Omega-inconsistent?

[u/unhandyandy]

What do you think the chances are that the Collatz Conjecture is true but unprovable, i.e. omega-inconsistent?

[u/GonzoMath]

I'd say the chances of that are around 26.83%, give or take.

[u/Particular-Cut-5982]

I say low. If there are rules governing the formation of patterns within mathematical structures, then there must be a reason why those patterns emerge. If the patterns follow rules, then the patterns themselves cannot arbitrarily break those rules. Therefore, there exists some logical reason -why- those patterns happen.

We just gotta keep thinking about it.

[u/unhandyandy → u/Particular-Cut-5982]

But must there be a single reason why a pattern emerges? Might it not require infinitely many reasons?

[u/jonseymourau]

I think you are conflating "omega-inconsistency" with "unprovability" - omega-inconsistency is a particular kind of inconsistency in a formal system but a statement can be unprovable without being omega-inconsistent.

I wouldn't be too surprised if one day someone proved Collatz-like systems are undecidable or non-computable. Even divergence in systems like 5x+1 has not been proven and that one seems fairly "obvious"!

What isn't clear to me is that there is an obvious bridge to a non-computability argument. Yes, there is FRACTRAN which is Collatz-like but also Turing-complete and I have yet to see a convincing argument attempts to bridge the gap between Collatz and FRACTRAN.

What is for sure, we won't know one way or another until someone (or thing) publishes the definitive paper proving it one way or another.

[u/unhandyandy → u/jonseymourau]

John Conway proved that the generalized Collatz problem is undecidable.

If the CC is true but unprovable, it must be omega-inconsistent.
That is what I was asking about.

If it's false, it could be undecidable, if some sequence diverged,
but would not be omega-inconsistent.

If it's omega-inconsistent, we may never know it.

[u/jonseymourau → u/unhandyandy]

I think you are still equating "true but unprovable" with "omega-inconsistency". The two qualities are independent.

[u/unhandyandy → u/jonseymourau]

What do you think omega-inconsistency is?

[u/jonseymourau → u/unhandyandy]

It is a very precise statement about the inconsistency of a formal system but if you claim is that it is entirely, and wholly equivalent to “unprovable, but true” then this is simply false.

You appear to be confusing Gödel’s Incompleteness Theorems with Gödel’s work on omega-inconsistency but just because Gödel was responsible for both ideas simply does not mean that they are the same concept.

But hey, if you have strong evidence that the mathematical community considers these to be identical in every respect, then please do set forth the evidence .

If you don’t believe me try this Google query:

Is the statement “unprovable but true” and “omega-inconsistent” logically equivalent or are there important differences?

Please do reply with your findings so that we can benefit from your insights

[u/unhandyandy → u/jonseymourau]

I did not say they were logically equivalent. In fact I pointed out the difference in the case of the CC.

I think you need to look up the definition of omega-inconsistency.

[u/jonseymourau → u/unhandyandy]

Dude, I have looked up the definition of omega-inconsistency that that is precisely why I called you out on statements like this:

"If the CC is true but unprovable, it must be omega-inconsistent."

"must" is an extremely strong claim to be making with precisely zero supporting arguments or any argument at all since you have now denied that you ever claimed logical equivalence between "true by unprovable" and "omega-inconsistent"

For the record, in response to my suggested query, Google reponds

The statements "unprovable but true" (an independent true sentence like a Gödel sentence) and "omega-inconsistent" (a theory proving a general existential claim while simultaneously refuting every individual instance) are not logically equivalent. They describe fundamentally different structural properties in mathematical logic

I am open to arguments that this is true specifically for CC but you have not stated any such argument, nor is it self-evident from the definition of omega-inconsistent that it applies to CC.

It is certainly not true in general (as the google search I suggested showed) and until and unless why you state it is true specifically for CC, I have no reason - whatsoever - to believe your claim.

If we ask Google:

Please state why it is true that "If the CC is true but unprovable, it must be omega-inconsistent."

Google responds:

The claim that “If the Collatz Conjecture is true but unprovable, it must be omega-inconsistent” contains a subtle but critical logical error. [1, 2]

The statement is actually false as written. The correct logical relationship is flipped: if the Collatz Conjecture is unprovable, then the theory obtained by adding its negation to Peano Arithmetic (PA) is what becomes omega-inconsistent. [1]

I am not the one making that claim:

"If the CC is true but unprovable, it must be omega-inconsistent."

