interpretative ideas forged by Gödel's Incompleteness Theorems, — Aryamoy Mitra
Fastest way: Godel's theorems are rigorous arguments with a rigorously determined subject matter. We understand, then, or not, or to some degree which itself requires gauging. Interpretation, on the other hand, is all the area outside the rigor, and while some fun, even poetically interesting, most is nonsense. — tim wood
Thanks - that's what the literature suggests, too.1) Any arithmetic model? No. — tim wood
Once again, a presumed extension of the first answer.2) Crisis at the heart? No — tim wood
3) Transcending systems? No. Although one can think about the system as a system, from "outside" of the system. In this instance called meta-mathematical thinking. And I think that one solves all Godelian problems by adding as axioms the critical sentences. In any case, there is no "Royal road to knowledge" certainly with respect to Godel's theories and thinking, there's just the work of it. — tim wood
I think the idea is that any system of sufficient power is subject to Godelian self-referential sentences that are in that system unprovable, but provable in an expanded system that incudes the sentence as an axiom, that extendibility into the transfinite. But if you run with this, your almost certainly running into speculation, which Godel really is not about. Your choice: riff on it in some ultimately nonsensical way or read a book or two. Godel's 1931 undecidability paper in English is very readable.Isn't there, however, a self-perpetuating element to Gödel incompleteness (that is, irrespective of how many new axioms one defines a set or structure with, an unprovable sentence can always be derived within it)? — Aryamoy Mitra
In my view, no.Secondly, doesn't the absence of self-consistency foreshadow, that an intractable crisis permeates the heart of all (conceivable) logical architectures? — Aryamoy Mitra
systematically finite — Aryamoy Mitra
characterized by Peano Arithmetic — Aryamoy Mitra
simultaneously unprovable — Aryamoy Mitra
entirely bereft of composite statements — Aryamoy Mitra
ceases to unequivocally demonstrate its own mathematical consistency — Aryamoy Mitra
when accorded a finite list of axioms or postulates, do there always result either indeterminate, or unprovable truth values — Aryamoy Mitra
absence of self-consistency — Aryamoy Mitra
finite set of elementary axioms — Aryamoy Mitra
in order to illustrate a logical system's consistency, one must transcend the system entirely — Aryamoy Mitra
Isn't that tantamount, for instance, to asserting that all finitely synthesized constructs of reasoning, are by their existence inconsistent? — Aryamoy Mitra
how many new axioms one defines a set or structure with, an unprovable sentence can always be derived within it — Aryamoy Mitra
Where do you find such terminology in discussions of incompleteness? Where did you read such things? — TonesInDeepFreeze
If a theory is recursively axiomatizable, sufficiently arithmetically expressive, and consistent (let's call these 'G-theories'), then it is incomplete, no matter whether the set of axioms is finite or infinite. Some theories that are not G- theories are complete (and some are finitely axiomatizable). — TonesInDeepFreeze
What lack of consistency? Incompleteness doesn't say that e.g. PA or ZFC are inconsistent. Rather, a proof of consistency is not available within their own systems. — TonesInDeepFreeze
Some theories are recursively axiomatizable with even an infinite set of axioms. For some reason you are stuck on a notion of finite axiomatization that is not relevant in this regard. — TonesInDeepFreeze
Also, 'elementary' has a technical meaning different from your use. — TonesInDeepFreeze
You shouldn't generalize about "logical systems" but rather you should be accurate by addressing just G-theories in this context. And the answer is yes; to prove the consistency of a G-theory we have to do that in some other theory (which itself could be another G-theory). For example, Z set theory proves the consistency of PA. — TonesInDeepFreeze
I think the idea is that any system of sufficient power is subject to Godelian self-referential sentences that are in that system unprovable, but provable in an expanded system that incudes the sentence as an axiom, that extendibility into the transfinite. But if you run with this, your almost certainly running into speculation, which Godel really is not about. — tim wood
Neither of them are terminological, in the first place. — Aryamoy Mitra
What lack of consistency? Incompleteness doesn't say that e.g. PA or ZFC are inconsistent. Rather, a proof of consistency is not available within their own systems.
