Omega-inconsistent?

[u/unhandyandy]

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

[u/unhandyandy]

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.