Godel - Completeness of axiomatic systems

**Adam Mclean** · Jul27-04, 08:24 AM

I realised I failed to understand that

1 (real) is not equal to 1 (natural number)

These are different mathematical objects. Thus Godel incompleteness is not inherited by the Reals from the Natural numbers.

Thus all the agonising by philosophers of physics about
the implications for the incompleteness of formal
mathematical systems (as articulated by Godel)
for modern physics is pointless, as it seems that all of the
relevant mathematics used in contemporary physics
involves real and complex numbers and thus incompleteness
does not apply there.

**quartodeciman** · Jul27-04, 11:37 AM

The real number 1 and the unsigned integer 1 are constructed differently. But there is an embedding isomorphism to carry a minimal ring of real numbers generated by 0 and 1 onto the signed integers. This entails the equivalent of a least upper bound postulate, which everyone who enfranchises basic real calculus insists upon. Since physicists are in this group of math users, then the supposed "problem" has returned.

But scientists like physicists are users of math, not slaves to it. The Gödel undecidability/incompleteness theorems don't really pose a problem, except to those who are obsessed with all the talk of some Theory Of Everything from which absolutely every true statement can be derived. That is silly. If the physicist can't find a convenient result, a good unkosher "trick" can do the job. Remember, producing scientific information is the goal, not working out all the deductive consequences. Despite all the TOE talk, physics (yes, theoretical physics!) remains a practical enterprise.
---
Thomas Edison waited while one of his lab assistants tried to calculate the inner volume of an oddly-shaped container using integral calculus. Finally, Edison's patience was lost. He swore and grabbed the container, filled it full of water and poured it into a graduated cylinder. Now that is science!

**chronon** · Jul27-04, 12:56 PM

Originally Posted by Adam Mclean

1 (real) is not equal to 1 (natural number)

I would say that they are the same (i.e. any philosophising about their differences is irrelevant). The reason incompleteness doesn't follow through is that there is no way to say anything about a general integer in real closed field theory.

Originally Posted by Adam Mclean

Thus all the agonising by philosophers of physics about
the implications for the incompleteness of formal
mathematical systems (as articulated by Godel)
for modern physics is pointless, as it seems that all of the
relevant mathematics used in contemporary physics
involves real and complex numbers and thus incompleteness
does not apply there.

No. If this were really the case then physics would be much easier.

Do black holes lose information?: You've got the equations, so just do the calculation. Does string theory predict the mass of the electron? - you could just work it out. Real closed field theory only lets you make statements about polynomial equations, not the differential equations of physics (hence my post above).

**master_coda** · Jul27-04, 01:33 PM

Originally Posted by chronon

No. If this were really the case then physics would be much easier. Do black holes lose information?: You've got the equations, so just do the calculation. Does string theory predict the mass of the electron? - you could just work it out. Real closed field theory only lets you make statements about polynomial equations, not the differential equations of physics (hence my post above).

Of course, just because you have completeness, and even an algorithm for determining if any given statement in the system is true or false, it doesn't necessarily follow that problems are now easy to solve. The elation of proving something is solvable quickly fades when it is proven that it will take a trillion years to compute the solution.

**Adam Mclean** · Jul28-04, 04:49 AM

Originally Posted by quartodeciman

But there is an embedding isomorphism to carry a minimal ring of real numbers generated by 0 and 1 onto the signed integers. This entails the equivalent of a least upper bound postulate, which everyone who enfranchises basic real calculus insists upon. Since physicists are in this group of math users, then the supposed "problem" has returned.

I am not sure I understand this. Surely a least upper bound postulate requires a second-order logic, and thus
immunises us from incompleteness problems.
Can such an isomorphism bring with it the problems of formal systems that Godel discovered?

Can you please explain this further?

**Adam Mclean** · Jul28-04, 05:31 AM

Originally Posted by master_coda

The elation of proving something is solvable quickly fades when it is proven that it will take a trillion years to compute the solution.

The undecidability of a formal system is very different from the decidability of particular propositions that may take a trillion years. After all, to a mathematician as opposed to a physicist, a trillion years is as far removed from infinity as the few seconds it would take to calculate 1 + 1 on a calculator.

**master_coda** · Jul28-04, 08:43 AM

Originally Posted by Adam Mclean

The undecidability of a formal system is very different from the decidability of particular propositions that may take a trillion years. After all, to a mathematician as opposed to a physicist, a trillion years is as far removed from infinity as the few seconds it would take to calculate 1 + 1 on a calculator.

Of course, I never said that decidability wasn't a good thing. Just that it isn't necessarily very useful for physics -> I was replying to a remark that completeness would make physics much easier.

