PeteOlcott wrote: ↑Sat Mar 30, 2019 6:08 am
Scott Mayers wrote: ↑Sat Mar 30, 2019 5:27 am
I still need to first know what your own answer to, "Is there a sentence that proves itself is not provable?" Is your paper for a "yes" a "no", "both" or "neither"?
The model of Truth that Prolog has is the correct model of Truth.
If it is provable then it is true.
It only counts as false if its negation is provable.
If it is not provable then it is either unknown, false or erroneous.
If this slightly more complex expression including the Liar Paradox is false:
∃F ∈ Formal_Systems ∃G ∈ Language(F) (G ↔ ~(F ⊢ G))
Then Gödel's 1931 Incompleteness, and Tarski Undefinability have both been refuted.
If this: G ↔ ~(F ⊢ G) is self-contradictory then the above expression is false.
Within this formalized notion of truth: ∀F∀x True(F, x) ↔ (F ⊢ x)
This expression can only be true iff it is not true: G ↔ ~(F ⊢ G)
Okay, then you are depending upon the nature of Prolog to provide the proof. This sounds similar to what
logik mentioned a while back on another thread using his python. My concern is that when dealing with proofs about logic itself, you can use a programming language that is designed to test some system's logic, you have to TRUST the logic that system is using. I can't judge the nature of a Prolog's functioning logic (nor of Python that
logik was using) in order to know what its outputs of given inputs are. That is, anyone can design a machine such that it produces any output exists. So I can't trust a proof based on the 'authority' of some program's inputs and outputs. Those programs are only functionally useful for those already having faith in them for whatever reason.
Compare this to trusting some 'authority' of some religion. If one want to prove that their particular god exists, they can easily find some 'authority' who will respond positively to the question, "does God exist?"
Another example? A magician can "prove" something 'true' that in other contexts would not be. Thus, should the FACT that a magician can 'prove' to its audience that they can make a person disappear in one location and reappear in another suffice to prove THAT such a reality is true? Then reverse this also. If you technically can't get the magician to prove that their act is just a trick, does their
principle behavior to refuse doing this require we trust the presentation honest? That is, in principle, the magician can be unable to PROVE to us that their presentation of making people appear and disappear is a trick based upon their restricted principle of keeping their tricks a secret.
My point is you can't rely on the FAITH of a program for your counter-proofs against the original theories about incompleteness. I understood Prolog to be for those already trusting the prior logic but used to test NEW programs? I notice in my texts, that this is more functionally used in "discrete math" texts aimed for the computer science student's perspective, but NOT without a prior 'faith' in the depths of philosophical logic. I have one text that begins with presenting the first-order languages. If the creators of Prolog knew their own program sufficed to disprove the incompleteness theorems they are dependently trusting (because the system's discrete logic is itself restricted to binary logic underneath its hood), why were they not so smart as to have appealed to a disproof themselves of this before?
So let's take this apart without using the programs. Next, I see your use of predicate logic is trusted as you had faith in it as input in Prolog. Note that Godel himself PROVED the completeness and consistency of the Predicate logic, that Propositional logic is embedded in it as its precursor and proven complete and consistent, but that it leaves out any particular set theory, that might be included in Prolog's core logic. So perhaps we can look at what argument you might be wanting to present with a trust in Predicate logic? But wait, ....if Godel used these and was credited upon also the incompleteness theorems of his own, why would his own theorems stand unless we disprove that Predicate calculus or the propositional logic it is founded upon is false. You can't trust the 'child' logic if the 'child' is dependent upon the 'parent' logic that is itself questionable.
So we can't have faith in using a predicate logic that Godel USED to prove his own incompleteness theorem ....especially knowing that he was also the one that proved the predicate logic complete and consistent! That is you can't use the expression,
∃F ∈ Formal_Systems ∃G ∈ Language(F) (G ↔ ~(F ⊢ G))
without noticing that it is itself a Predicate logic that Godel specifically had a role in history of providing closure with. Based only on the FACT that he did this should raise questions about why you should trust USING his own system of arguments to disprove his own latter theorems about incompleteness, right?
Of course, it is wise to show how someone is being 'hypocritical' by USING their own logic against them, right?! So, pretend you have a 'proof' that proves Godel's thinking was faulty by using his own system of logic against him. You would then have a proof that his own original foundations are possibly up to question, like the predicate calculus.
Certainly if someone told me they have a penthouse on the eightieth floor of some hi-rise, they are implying that there is a foundation beneath them that is solid. If an architect building some hi-rise himself was living in that penthouse, like say Godel himself, was asserting he had a proof that you cannot 'completely' build all numbers of floors above him, that he would be rather odd to be implying that this means their is no completed foundation beneath him. Now if his own credibility is at stake for using his foundation to prove his theorem that all floors above him cannot be 'proven' to be build-able, he mustn't be implying that you can't continue to build another floor higher but that you can't PREDICT ahead of time a list of all the floors that can't be built above him.
So I need to know what you first think that the 'incompleteness' theorems are implying. If you are using Godel's reasoning against him, I need to know if you interpreted his argument correctly. To disprove his own theory built upon his foundation should this require you prove that you have some formula that can list the floors above him that cannot be built?
HOW can you have a third floor if you don't have the two floors beneath it first? Turing's tape presumed you AT LEAST can have two values printed on it. Boolean algebra was already trusted by Turing. AND it IS at minimally necessary or even his own logic can't be trusted.
