Galilean principle of relativity and Gödel's incompleteness theorems

[whosapopstar?]

Here is a question, that is so many levels above my analytical, logical, mathematical and physics skills (which sum up, in my case, to no more than popular science and science fiction reading), so the only reason that i am still asking this question, is that, not asking a question, seems to me, to be an act of even more foolishness.

Now,
Isn't there some kind of unsuitability, between the Galilean principle of relativity, and Gödel's incompleteness theorems?

I ask this question, since it seems to me (and i am probably, oops, wrong, well, one more time) that the Galilean principle of relativity, either says, that there can be no change, in known physical laws, at different inertial frames, and then, this means, that logically, the Galilean principle of relativity, is trying to negate something, using a set of rules, but doing so, only within that specific set of rules, or either that the Galilean principle of relativity says, that all the known and unknown physical laws, stay the same, within different inertial frames, and that means, that every new law, can be proven, only using past known set of laws/rules.

Isn't it so, in this sense, that the Galilean principle of relativity, is conjecturing, just what Gödel's has proved as a false (or an incomplete?) conjecture?

 

[pervect]

Goedel's theorem is about whether or not proofs can exist in an axiomatic system.

I don't see how you're going to apply it to SR - it's not an axiomatic system, and as physicisits, we do measurements, not mathematical proofs. When the measurements are in agreement with the theory, we say the theory is confirmed, or at least not refuted.

If you want to apply Goedel's theorem and somehow replace "proof" by "measurement", you'd have to start by reducing measurements to integers. And I don't think there is such a mapping.

Goedel's clever idea was to point out that proofs must be able to be written down, and, hence can be encoded by a (very larger) integer.

Measurements aren't this simple.

Goedel went on to show that there are equations whose solution set is the set of proof-numbers that the equations have no solution. But you need the key step of being able to reduce proofs to integers to accomplish this.

 

[someGorilla]

Here is a question, that is so many levels above my analytical, logical, mathematical and physics skills (which sum up, in my case, to no more than popular science and science fiction reading), so the only reason that i am still asking this question, is that, not asking a question, seems to me, to be an act of even more foolishness.

Now,
Isn't there some kind of unsuitability, between the Galilean principle of relativity, and Gödel's incompleteness theorems?

I ask this question, since it seems to me (and i am probably, oops, wrong, well, one more time) that the Galilean principle of relativity, either says, that there can be no change, in known physical laws, at different inertial frames, and then, this means, that logically, the Galilean principle of relativity, is trying to negate something, using a set of rules, but doing so, only within that specific set of rules, or either that the Galilean principle of relativity says, that all the known and unknown physical laws, stay the same, within different inertial frames, and that means, that every new law, can be proven, only using past known set of laws/rules.

Isn't it so, in this sense, that the Galilean principle of relativity, is conjecturing, just what Gödel's has proved as a false (or an incomplete?) conjecture?

Please remove the comma key from your keyboard

 

[AlexGTV]

I don't see the connection here.

The analogy is that the Galilean principle is equivalent to a mathematical axiom. You can't prove the principle as you can't prove an axiom. But you accept both as self-evident.

Goedel didn't say anything about axioms. He proved that there are mathematical statements, deriving form axioms, that can't be proved or disproved regardless of what axioms we will choose. He proved that if math is consistent (that there are no contradictions) it is necessarily incomplete. Since we believe it is consistent with Goedel we have the proof that it is incomplete.

 

[whosapopstar?]

Has such a mapping already been attempted in the past?
If not, because it is not feasible, can you explain in more simple words, why it is not feasible?

 

[Mentz114]

Has such a mapping already been attempted in the past?
If not, because it is not feasible, can you explain in more simple words, why it is not feasible?

As Pervect has told you, physical laws do not form a formal logical system of propositions and deductions - which is what Godels theorem is about. Applying it to relativity would be like using a spanner on a woodscrew.

 

[bcrowell]

A good book on this topic is Torkel Franzén, Godel's Theorem: An Incomplete Guide to Its Use and Abuse. Godel's theorem has no interesting implications for physics.

 

[whosapopstar?]

This book seems to be very rare - couldn't find even a summary of it, only a few library index references.

Can't this incompatibility or disinterest between logics and physics measurements, be explained in simple words? A very basic explanation, the sort of explanations presented in popular science reading, which almost everyone can understand? Why can't a physics law of nature, that is derived from measurements, be considered an axiom? Can't it be explained as simply as explaining ocean tide and ebb or explaining why the notion that only one line can connect 2 dots, is considered an axiom (please do fix any inaccuracy)?

 

[pervect]

I'd suggest reading "Godel Escher Bach, an Eternal Golden Braid" for more info about Goedel's theorem at a semi-popular level.

And I thought my previous explanation WAS simple :-)

If I may suggest, start asking any questions you have about special relativity here, and any questions you may have about Goedel's theorem in the math forums. Rather than worry about how two things you don't understand may be related, start understanding first one thing (Godels theorem) then the other thing (SR) - or vica versa, the order doesn't mattter.

Then once you understand BOTH, you'll be in a position to fruitfully start worrying about whether or not they are related somehow. You might also wander off into http://en.wikipedia.org/w/index.php?title=Zermelo–Fraenkel_set_theory&oldid=522946338 for more info about ZFC, somwhere along the line, if you get more into the math end than the physics end.

 

[whosapopstar?]

By coincidence i am reading just now 'Godel Escher Bach!...' Very interesting!

i will try to learn the subject according to your further instructions and return if any questions come up.

Thanks.

 

[haushofer]

There is something about Godel, by Berto, is also very nice to read.