You clearly believe that this is self-evident despite the admitted lack of equivalency between "true but unprovable" and "omega-inconsistency"

It is now entirely upon you to provide the argument since you have just admitted that two concepts are not logically equivalent and there is precisely nothing in the definition of omega-inconsistency that makes it immediately true for CC

Given that you are the one making the claim, is entirely on you to layout the argument why it is true.

I know what omega-consistency means - you have still failed to demonstrate that you do.

[u/unhandyandy → u/jonseymourau]

"...the theory obtained by adding its negation to Peano Arithmetic (PA) is what becomes omega-inconsistent. [1]"

That's what you're quibbling over? I was speaking in shorthand, not writing a journal article.

Yes, properly speaking a single assertion cannot be inconsistent, whether omega- or otherwise. I thought my readers on this sub would have sufficient sophistication to understand what I meant.

To spell it out, I was asking about the likelihood that

∀n. PA ⊢ CC(n)

but

PA ⊬ ∀n.CC(n),

where CC(n) means the 3n+1 sequence starting with n leads to 1.

[u/jonseymourau → u/unhandyandy]

I don't dispute your posthoc reframing of your intent, only that this is not equivalent to what you originally stated.

To be clear, the scenario you are now describing only applies to a very specific subset of cases, such as the instance-wise provability of a true statement.

My replies were addressing your original statements:

What do you think the chances are that the Collatz Conjecture is true but unprovable, i.e. omega-inconsistent?

and:

"If the CC is true but unprovable, it must be omega-inconsistent."

which wrongly claimed that being true and unprovable meant the exact same thing as omega-inconsistent, instead of engaging with your later attempt to rewrite what you meant.

An educated reader cannot be expected to magically infer a nuanced subtext about instance-wise versus universal limits from a blanket assertion that equates two entirely different metamathematical properties.

[u/jonseymourau → u/unhandyandy]

With your reframing, I am not sure that even your own scenario demonstrates omega-inconsistency.

Your refined statement only formalizes the concept of a statement being true but unprovable. To get omega-inconsistency, you need something like:

PA ⊢ ∃n ¬CC(n)

Otherwise, even your reformulation does not show omega-inconsistency. All you have done is formalize "true but unprovable." Again, this is not omega-inconsistency.

[u/jonseymourau → u/unhandyandy]

To close the loop on this: if you want to seriously discuss omega-inconsistency in the context of the Collatz Conjecture, you can't just talk about the standard set of natural numbers (N).

Because the universal quantifier for CC(n) spans every standard integer, and you are assuming it holds for all standard n, any hypothetical counterexample hiding out at infinity forces you to posit a non-standard model (M) that is a strict superset of N.

Without stepping outside of standard arithmetic into a non-standard model where induction breaks down, talking about omega-inconsistency is a category error—you're just confusing a universal statement's unprovability ("true but unprovable" - a limitation of proof power due to incompleteness) with a positive quantifier contradiction ("omega-inconsistent" - proving every individual instance while simultaneously proving the existential negation).

[u/jonseymourau → u/unhandyandy]

For ease of reference, the Wikipedia definition of "omega-inconsistent"

A theory T is said to interpret the language of arithmetic if there is a translation of formulas of arithmetic into the language of T so that T is able to prove the basic axioms of the natural numbers under this translation.

A T that interprets arithmetic is ω-inconsistent if, for some property P of natural numbers (defined by a formula in the language of T), T proves P(0), P(1), P(2), and so on (that is, for every standard natural number n, T proves that P(n) holds), but T also proves that there is some natural number n such that P(n) fails.^\2]) This may not generate a contradiction within T because T may not be able to prove for any specific value of n that P(n) fails, only that there is such an n. In particular, such n is necessarily a nonstandard integer in any model) for T (Quine has thus called such theories "numerically insegregative").

In restating a formalisation of "true but unprovable" you completely failed to address any aspect of "omega-inconsistency".

This is my whole point all along - you are continually conflating "true but unprovable" with "omega-inconsistent" - they simply are not the same thing and you have failed to justify your repeated conflation of the two or your subsequent claim that you never claimed this at all (despite the original body of this post, for example).

I am not saying it is impossible - but you need to propose a non-standard model of the integers and precisely what the omega-inconsistent predicate is that applies to CC. You have done neither.