If you actually need to solve a problem, and not just show that it is solvable, then the fact that the problem is solvable given an unlimited amount of time isn't good enough. It would be nice to be able to solve a problem before the end of all life in the universe.

**quartodeciman** · Jul28-04, 12:40 PM

Originally Posted by Adam Mclean

Surely a least upper bound postulate requires a second-order logic, and thus immunises us from incompleteness problems.

Or else it requires an uncountable axiomatic basis.

I don't know that it avoids undecidability/incompleteness. There just isn't a clean demonstration of them in the calculus real number system AFAIK.

Can such an isomorphism bring with it the problems of formal systems that Godel discovered?

Maybe we never do get rid of obstacles to a perfect computational/noncomputational scheme. But I don't think any of this holds up the natural development of physical thought. By the time anyone has answered something like this the theorist has moved on to complex, quaternion, octonion and other more elaborate and expansive (and maybe even more specialized) logistical systems.

**Adam Mclean** · Jul28-04, 04:35 PM

Originally Posted by quartodeciman

Maybe we never do get rid of obstacles to a perfect computational/noncomputational scheme. But I don't think any of this holds up the natural development of physical thought. By the time anyone has answered something like this the theorist has moved on to complex, quaternion, octonion and other more elaborate and expansive (and maybe even more specialized) logistical systems.

I think that the complex numbers, quaternions, octonions, and so on, are merely defined as extensions of the real numbers, with additional axioms which define the i, j, k and the way of doing additions and multiplication of complex numbers or quaternions. Thus they will not change the completeness of the reals. As I recall quaternion multiplication is non-commutative, but this does not impact upon the real number addition. When dealing with complex numbers or quaternions one has to be aware that the operations + and * for these additional numbers to the system are not the same as + and * for adding or multiplying reals. The system of the complex numbers as well as the quaternions includes the real, numbers merely adding an extra layer of structure to it. As far as I understand this extra layer of structure will require second order logic - thus one can create the equivalent of cauchy sequences in these additional layers. Thus they do not import a first order logic layer that could create Godel incompleteness.

Perhaps there are other mathematical systems that can be defined on top of the axioms of the real numbers that can introduce Godel incompleteness. I have not studied Lie Groups but they seem to stitch together reals or complex numbers with groups. Groups are defined through a first order logic so are capable of exhibiting Godel incompleteness. Could this incompleteness be imported from groups into this extension of the reals that is a Lie Group. Or am I way off beam here ?

**Hurkyl** · Jul28-04, 04:49 PM

Any statement you make about the complex numbers or quaternions (and presumably the octonions) can be rephrased as a statement about real numbers. For example:

If x is a nonzero complex number, then there is a complex number y such that x * y = 1.

vs

If the real numbers m and n are not both zero, then there exist real numbers p and q such that (m * p - n * q) = 1 and (m * q + n * p) = 0.

(The correspondence is x <-> m + ni and y <-> p + qi)

**quartodeciman** · Jul28-04, 10:05 PM

The property that any polynomial function of degree > 0 over the complex field possesses at least one value from the same field that assigns that function a value of 0 does not easily translate to a corresponding property of the real field.

**Hurkyl** · Jul29-04, 06:27 AM

It may not easily translate, but the point is that it does translate, and in a straightforward manner.

**quartodeciman** · Jul29-04, 11:21 AM

???

Something like two polynomial functions over the real field coordinated so as to guarantee real field solutions for each of their corresponding equations? I don't recall seeing this being exploited to demonstrate the fundamental theorem of algebra.

**Hurkyl** · Jul29-04, 04:41 PM

I can't imagine how knowing completeness would assist in proving that any complex polynomial has a root. However, completeness does assist in proving that, say, any hypercomplex polynomial has a root, as follows:

The complex numbers and the hypercomplex numbers are both models of a complete theory.
Every complex polynomial has a root.
Therefore, every hypercomplex polynomial has a root.

**DrMatrix** · Aug5-04, 02:06 AM

Originally Posted by quartodeciman

The real number 1 and the unsigned integer 1 are constructed differently.

The reals and the integers are not constructed. The real numbers is a set which has certain properties. The set of integers is a set with certain other properties. You can construct a model for the integers using set theory. You can construct a model for the real numbers from the integers. The constructed model for the reals contains a model for the integers.

**DrMatrix** · Aug5-04, 02:10 AM

The real numbers form a complete ordered field. This use of complete is not the complete used by Godel. The axioms of the real numbers are not complete in the sense Godel uses the word complete.