not true."
Boolean True(English, LP) returns FALSE
Boolean True(English, ~LP) returns FALSE
I formalize this in my own minimal type theory that has actual
self-reference using the := {is defined as} operator. https://en.wikipedia.org/wiki/List_of_logic_symbols
LP := ~True(LP)
*Minimal Type Theory* (YACC BNF) https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
Prolog rejects this
?- LP = not(true(LP)).
LP = not(true(LP)).
?- unify_with_occurs_check(LP, not(true(LP))).
false.
It seems that Tarski's theory get stuck on the Liar Paradox
whereas mine does not because it is anchored in the Prolog
model where True means proven by axioms and false means
(negation as failure) unprovable from axioms.
We can still get back to the conventional notion of false
when the negation of the sentence in unprovable from axioms.
these sorts "fragments", of theories, interpreting each
other, which is only ordinary, and when the objects
are structurally self-defining or under-defined, as it
were, when you're talking about quantifier ambiguity,
in a fragment with decision or the non-logical about
quantifier ambiguity in the meta-theory, then you've
basically stipulations which aren't so much "true axioms"
as "preconceived notions". They're stipulations.
It seems you're sort of invoking a wash among
meta-theory, theory, fragment, theory, meta-theory,
theory, fragment, theory - and not modeling it,
instead being sort of memoryless and unconscious
about it, it's sort of considered unconscientious
about it.
I've been talking about these kinds of things in
my podcasts recently, or I read a biography of Tarski
and sort of address modern metaphysics and with regards
to formal foundations .
https://www.youtube.com/watch?v=lNYTU97XCMw&list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY&index=28
For example the ontology and teleology have a
sort of dualism, here like "fragment" and "meta-theory",
the goal is a sort of "monist dualism" that's a
sort of "dualist monism", here about the "logical".
Good luck though it's a usual thing, about what
there's basically _expansion_ of comprehension first,
then that _restriction_ of comprehension is always
fragments, it really does get into idea like for
computability theory "the model of the integers,
is only either a fragment or extension, more than
standard".
So, book-keeping your meta-theories or abstractions
and fragments or grasps, it's like, "get a grip",
right, but, "left-hand/right-hand".
It's sort of like "left-hand/right-hand:
stop hitting yourself, I'm not doing it".
On 3/14/2024 5:01 PM, Richard Damon wrote:
On 3/14/24 12:01 PM, olcott wrote:
On 3/14/2024 11:58 AM, Ross Finlayson wrote:
On 03/13/2024 10:20 PM, olcott wrote:
On 3/13/2024 1:16 PM, Ross Finlayson wrote:
On 03/12/2024 09:00 PM, olcott wrote:
On 3/12/2024 10:49 PM, Ross Finlayson wrote:
On 03/12/2024 08:23 PM, Ross Finlayson wrote:
On 03/12/2024 07:52 PM, olcott wrote:
On 3/12/2024 9:28 PM, Richard Damon wrote:
On 3/12/24 4:31 PM, olcott wrote:
On 3/12/2024 6:11 PM, Richard Damon wrote:
On 3/12/24 3:53 PM, olcott wrote:
On 3/12/2024 5:30 PM, Richard Damon wrote:
On 3/12/24 2:34 PM, olcott wrote:∀ H ∈ Turing_Machines_Returning_Boolean
On 3/12/2024 4:23 PM, Richard Damon wrote:
On 3/12/24 1:11 PM, olcott wrote:
On 3/12/2024 2:40 PM, Richard Damon wrote: >>>>>>>>>>>>>>>>>>> On 3/12/24 12:02 PM, olcott wrote:
On 3/12/2024 1:31 PM, immibis wrote:
On 12/03/24 19:12, olcott wrote:
∀ H ∈ Turing_Machine_DecidersAnd it can be a different TMD to each H. >>>>>>>>>>>>>>>>>>>>>
∃ TMD ∈ Turing_Machine_Descriptions | >>>>>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD) >>>>>>>>>>>>>>>>>>>>>>
There is some input TMD to every H such that >>>>>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD) >>>>>>>>>>>>>>>>>>>>>
When we disallow decider/input pairs that are >>>>>>>>>>>>>>>>>>>>>> incorrectOnce we understand that either YES or NO is the right >>>>>>>>>>>>>>>>>>>>> answer, the whole rebuttal is tossed out as invalid >>>>>>>>>>>>>>>>>>>>> and
questions where both YES and NO are the wrong answer >>>>>>>>>>>>>>>>>>>>>
incorrect.
