This is a legitimate, unbiased thought experiment looking for honest introspection. Let me summarize and then present two concise diametrical responses:
A century ago, a brilliant young mathematician proved that any system of logic will have true statements that cannot be proven. That is, there exist fundamental truths that you cannot use logic to prove. To be clear, this is not a logical trap or a self-referential paradox. It is a universally accepted fundamental proof in mathematics. "Completeness" in mathematics means that every question will have an answer, which is what Göedel proved is fundamentally not true.
Response 1: faith strengthening. Mathematicians have shown that no matter how advanced science and mathematics ever become, there will always be fundamental gaps in human understanding -- truths that we have to accept on faith alone. There will always be unprovable truths, and it is beyond all possible human knowledge to know which ones or why.
Response 2: faith weakening. How can an Omnipotent Deity even exist if such a Deity is unable to explain any particular concept? If an all-powerful God cannot explain something that is true, then there cannot exist an all-powerful God. The explanation for why or how something is true would be beyond even God's comprehension. Such a Deity cannot exist (or at least your "God" cannot be omnipotent).
For example, "no statement can refer to itself in the context of provability." Would this technically make it incomplete if the axiom itself is banning the statement?
He was an original member of the Vienna Circle before he developed his famous theorems. Does Gödel’s Incompleteness Theorems Disprove or Undermine the Verification Principle that was associated with verificationism and Logical Positivism?
When I was studying Gödel's results and their proofs I implicitly supposed that the background logic for the formal system in question is the classical first order logic with equality. But recently I found the following statement on Stanford Encyclopedia of Philosophy:
>Obviously, it is assumed that our formal systems are also equipped with a system of rules of inferences (and possibly some logical axioms), usually some standard system of classical logic (though the incompleteness theorems do not essentially presuppose classical logic, but also apply to systems with, e.g., intuitionistic logic).
So at least one of the axioms of the classical logic is not needed to prove incompleteness. It makes me wonder what exactly is required for the logic and inference rules to carry out the incompleteness results? Is it true that all we need are axioms for disjunction and equality, explosion principle for negation and modus ponens?
''not everything that is true can be proven''. Is that basically it? How does this help us?
I thought some people here might like this discussion piece and that it might, to some extent, clarify what’s going on with paraconsistent mathematics as an alternative to classical mathematics.
This might seem like it's coming out of left field but as I was going over this rather tragic part of mathematics I couldn't help but feel disturbed. Gödel's work in mathematics reveals that not all true statements can be proven to be true, that we can never know truly if our system of mathematics is internally consistent (If there is some paradox, or if you can prove A and not A are both true than it is inconsistent), and math isn't definite (leads to the Halting problem in computers, essentially in many situations it's impossible to determine whether something will terminate or not simply from the input). Since math is just a small abstraction of pure logic, and God is rational, Having limits to what can be logically known in maths, having limits in what reason can determine alone sort of irks me because it makes me wonder if that somehow limits God.
This isn't some can God make a square circle rubbish, at least with the Halting problem I feel sort of okay because God looking from eternity can just see the outcomes rather than have to reason it, but I was just wondering if anyone here has any background in mathematics and could explain why math can be incomplete and what it means/doesn't mean for God.
How much and what math do I need to really understand Gödel's incompleteness theorem? I'm not talking about understanding it at high level on a youtube video, more like being able to understand it at the level where I can recreate the proof myself.
Gödel on Wittgenstein:
> It is indeed clear from the passage that you cite that Wittgenstein did not understand my Theorem (or that he pretended not to understand it). He interprets it as a kind of logical paradox, while in fact it is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics).
I assume experts have analyzed this dispute, was Gödel correct?
Follow-up question: I often see Wittgenstein described as a philosopher and logician, as someone 'working on logic' and so on, he also spoke about the joy of solving math puzzles for fun, and enjoys this "logical positivism" sciency flair, people quote his comments on logic and math etc. But I find it very difficult to find any tractable contributions of W to formal logic, or really just any evidence of his competence in this field other than Russell saying he's clever. As far as I know he never proved a theorem, had no significant formal education in mathematical logic, and has no noteworthy contributions to logic. With Gödel, on top of this, thinking he didn't understand his results (which are now taught to undergraduate students), does this cast some doubt on the characterization of Wittgenstein as a logician? If he couldn't understand Gödel's results, do his thoughts on logic and math come from a place of sufficient competence and knowledge?
I've been reading about complex systems theory, and in most of the books I've read that try to provide a non-technical overview of the field Gödel and Turing are mentioned. The simplified, non-technical explanation of Gödel's theorem (I may be missing a lot here) that I've gleaned from these books - and this is surely a pretty ridiculous over-simplification - is that Gödel basically found a way to formalize the following self-referential proposition: "This proposition is unprovable." Given that this is kind of like Russel's "This statement is false" paradox, there's no way to determine the truth value of the proposition within the formal logical system, and so therefore that system is incomplete.
