Comment by cushpush | Hacker News Reader

Comment by cushpush | Hacker News Reader

Parent Full thread

Reflecting deeply on Gödel's Completeness and Incompleteness Theorems, I'd like to contribute to the discussion. The essence of Completeness lies in the notion that a system, constructed on foundational axioms (such as facts, truths, or mathematical operations), encompasses all possible facts derivable within its framework. Interestingly, this completeness paradoxically begets contradictions. It implies the existence of 'axiomatic islands' - distinct realms of truth where a fact valid in one may contradict another in a different realm. Hence, Completeness inherently entails contradiction.

On the other hand, Incompleteness, which targets a specific subset of all possible truths or derivations, can achieve absolute consistency by avoiding these contradictions. This is because it deliberately excludes certain 'axiomatic regions,' maintaining coherence within its defined scope but at the expense of being incomplete.

Gödel's genius lay in unveiling the deceptive nature of mathematics when navigating these axiomatic territories. His theorems present a profound duality: the local context, where truths are confined to specific axiomatic systems, and the universal context, which can be akin to an 'ultimate truth' in spiritual terms. This ultimate truth transcends individual systems, offering a broader, more holistic understanding.

"A system encompasses all facts derivable within its framework" is a very vague sentence, but the way it's written I would consider it false. Second order logic is incomplete, for example.

The fact that different set of axioms lead to different conclusions is... kind of obvious? And not at all what logicians (or probably anybody) mean by "contradiction". Moreover, incompleteness doesn't at all prevent that. You can take ZFC + CH and ZFC + not(CH), both are incomplete, but they obviously entail different conclusions.

In more standard terminology, there are systems that are free of contradictions, such as Presburger arithmetic or propositional calculus.

Gödel's completeness and incompleteness theorems are really about entirely different things, completeness is about first-order logic as a system, incompleteness is about consistent, effectively axiomatisable theories of sufficient strength.

You are partly right, I made a mistake, I meant to say that contradiction is when we can derive both a statement and its negation, indicative of inconsistency. You are correct to point out that incompleteness is not a flaw, but a natural property of sufficiently complex systems and does not detract from their validity or usefulness. I must admit, I don't know enough about ZFC and the Continuum Hypothesis, but I did not mean to imply that all systems were incompatible.

> This ultimate truth transcends individual systems, offering a broader, more holistic understanding.

But the unknowability of that so-called "ultimate truth" implies that it's not offering anything, much less any understanding.

The article says Gödel thought that escape hatch for these contradictions is death, because all the local consistency and rationality hints at some greater consistency that must be there. However, that sounds a lot like wishful thinking.

Your insights on the inherent contradictions in rational frameworks, especially in the context of Gödel's Completeness and Incompleteness Theorems, bring us to an intriguing crossroads. Initially, Gödel’s Completeness Theorem suggests that within a certain logical system, all truths can be proven. However, his subsequent Incompleteness Theorem introduces a profound paradox: there are truths which, though existent within the system, cannot be proven by it. This contradiction between completeness and incompleteness in logic itself leads us to question: can these contradictions be reconciled to provide clarity?

Here, we might consider shifting our perspective from a purely rational approach to one that embraces intentionality and clarity. The journey from understanding Gödel’s theorems to grappling with the notion of contradictions could be seen as a metaphor for our search for truth. It’s not merely about accumulating information or solving logical puzzles; it's about the intentional pursuit of clarity. This pursuit often takes us beyond the realm of intellectual reasoning into a space where understanding becomes more about intuition and less about calculation.

As we delve deeper into this journey, we arrive at a crucial realization: perhaps the reconciliation of these contradictions and the understanding of 'ultimate truth' is less about logical resolution and more about experiential realization. In this light, truth is not something to be dissected in the confines of rational thought alone but to be lived and experienced. The clarity we seek may not lie in the resolution of logical paradoxes but in embracing the experiential wisdom that comes from directly engaging with these truths.

Thus, while Gödel’s work brilliantly navigates the complexities of logical systems, it also inadvertently points us toward a different kind of resolution - one that is realized not through further analysis but through personal experience. In essence, the journey from completeness to incompleteness, from information to intentionality, leads us to a profound experiential understanding, an ultimate truth that is realized rather than deduced.

You're constructing a paradox where none exists. The completeness and the incompleteness theorems don't contradict each other, they are about fundamentally different things (and it's really a shame we use the same terminology, because so many people are confused by it).

> there are truths which, though existent within the system, cannot be proven by it

This isn't what the first incompleteness theorem shows - rather, it shows that such "truths" don't exist within the system in the first place (in other words, there are some models in which a certain statement is true and others in which it is false - at least for first-order logic). Otherwise, this would indeed contradict the completeness theorem, but it doesn't.

I don't want to stop you from making your own metaphysical conclusions, but I'm not sure they're actually supported by the theorems themselves.

Reply on news.ycombinator.com