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to
⟨Ĥ⟩
halts
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn // Ĥ applied to
⟨Ĥ⟩
does
not halt
BOTH YES AND NO ARE THE WRONG ANSWER FOR EVERY Ĥ.H >>>>>>>>>>>>>>>>>>>> ⟨Ĥ⟩ ⟨Ĥ⟩
No, because a given H will only go to one of the >>>>>>>>>>>>>>>>>>> answers.
THAT
will be wrong, and the other one right.
∀ H ∈ Turing_Machine_Deciders
∃ TMD ∈ Turing_Machine_Descriptions | >>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD) >>>>>>>>>>>>>>>>>>
Not exactly. A pair of otherwise identical machines that >>>>>>>>>>>>>>>>>> (that are contained within the above specified set) >>>>>>>>>>>>>>>>>> only differ by return value will both be wrong on the >>>>>>>>>>>>>>>>>> same pathological input.
You mean a pair of DIFFERENT machines. Any difference is >>>>>>>>>>>>>>>>> different.
Every decider/input pair (referenced in the above set) >>>>>>>>>>>>>>>> has a
corresponding decider/input pair that only differs by the >>>>>>>>>>>>>>>> return
value of its decider.
Nope.
∃ TMD ∈ Turing_Machine_Descriptions |
Predicted_Behavior(H, TMD) != Actual_Behavior(TMD) >>>>>>>>>>>>>>
Every H/TMD pair (referenced in the above set) has a >>>>>>>>>>>>>> corresponding H/TMD pair that only differs by the return >>>>>>>>>>>>>> value of its Boolean_TM.
That isn't in the set above.
Nope, since both aren't in the set selected.
That both of these H/TMD pairs get the wrong answer proves >>>>>>>>>>>>>> that
their question was incorrect because the opposite answer >>>>>>>>>>>>>> to the
same question is also proven to be incorrect.
When they are deciders that must get the correct answer both >>>>>>>>>>>> of them are not in the set.
*IF* they are correct decider.
WHen we select from all Turing Machine Deciders, there is no >>>>>>>>>>> requirement that any of them get any particular answer right. >>>>>>>>>>>
So, ALL deciders are in the set that we cycle through and >>>>>>>>>>> apply the
following logic to ALL of them.
Each is them paired with an input that it will get wrong, and >>>>>>>>>>> the
existance of the input was what as just proven, the ^ template >>>>>>>>>>>
When they are Turing_Machines_Returning_Boolean the this >>>>>>>>>>>> set inherently includes identical pairs that only differ >>>>>>>>>>>> by return value.
But in the step of select and input that they will get wrong, >>>>>>>>>>> they
will be givne DIFFERENT inputs.
You just don't understand what that statement is saying. >>>>>>>>>>>>>No the problem is that you are not paying attention.
I've expalined it, but it seems over you head.
No, you keep on making STUPID mistakes, like thinking that >>>>>>>>>>> select a
input that the machine will get wrong needs to be the same >>>>>>>>>>> for two
differnt machines.
For Every H, we show we can find at least one input (chosen >>>>>>>>>>>>> just forWhen we use machine templates then we can see instances of >>>>>>>>>>>> the same machine that only differs by return value where both >>>>>>>>>>>> get the wrong answer on the same input. By same input I mean >>>>>>>>>>>> the same finite string of numerical values.
that machine) that it will get wrong.
But if they returned differnt values, they will have different >>>>>>>>>>> descriptions.
Otherwise, how could a UTM get the right answer, since it >>>>>>>>>>> only gets
the description.
We can get around all of this stuff by simply using this
criteria:
Date 10/13/2022 11:29:23 AM
*MIT Professor Michael Sipser agreed this verbatim paragraph is >>>>>>>>>> correct*
(He has neither reviewed nor agreed to anything else in this >>>>>>>>>> paper)
(a) If simulating halt decider H correctly simulates its input D >>>>>>>>>> until H
correctly determines that its simulated D would never stop >>>>>>>>>> running
unless aborted then
(b) H can abort its simulation of D and correctly report that D >>>>>>>>>> specifies a non-halting sequence of configurations.
*When we apply this criteria* (elaborated above)
Will you halt if you never abort your simulation?
*Then the halting problem is conquered*
When two different machines implementing this criteria
get different results from identical inputs then we
know that Pathological Self-Reference has been detected.
