Galilean principle of relativity and Gödel's incompleteness theorems

In summary: 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...Please remove the comma key from your keyboardIn summary, Gödel's incompleteness theorem states that there are statements that can never be proven or disproven, within the confines of a formal logical system. This is in contrast to the Galilean principle of relativity, which states that physical laws remain the same, regardless of an observer's location in space. It is not feasible to apply Gödel's theorem to relativity, as it is not an axiomatic system.
  • #1
whosapopstar?
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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?
 
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  • #2
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.
 
  • #3
whosapopstar? said:
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 :devil:
 
  • #4
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.
 
  • #5
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?
 
  • #6
whosapopstar? said:
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.
 
  • #7
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.
 
  • #8
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)?
 
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  • #9
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.
 
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  • #10
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.
 
  • #11
There is something about Godel, by Berto, is also very nice to read.
 

1. What is the Galilean principle of relativity?

The Galilean principle of relativity, also known as Galilean invariance, is a concept in physics that states that the laws of motion are the same for all observers in uniform motion. This means that the laws of physics remain unchanged regardless of the observer's frame of reference.

2. How does the Galilean principle of relativity relate to Einstein's theory of relativity?

The Galilean principle of relativity was replaced by Einstein's theory of relativity, which includes both the special and general theories. These theories state that the laws of physics are the same for all observers in any state of motion, including acceleration. This is known as the principle of relativity.

3. What is Gödel's incompleteness theorem?

Gödel's incompleteness theorem, also known as Gödel's first incompleteness theorem, is a mathematical theorem that states that any formal system of mathematics will contain statements that cannot be proven within that system. In other words, there will always be true statements that cannot be proven using the rules and axioms of a given mathematical system.

4. How do Gödel's incompleteness theorems relate to the foundations of mathematics?

Gödel's incompleteness theorems have significant implications for the foundations of mathematics, as they show that there are limitations to what can be proven within a formal mathematical system. This challenges the idea that mathematics is a complete and consistent system that can be used to describe all of reality.

5. What impact do Gödel's incompleteness theorems have on the philosophy of science?

Gödel's incompleteness theorems have sparked much debate and discussion in the philosophy of science. They call into question the idea of a complete and consistent system of knowledge, and suggest that there will always be limitations to our understanding of the world. They also challenge the concept of objective truth, as it suggests that there are always statements that cannot be proven or known with certainty.

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