— TonesInDeepFreeze
That is literally, what 'self-consistency' denotes (demonstrating a self-referential consistency, from within a system). — Aryamoy Mitra
axiomatic finiteness — Aryamoy Mitra
whether [...] Gödel incompleteness extends outside Peano Arithmetic. — Aryamoy Mitra
What does a non-elementary axiom entail? — Aryamoy Mitra
might the Gödelian constraints on certain proof-statement correspondences in formal languages, lend itself to the underlying logical edifice? — Aryamoy Mitra
'self-consistency' is not ordinarily used in the sense of "proves its own consistency". Rather, 'self-consistency' is just a longer phrase for 'consistency'.
A theory is consistent if and only if there is not a formula such that the theory proves both the formula and the negation of the formula. — TonesInDeepFreeze
Do you mean that there are only a finite number of axioms? Or do you mean that the axioms entail that there are only finite sets, or something like that? — TonesInDeepFreeze
Do you mean to ask whether there are theories stronger than PA that are incomplete. Yes.
Or theories with all the axioms of PA plus more axioms and that are incomplete? Yes. — TonesInDeepFreeze
I don't know what you mean by 'proof-statement correspondences' nor what you mean by 'underlying logical edifice'. — TonesInDeepFreeze
Sincerely I say that your understanding of this subject would depend on familiarizing yourself with good books or articles on it, and with that you would have recognizable terminology in which to couch your questions about it. — TonesInDeepFreeze
'Godel's Theorem: An Incomplete Guide To Its Use And Abuse' - Torkel Franzen
Probably the best book ever written for introducing the subject of incompleteness for everday readers. — TonesInDeepFreeze
a finite number of axioms — Aryamoy Mitra
First-order logic (unless I'm mistaken) is a corollary of propositional logic — Aryamoy Mitra
[First-order logic] quantifies the interrelations between its subjects - as opposed to delineating them with logical connectives. — Aryamoy Mitra
With a 'logical edifice', I was referring to a set of ideas that stemmed from propositional conventions, which were then affixed with arithmetic operators. Won't any constraints on the latter, inclusive of Gödel incompleteness, emerge for the former (propositional ideas)? — Aryamoy Mitra
No, the opposite. First-order logic subsumes propositional logic. — TonesInDeepFreeze
I cannot make sense of that. I don't know what you mean by "a set of ideas that stemmed from propositional conventions, which were then affixed with arithmetic operators". — TonesInDeepFreeze
First-order logic, in most contexts, cannot exist without an underlying propositional logic (once again, unless I'm mistaken). — Aryamoy Mitra
'adding' and 'subtracting' propositional conditions from one another — Aryamoy Mitra
I still don't understand how something can be "true" and "unprovable" — TheMadFool
Incompleteness theorems.There are lots of people who think Godel's undecidable theorems are applicable to practically anything they can think of - and they're not. — tim wood
If the theorem F1 is unprovable — TheMadFool
I think there are vast more of those mathematicians who push the theorem to the sidelines close to the border of logic and logical inquiry and insist that it has nothing to do with anything else in the field of math than what the theorems state — ssu
Here by 'provable' and 'unprovable' we don't mean absolutely unprovable (i.e. not provable from any set of axioms) since there are no absolutely unprovable formulas. Rather we mean unprovable in whatever theory is in question (let's say it's PA for simplicity of exposition). — TonesInDeepFreeze
You are completely confused about this subject. — TonesInDeepFreeze
Here by 'provable' and 'unprovable' we don't mean absolutely unprovable (i.e. not provable from any set of axioms) since there are no absolutely unprovable formulas. Rather we mean unprovable in whatever theory is in question (let's say it's PA for simplicity of exposition). — TonesInDeepFreeze
Sentences are not true or false in a theory. — TonesInDeepFreeze
What is the source you read about this subject? — TonesInDeepFreeze
you didn't answer my question. Why? Please reread your reply to my question and my response to it. — TheMadFool
the Godel sentence becomes, "proven" (as true) AND "unprovable". Isn't this a contradiction? — TheMadFool
I'll leave it to you to connect the dots. — TheMadFool
Here by 'provable' and 'unprovable' we don't mean absolutely unprovable (i.e. not provable from any set of axioms) since there are no absolutely unprovable formulas. Rather we mean unprovable in whatever theory is in question (let's say it's PA for simplicity of exposition).
— TonesInDeepFreeze
Sentences are not true or false in a theory.
— TonesInDeepFreeze — TheMadFool
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