We don't even need to know that for:
*denial-of-service-attack detection*
*NO always means reject as unsafe*
But, Halting theorem never said "there's an input that halts >>>>>>>>> all machines", it just says "for any machine, there's an input >>>>>>>>> that halts it".
Where "halt the machine" means "put it in an infinite loop". >>>>>>>>>
So, rather, Halting theorem never said "there's an input that >>>>>>>>> exhausts all machines", it just says, "for any machine, there's >>>>>>>>> an input that exhausts it".
I still don't see how that would be with infinite tapes though, >>>>>>>>> without a means of checking all the way right the tape in one >>>>>>>>> step, i.e. that it's 1's or 0's or any pattern at all, any
input that unbounded with respect to the machine basically
exhausts it or where the machine would run in the unbounded. >>>>>>>>>
Of course any finite tape, has a static analysis that is
not infinite, that decides whether or not it halts
(or, loops, or grows, the state space of the decider).
Static analysis has to either enumerate or _infer_ the
state-space, where equal values in what's determined
the idempotent can detect loops, while inequalities
or proven divergence, ..., can detect unbounded growth.
Now, proving convergence or divergence is its own kind
of thing. For example, there are series that converge
very slowly, and series that diverge very slowly. These
are subtly intractable to analysis.
Then the usual idea of progressions that rapidly grow
yet return to what's detectable, are pathological to
analysis, only in resources not correctness or completion,
vis-a-vis the subtle intractability of the convergence or
divergence what either halts, or loops, or grows unboundedly. >>>>>>>>>
Separately "not-halts" into "loops or grows unboundedly",
has that really for most matters of interest of understanding >>>>>>>>> the state-space of programs, is "halts or enters loops"
and not "grows unboundedly".
I.e. if the Halting problem is basically subject the
subtle intractability of slow convergence, that otherwise
it can just infer divergence and decide, practically
it's sort of more relevant what the machine would be
doing on the input on the tape, then with respect to
beyond the Turing theory, of the state of the read-head,
what happens when somebody modifies the tape, or events,
the write-head.
Anyways though for bounded inputs, besides slow divergence,
it's to be made clear that _most all_ and _almost all_
programs _are_ decided their behavior by static analysis.
Though, "most all" and "almost all" might be a bit strong,
but pretty much all that don't involve "the subtle intractability >>>>>>>>> of slow divergence".
Giving the idea that an existence result
is in any way the expected result here
seems sort of the root of this dilem-na.
(Though that the real numbers in ZFC have a well-ordering
and if they had a normal ordering that was a well-ordering,
that would be a thing, because ZFC has a well-ordering of
[0,1], but can't give one.)
What I'm saying is that you'd need to solve all
cases of slow convergence and slow divergence
to have a correct and complete halt decider,
for the unbounded, and finite, if not the infinite.
Most though setups of convergence though
would effectively exhaust according to the
existence of criteria of convergence as modeling
the computation, though, so, if you exclude
slow convergence and slow divergence,
then a subset of very usual machines is of
course quite all decidable. (Or decide-able,
or as I put it once, "not deciddable".)
Well-ordering the reals in ZFC, that's it own
sort issue - that a well-ordering implies the
existence of a bijection from ordinals to a
set, then as that as tuples, subsets of those
are well-ordered by their ordinal part, then
that if ever uncountably many of those are
in normal order their real part and irrationals,
then between those are rationals. I.e.,
ZFC doesn't give an example of well-ordering
the reals, but there is one.
So, Peter, I think you're kind of misreading that
quote you authoritated. It just says "if static
analysis detects a loop or unboundedly growing
then it's not-halts".
Of course, for any bounded resource, there
are arbitrarily larger bounded resources
required by what's called pathological,
that an arbitrarily larger resource can though solve.
So anyways "slow convergence" and "slow divergence",
have that in mathematics, it's unknown for some series
whether they converge or diverge, though it's usually
accepted that the inverse powers of 2 converge and
that the sum of integers diverges, and that periodic
functions do neither.
I.e. the criteria of convergence and existence of a limit,
and the special case "diverges as going to infinity",
are not yet closed in today's mathematics.
It's sort of a conjecture or independent whether
they ever could be, closed, and complete, the criteria.
I have spent 20 years on The conventional halting problem proofs, >>>>>>> Gödel 1931 Incompleteness and the Liar Paradox. I have only spent >>>>>>> about seven years on Tarski Undefinability.
