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Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)
Posted: Fri Apr 26, 2019 2:53 pm
by PeteOlcott
Here is the simplest way to explain my system
(1) If there is a (connected sequence of valid deductions from true premises to a true conclusion) then X is TRUE: (F ⊢ X).
(2) If there is a (connected sequence of valid deductions from true premises to the negation of the conclusion) then X is FALSE: (F ⊢ ¬X).
(3) Otherwise X UNSOUND
We use the Axioms of formal systems in place of true premises and the conclusion of a formal proof is called a consequence.
Re: Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)
Posted: Fri Apr 26, 2019 5:14 pm
by wtf
PeteOlcott wrote: ↑Fri Apr 26, 2019 2:53 pm
(1) If there is a (connected sequence of valid deductions from true premises to a true conclusion) then X is TRUE: (F ⊢ X).
How do you know your premises are true?
If by "true" all you mean is that they are your premises, then you are simply redefining the word "validity" with your own term "truth."
You've already said you don't believe in objective truth. Is that correct?
So if person X assumes that 2 + 2 = 4 and Y assumes that 2 + 2 = 5, then both of these things are true.
Is that your position?
Re: Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)
Posted: Fri Apr 26, 2019 5:55 pm
by PeteOlcott
wtf wrote: ↑Fri Apr 26, 2019 5:14 pm
PeteOlcott wrote: ↑Fri Apr 26, 2019 2:53 pm
(1) If there is a (connected sequence of valid deductions from true premises to a true conclusion) then X is TRUE: (F ⊢ X).
How do you know your premises are true?
I know that my premises are true because they are semantic tautologies:
https://www.britannica.com/topic/tautology
Tautology, in logic, a statement so framed that it cannot be denied
without inconsistency. Thus, “All humans are mammals” is held to
assert with regard to anything whatsoever that either it is a human
or it is not a mammal. But that universal “truth” follows not from
any facts noted about real humans but only from the actual use of
human and mammal and is thus purely a matter of definition.
Re: Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)
Posted: Fri Apr 26, 2019 7:10 pm
by wtf
PeteOlcott wrote: ↑Fri Apr 26, 2019 5:55 pm
I know that my premises are true because they are semantic tautologies:
You ignored my question.
Person X stipulates the axiom 2 + 2 = 4.
Person Y stipulates the axiom 2 + 2 = 5.
Are both axioms true by virtue of being axioms?
Re: Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)
Posted: Fri Apr 26, 2019 7:47 pm
by PeteOlcott
wtf wrote: ↑Fri Apr 26, 2019 7:10 pm
PeteOlcott wrote: ↑Fri Apr 26, 2019 5:55 pm
I know that my premises are true because they are semantic tautologies:
You ignored my question.
Person X stipulates the axiom 2 + 2 = 4.
Person Y stipulates the axiom 2 + 2 = 5.
Are both axioms true by virtue of being axioms?
I think that most people would agree that person Y would be a liar for
directly contradicting mutually agreed upon conventions.
If person Y could come up with some set of axioms such as PA that
make their stipulation coherent (such as the distinction between
Euclidean and non-Euclidean geometry) then it would be acceptable.
Simply rearranging the finite strings that are associated with elements
of the order set of natural numbers would not count because it would
not change the actual axioms it would only provide an alternative
way that these same axioms could be encoded.
Re: Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)
Posted: Fri Apr 26, 2019 7:59 pm
by Logik
PeteOlcott wrote: ↑Fri Apr 26, 2019 7:47 pm
I think that most people would agree that person Y would be a liar for
directly contradicting mutually agreed upon conventions.
I think most logicians would agree that this is a bandwagon fallacy.
PeteOlcott wrote: ↑Fri Apr 26, 2019 7:47 pm
If person Y could come up with some set of axioms such as PA that
make their stipulation coherent (such as the distinction between
Euclidean and non-Euclidean geometry) then it would be acceptable.
All unique axioms lead to such distinctions.
PeteOlcott wrote: ↑Fri Apr 26, 2019 7:47 pm
Simply rearranging the finite strings that are associated with elements
of the order set of natural numbers would not count because it would
not change the actual axioms it would only provide an alternative
way that these same axioms could be encoded.
I thought you said that the only property an axiom has is "truth". You are now telling us that axioms have "encoding". Like encoding meaning, or what?
Almost as if you can't decide if Tarski's is refuted or not.
Re: Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)
Posted: Fri Apr 26, 2019 11:52 pm
by wtf
PeteOlcott wrote: ↑Fri Apr 26, 2019 7:47 pm
wtf wrote: ↑Fri Apr 26, 2019 7:10 pm
Person X stipulates the axiom 2 + 2 = 4.
Person Y stipulates the axiom 2 + 2 = 5.
Are both axioms true by virtue of being axioms?
I think that most people would agree that person Y would be a liar for
directly contradicting mutually agreed upon conventions.
Then you agree that the truth value of a stipulative definition is subject to verification by conventions, authorities, or facts outside of the definition itself. Is that what you just said? You just defeated your entire argument.
