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According to Godel's incompleteness theorem, any formal system can never deduce all the truths about the set of natural numbers. Hence, to deduce more truths than we were able to before, we extend our formal system.

Suppose we're initially in Robinson's arithmetic

In this theory, "every number is either even or odd" is an undecidable statement. Since we intuitively know that every natural number is even or odd, we want to restrict to the theories that can prove that statement to be true, rather than leave it undecided. So we extend this theory to Peano arithmetic.

Now we are in Peano arithmetic

Here, the Goodstein theorem is an undecidable statement. We extended this theory to ZFC which proved this theorem to be true. But do we really know that Goodstein's theorem is true? I mean, do we know that a computer will not find a counterexample to Goodstein's theorem?

And how did we know that extending Peano arithmetic to ZFC was the right choice in further "pinning down" the natural numbers?

Now we are in ZFC

For the sake of discussion, let's say the Collatz conjecture or the Riemann Hypothesis is undecidable in ZFC. Can't we then make the Collatz conjecture either true or false depending on what we want, with no obvious choice? How would we then know how to further extend ZFC to further "pin down" the natural numbers?

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  • $\begingroup$ No , we do not actually KNOW that Goodstein's theorem is true , but if we accept that ZFC is consistent together with some concepts with infinite ordinals , it is true. So, we can safely assume that it is true. It is by the way equiavalent to the consistency of the peano axioms, so there is overwhleming evidence for the truth. If there were a counterexample , we could in principle find it , but not in practice. The computational time would just be FAR FAR FAR FAR too long. $\endgroup$
    – Peter
    Commented Jun 18, 2023 at 17:24
  • $\begingroup$ The Riemann hypothesis is at least semidedicable : If it is false, there is a proof in ZFC that it is false. So, if we could show with a stronger theory , that it is independent of ZFC, we would have proven it. The situation is different in the case of the Collatz conjecture. A possible counterexample is not obviously provable , if it is a divergent sequence. $\endgroup$
    – Peter
    Commented Jun 18, 2023 at 17:28

1 Answer 1

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This is a good Question , though it is based on incorrect View-Point or misinterpretation.

Here is My way or Interpretation to look at it :

[[ EDITORIAL COMMENT :
This Answer has become too long , in the effort to make it Compete.
I am hesitant to Post it , yet I do not want to abandon it : I will just say that going over (A) & (B) may be enough.
In a quick reading (C) , (D) & (E) may be skipped ]]

(A) Describing the Moon:

When we see the moon , we always see the Same Side , the other Side is not visible.
We may make many "theories" about what the other Side could be. Maybe it is totally flat (Moon is not a Ball !) , may be it is like Mars or other Planets (Moon is a Ball) , maybe it is conical , with a tall mountain , maybe Aliens are there , Etc.

We might make higher quality , higher resolution , close-up Images while looking at the visible Side , to update our "theories".
If we can never see the other Side (Moon is unknowable , undecidable , incomplete) , then we can never know which "theory" is Correct.
We can only say which is more likely , which is unlikely , which we agree with , which we reject , Etc.

When we are eventually able to send Space-Ships & astronauts , we can see the other Side (Moon is not unknowable , not undecidable , not incomplete) & hence know which is the Exact Correct theory , thereby rejecting all other theories.

The Current Question is based on that line of thinking.

(B) Describing the Natural Numbers :

In the Case of Natural Numbers , the Case is entirely Different.
There is no Set of Natural Numbers "out there" , which we trying to view through Axioms.
Here , the Axioms "create" the Set.

When Robinson gives some Axioms , then that will "create" the "output" Set or theory X , which is incomplete , having undecidable Sentences , hence unknowable in a sense.

When Peano gives more Axioms , then that will "create" the new "output" Set or theory Y1 , which is still incomplete , having undecidable Sentences , hence unknowable in that sense. Y1 will be (must be) consistent with X.
Here , Goodstein theorem is not Provable within X & Y , though Provable within ZFC. Proving that is via various methods.

"Do we really know that Goodstein theorem is true? Do we know that a computer will not find a counterexample to Goodstein theorem?"
A Computer can not get the Counter Example. When a Computer gets the Counter Example , that will Contradict what was Proven till now. We might have to update the theories , though that is unlikely.
Is it true ? It is neither true nor not true in the Peano creation , though it is true in the ZFC creation.

When we include ZFC , it will make or "create" a new theory Z1 which is consistent with Y1 & X.
We are free to instead include some alternate Axiom , which will make or "create" Z2 where Goodstein theorem is not true , though it must be Consistent with Y1 & X.

We might even take out Peano & use some other Axiom which will "create" theory Y2 where Goodstein theorem is not true.

Mathematicians get to make the Axioms which may or may not be useful. Interesting & useful Axioms get used , the others are thrown out.

We can never send Maths Ships (Space Ships) which will take Mathematicians (astronauts) to the Hypothetical "Space" where Natural Numbers live , where Observation will tell us which theory (Robinson / Peano / ZFC / ??????) is the One True Theory of Natural Numbers. Axioms "create" these Entities , Axioms + logic gets to decide what behaviour these Entities will have.

All Consistent Mathematical theories are Equal , Some are Interesting / Nice / Insightful / useful !

(C) Smart Phones :

Constantly , Smart Phone makers add functionality like Quad Sim , Quad Camera , Bluetooth , Screen Sharing , Audio Sharing , Side Display , IOT , Etc. There is no way we can tell which maker is moving in the right Direction , to make the Ultimate Ideal Smart Phone Design made by Cosmic Entity.
Unnecessary functionality gets thrown out , the useful things get accepted.
There is no Concept of the Ultimate Smart Phone which makers are trying to achieve or emulate.
Likewise , there is no Ultimate theory of Natural Numbers which we are trying to grasp via various Axioms.

(D) Matrix Multiplication :

When we first encounter Multiplication of Matrix A ($m \times n$) by Matrix B ($n \times m$) to get Product ($m \times m$) , we can question whether that is the Correct way to Multiply in the Matrix Case.
That is just a useful convention to call it Multiplication & that method is generally useful in various Areas.
The Cosmic Entity will not tell us the Correct way , there is no Mathematical Space where we can visit to figure out whether that is really Multiplication & thus figure out whether we are using the Correct method to Multiply.
The Current Name is a Convention , the Current Method is a Definition. It is useful in general.
Likewise , the output of Set theory Axioms by Robinson & Peano & ZFC is conventionally known by the name Natural Numbers. These are Currently Interesting & useful. These may get replace later because those are more Interesting or more useful Axioms , not because the newer Axioms are more Accurate in Describing the Hypothetical Natural Numbers.

(E) Summary Statistics :

There are many Conflicting way to Summarize Data : mean , median , mode , Etc.
Which is the One Correct Way ? There is no such way. Each is useful in certain Areas. We can not visit some Statistics World to check what Cosmic Entities are using to Summarize Data.
We can just check Parameters like Convention , Interestingness , Intuitiveness , usefulness.
Likewise , we make Interesting or useful Set theories , not attempting to match the Perfect Natural Numbers , where-ever they may Exist !

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