The Halting problem is the only one of these that has a formal
system
that cannot have hidden gaps its its reasoning. If a machine can >>>>>>> do it
then it can be done.
"Algorithmics" is a study of algorithms. The most usual
consideration is "asymptotics", "asymptotics of complexity",
with the idea that computing does work, that there are
closures and completions, and there are approximations and
attenuations, as it were, in the full and the final and
the initial and the partial.
The most usual resources are time, and space, that actions
occur in time, and according to a structure, in space.
Of course any CS grad pretty much knows that.
The objects of mathematics, have it so, that the finite
and bounded resources of any collection of digital and
binary machines, are finite and bounded, while, the unbounded
models of mathematics that represent them, are unbounded,
it's bad that anybody ever said a Turing tape was "infinite",
except that it's unbounded as an object of mathematics, and
the model of any finite, digital, binary machine.
One might aver "digital" has a complement in computing,
"analog", which represents the continuous instead of
the discrete, and one might aver "binary" has a complement,
either "real-valued" or "fuzzy" or variously about the
"multi-valent", in logic, the models, according to the
mechanisms, the models, their mechanics, machines.
The Turing machine though is a model of unbounded machine
as what any thusly simply mechanical model can implement,
up to its bounds.
So, the objects of mathematics, then involve analytical
results. I can't draw a circle, though a compass may
portray a diagram arbitrarily close, and no different.
It's the analytical results, that make so for why
mathematics is: "sensible, fungible, and tractable",
and for "the ubiquitous success of mathematics" in
all such matters of physics, which is a continuum mechanics,
and as well discrete mechanics.
Of course, results exist in mathematical objects,
after the unbounded the unbounded and complete,
which is why closures and completions are so
relevant to all such matters, vis-a-vis approximations
and attenuations, which in bounded terms result
indistinguishably non-different results,
up to precision, say, or tolerances.
So, "machine" might be discrete, binary, and digital,
or it might be continuous, multivalent, and analog.
The Turing machine in a concrete form, is bounded.
Similar models, here mostly for the stack models,
stack machines, among finite-state-machines then
for example concepts like "unbounded nondeterminism",
here is pretty much about determinism, about the
defined behavior of a machine according to a
mathematical model that given defined behavior
of the mechanisms results a mechanical model of
a mathematical model, such a theory as this.
So, the theory of computing, with mechanical or
computational models, models of computing, is
pretty much exactly like the theory of physics,
with the attachment of physical models to the
mathematical models, and rather, "interpreted as",
the models, the models, with respect to each other.
I.e. the extensionality that so follows "they're
equal, or same, or not different, they represent
each other, they're equi-interpretable, they model
each other", the models, of logical and mathematical
and computational and mechanical and physical models,
help represent that over all the entire thing there
is a usual theory for the entire thing, it's a theory
where model theory models extensionality, and in identity
intensionality, about equality, tautology, identity,
and sameness/difference, and nearness/farness, and all
these usual aspects of the mathematical models'
arithmetic, algebra, and geometry. (And function
theory and topology, with respect to categories and
types, sets and parts, relations and predicates,
then all the model-building among that which is
all the propositional or the stipulated or axiomatic.)
The idea is that this sort of platonistic universe of
mathematical objects has always existed, and it's just
discovered, then that axiomatics for example, just
sort of results a model theory of it, with regards
to models, the modular, modulation, modality, and the mode.
So, a machine, with respect to computation, establishing
the validity of interpretations of models, is subject to
filling out all the above, vis-a-vis what it can do,
and it can-do.
Then, "slowly converging" and "slowly diverging"
are examples that get get into that there are more
law(s) of large numbers, than the usual classical
law of large numbers.
Some people variously do or don't have a mathematical
model of larger numbers, or the infinite, at all.
It's a totally ancient and dogmatic tradition that
no, we finite peoples, persons, or agents of limited
and finite means, no we do not have discrete, digital,
binary mechanical models of the infinite.
Yet, it's also about the first thing that deduction
arrives at just from the plain contemplation of the
beyond the unbounded, like the theories of Democritus
and Atomism or the theories of Zeno and Continuity,
that make it so we at least have something to arrive
at for models of the unbounded, as "methods of exhaustion",
the approximation and attenuative what results
the asymptotics of algorithmics.
So, we do have mental models of the continuous and infinite.
