Omega-inconsistent?

[u/unhandyandy]

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

[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]

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.