Re: Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)
Posted: Sat Apr 27, 2019 12:08 am
by PeteOlcott
wtf wrote: ↑Fri Apr 26, 2019 7:10 pm
PeteOlcott wrote: ↑Fri Apr 26, 2019 5:55 pm
I know that my premises are true because they are semantic tautologies:
You ignored my question.
Person X stipulates the axiom 2 + 2 = 4.
Person Y stipulates the axiom 2 + 2 = 5.
Are both axioms true by virtue of being axioms?
If person Y could come up with some set of axioms such as PA that
make their stipulation coherent (such as the distinction between
Euclidean and non-Euclidean geometry) then it would be acceptable.
Then we would have two distinct formal systems each having their own
notion of truth relative that that formal system. It cannot contain any
contradictory axioms.
Re: Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)
Posted: Sat Apr 27, 2019 12:12 am
by PeteOlcott
Logik wrote: ↑Fri Apr 26, 2019 7:59 pm
PeteOlcott wrote: ↑Fri Apr 26, 2019 7:47 pm
If person Y could come up with some set of axioms such as PA that
make their stipulation coherent (such as the distinction between
Euclidean and non-Euclidean geometry) then it would be acceptable.
All unique axioms lead to such distinctions.
No you are wrong. We do not require a whole new formal system every time
that we add an axiom. We only need this when axioms form contradictions.
Re: Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)
Posted: Sat Apr 27, 2019 12:13 am
by PeteOlcott
Logik wrote: ↑Fri Apr 26, 2019 7:59 pm
PeteOlcott wrote: ↑Fri Apr 26, 2019 7:47 pm
Simply rearranging the finite strings that are associated with elements
of the order set of natural numbers would not count because it would
not change the actual axioms it would only provide an alternative
way that these same axioms could be encoded.
I thought you said that the only property an axiom has is "truth". You are now telling us that axioms have "encoding". Like encoding meaning, or what?
Almost as if you can't decide if Tarski's is refuted or not.
Every axiom must have an associated finite string. It cannot simply sit inside someone's head.
Re: Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)
Posted: Sat Apr 27, 2019 12:33 am
by wtf
PeteOlcott wrote: ↑Sat Apr 27, 2019 12:08 am
If person Y could come up with some set of axioms such as PA that
make their stipulation coherent (such as the distinction between
Euclidean and non-Euclidean geometry) then it would be acceptable.
I said nothing about PA. In fact these are
one-sentence axiom systems. The entirety of the axiom system is:
* 2 + 2 = 5.
There are no inference rules and no other axioms. There's only one axiom. I just wrote it down. There are no inferences other than 2 + 2 = 5. That's a perfectly valid deduction in this axiom system.
Now that is the axiom that I've formulated, using your idea of a stipulative definition. According to your theory, it must be true by virtue of being an axiom.
But you just told me that the truth value of my axiom is subject to verification based on external factors: what everyone believes, or what experts believe, or perhaps actual facts.
You have fatally contradicted your own theory. Axioms are neither true nor false in and of themselves.
Re: Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)
Posted: Sat Apr 27, 2019 1:02 am
by PeteOlcott
wtf wrote: ↑Sat Apr 27, 2019 12:33 am
PeteOlcott wrote: ↑Sat Apr 27, 2019 12:08 am
If person Y could come up with some set of axioms such as PA that
make their stipulation coherent (such as the distinction between
Euclidean and non-Euclidean geometry) then it would be acceptable.
I said nothing about PA. In fact these are
one-sentence axiom systems. The entirety of the axiom system is:
* 2 + 2 = 5.
There are no inference rules and no other axioms. There's only one axiom. I just wrote it down. There are no inferences other than 2 + 2 = 5. That's a perfectly valid deduction in this axiom system.
Now that is the axiom that I've formulated, using your idea of a stipulative definition. According to your theory, it must be true by virtue of being an axiom.
But you just told me that the truth value of my axiom is subject to verification based on external factors: what everyone believes, or what experts believe, or perhaps actual facts.
You have fatally contradicted your own theory. Axioms are neither true nor false in and of themselves.
No you are right that could be an axiom in its own formal system.
Because it contradicts many other formal systems it is excluded from them.
I think these things through more deeply as I get more feedback.
Re: Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)
Posted: Sat Apr 27, 2019 1:05 am
by wtf
PeteOlcott wrote: ↑Sat Apr 27, 2019 1:02 am
I think these things through more deeply as I get more feedback.
I am gratified to have made my point.
Re: Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)
Posted: Sat Apr 27, 2019 1:58 am
by PeteOlcott
wtf wrote: ↑Sat Apr 27, 2019 1:05 am
PeteOlcott wrote: ↑Sat Apr 27, 2019 1:02 am
I think these things through more deeply as I get more feedback.
I am gratified to have made my point.
Yes, your feedback and that of Logik has been useful.
Re: Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)
Posted: Sat Apr 27, 2019 1:18 pm
by Logik
PeteOlcott wrote: ↑Sat Apr 27, 2019 12:13 am
Every axiom must have an associated finite string. It cannot simply sit inside someone's head.
Sure. But the meaning of strings is different from system to system.
2+2 = 4 (decimal)
2+2 = 11 (ternary)
IF "=" means "semantic equivalence"
∴ 4 = 11
IF "=" means "symbolic equivalence" then it's false.