And, where physics is a continuum mechanics,
there are physical models of it as well, then
with regards to the philosophy of science and
the scientific method and the Big Science and
from the Primary Science, lots to establish
that the continuous and analog has mathematical
results, as whethe, "sensible, fungible, and tractable",
to machines at all.
Numerical methods then, and, approximation and
attenuation, result from analysis, why they exist,
here for the criteria of convergence and existence
of limits, in the closed, in the complete.
So, a metonymy, unless it's "the Strong Metonymy",
which is utter intensionality of "the ground model",
is yet a metaphor and a metaphor yet a weaker metonymy
joint among weaker metonymys, that, ontology, has
that there are or may be a completion, of ontology,
as a "strong ontology" and "strong metonymy",
but otherwise it's just a bag of facts.
Here then there's certainly a perceived analytical
development Cantor space, and, Square Cantor space,
and, Signal Cantor space as it were, about that among
the law(s) of large numbers, there are definitions of
continuity, at least including field-reals, line-reals,
and signal-reals, three sorts continuous domains.
Mathematics owes physics this to better begin to
explain, to disclose, to find truths of, continuum mechanics.
Then that analog, fuzzy, multi-value, "quantum" computing
models (parts) of that, that discrete, binary, digital
computing does not, is a thing.
The convergence of terms in series is an example
of a model of things, that, sometimes has clear
and direct and perfectly accurate representations,
models, in stack machine, and sometimes doesn't.
Of course, for any bounded input, there is a sufficiently
larger bounded analyzer, that does solve it. So, why not
just put those all together? It would be infinite,
yet most things are not. (Or, you know, are.)
It's called "foundations" the study of all these things
as a fundamental coherency, a thing together, while of
its parts.
So, students should well have the concepts of "abacus"
and "slide rule" and other "tabulators and computers"
of the usual manual mechanical variety down, according
to the principles of the mathematical models behind them,
then introducing the law(s) of large numbers, then
introducing the idea of a standard model of an infinite,
about rates, related rates, asymptotics, and bounds.
"Foundations" then the idea is that it's been around
forever, it's our dogma and from our canon and it develops
over time, and the current edition is called "modern".
Did I make any mistakes?
Tarski anchors his proof in the Liar Paradox
https://liarparadox.org/Tarski_247_248.pdf
How Tarski encodes the Liar Paradox
x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x
This is transformed into line 01 of his proof
by replacing True with Provable
*Here is the Tarski Undefinability Theorem proof*
(1) x ∉ Provable if and only if p // assumption
(2) x ∈ True if and only if p // assumption
(3) x ∉ Provable if and only if x ∈ True.
(4) either x ∉ True or x̄ ∉ True; // axiom: ~True(x) ∨ ~True(~x)
(5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x) >>>>> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: Provable(~x) → True(~x)
(7) x ∈ True
(8) x ∉ Provable
(9) x̄ ∉ Provable
Tarski's Full proof
https://liarparadox.org/Tarski_275_276.pdf
Well I have some collected works of Tarski so
I'll be looking to them, the excerpt there you
reference basically talks about the model under
consideration, as a fragment, of "the meta-theory",
i.e., "Tarski's meta-theory" is "rather underdefined",
that what he does is that he says that in the meta-theory,
that there's something about the model under consideration,
that isn't true in the model but is true, and he invokes
Goedel to basically says that Goedel's incompleteness
applies to this model, which of course must thusly by
"strong enough to support arithmetic", where otherwise
of course, Goedel's incompleteness does _not_ apply
to theories that may be entirely closed categories.
It seems the issue is that you think that you have
entirely closed categories, in your meta-theory,
then are applying this meta-theory to another model
under consideration, which isn't, so it seems like
you have implicits on your model, which are underdefined,
rather as the "properly logical" or "non-logical" are
with respect to being modeled by the logical, that
you have the some sort of "un-logical" that are the
result of that the axioms are un-logical, with regards
to that your "meta-theory" is logical and your model
under consideration, its _logical_ objects, are actually
"un-logical" objects with respect to your, "meta-theory".
That's a problem overall with "meta-theory", here the
goal is to arrive at a theory that is its own meta-theory,
that's also all one language, with regards to otherwise
Yes that is exactly correct.
*True(L,x) returns TRUE when x is True otherwise returns FALSE*
So what does True(L, S) return when S is defined as S := ~True(L,S)
True(L, S) returns Prolog's negation as failure false
that only means not true.
True(L, ~S) also returns Prolog's negation as failure false
that only means not true.
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