I'm an undergraduate studying computer science, so I only have a very limited math background (nothing beyond discrete math). But in thinking about Gödel's theorem, well, . . . I don't really get why it was so earth-shattering. I figure maybe my understanding of the theorem is too simple to fully grasp its implications?
I guess my thinking is this: isn't it possible that in a given logical system, Gödel's unprovable statement is the ONLY unprovable statement? Why does it matter so much? If it's the only unprovable statement, doesn't it seem like you could just ignore that one proposition in a logical system that could express a theoretically infinite (or at least very large) number of propositions? And just say something like "well, this system is complete except for this one dumb little statement Kurt came up with?" Or does the existence of the "This proposition is unprovable" possibly or necessarily imply that there is/might be other, more significant ways that any given logical system is incomplete? Or is my simplistic understanding of the incompleteness theorem so impoverished, unsophisticated, and generally off-base that I'm just not getting what Gödel proved and why it's important at all?
(this is a joke)
So you know how Ach'magut the Overseer basically wants to know everything? Well, it turns out he can't. Ever. It's not possible, mathematically, to know everything, because mathematics is incomplete. Any system of mathematics which includes arithmetic (i.e. 1+1, 3*5, 4^7) has statements which are true but impossible to prove.
The funniest part of this is that Ach'magut is smart enough to know this. He's got to have figured it out at this point, but he's a Great Elder and he has to keep trying. He's being eternally driven in a futile effort towards an unreachable goal.
Sometimes I pity the Great Elders more than the humans on the planet with them.
Next up: Urg'naut and the Conservation of Mass-energy
What are your thoughts on the common explanation of Gödel's theorem as meaning that there will always be statements that are true but unprovable?
Personally, I rather hate this explanation as it seems to me to be patently false: my understanding is that the theorem says that any formal language will result in contradictions and/or have statements which can be proven neither true nor false from within that language.
Equivalently, if a given formal language does not result in contradictions, there will be at least one statement within that language, call it S, such that there will be at least one model of that language where S is provably true and at least one model where it's provably false. The standard example is that the parallel postulate is independent of Euclid's other 4 postulates - there are consistent models of those four axioms where it's true -- Euclidean geometries -- and others where it's false -- non-Euclidean geometries.
Hence, seeing as any such statement will necessarily be true in some models and false in others, I really don't understand why incompleteness is so often characterized as meaning an incomplete system/language will contain true statements which are unprovable. What's with the emphasis on truth? We could as easily say it will contain statements which are false but can't be proven false, though that's not accurate either, as the true value of such a statement is simply independent of the relevant system.
However, I've heard this explanation so many times and often by people who really should know what they're talking about that it makes me wonder if I'm missing something. Anyone want to weigh in?
Is there an example of a very simple statement can not be proved nor disproved? (e.g. in Natural numbers?)
What would that be like? It doesn't fit my brain properly.
As part of Gödel's first incompleteness theorem, it is proven that it is possible to build a formula G such that:
- If Prv(x) is a formula that is true iff x is the Gödel number of a formula that is provable in the axiomatic system of arithmetic,
- and |G| denotes the Gödel number of G,
- then G ⇔ Prv(|G|) holds.
My question is, for the sake of curiosity, whether there has been any *effective construction* (likely by computer) of the formula G for some Gödel numbering scheme and realization of Prv?
Thanks,
Manuel
Digital Marxism is founded on intuitionistic logic
Intuitionistic logic is a first order logic
Gödels incompleteness theorem does not hold for first order logic
Gödels incompleteness theorem does then not hold for intuitionistic logic
Gödels incompleteness theorem does then not hold for Digital Marxism
Reviewing Gödel's Incompleteness Theorems recently, I've come across the following derivation, and am not sure where I'm going astray:
- PA |- Con(PA) -> G (Where G is the Gödel sentence. This is basically a restatement of the first incompleteness theorem inside of PA).
- PA |- G -> ~G (This is part of Gödel's proof)
- PA |- Con(PA) -> (G ^ ~G) (Combining 1 and 2)
- PA |- ~(G ^ ~G) -> ~Con(PA) (Contrapositive)
- PA |- (~G OR G) -> ~Con(PA) (De Morgan)
- PA |- ~Con(PA) (Applying Law of Excluded Middle)
I really doubt that PA |- ~Con(PA), but have no idea where I'm going wrong. Thank you.
I thought some people here might like this discussion piece and that it might, to some extent, clarify what’s going on with paraconsistent mathematics as an alternative to classical mathematics.
https://ojs.victoria.ac.nz/ajl/article/view/6926
Is the first order logic system the only logic system involved in the first and second Gödel's incompleteness theorems (https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems)? Not the other logic systems? (I guess so)
Thanks.
