3 ms·

Musn't SM be either incomplete or inconsistent, per Gödel's results? Can SM not represent integers?

archgoon 10y ago
> Musn't SM be either incomplete or inconsistent, per Gödel's results? No. The point of physical models is not to be able to answer all possible questions that you can ask about the models, that as you point out, is not possible. However, the point of a physical model is to simulate reality. This does not entail being able to answer arbitrary questions (in finite time) about it. Consider building something like a Nintendo Emulator. As a first pass, you might simply say "I just need to specify the registers and assembly instructions." You might miss out some assembly instructions, and when you pass in a program that uses them, your emulator would crash. You then add that in, and might find out that in order to do faithful reproduction, you need to model instruction execution time and so forth. In short, your building an emulator requires you to figure out 1) What the state of the Nintendo system needs to be (this may include a time variable). 2) How to correctly handle state transitions. However, even after you get a complete model of the Nintendo, this obviously doesn't let you answer questions like "If I were to pass in an arbitrary program to my emulator, would it halt on arbitrary inputs?" Physics is the same. We want to figure out how to specify the "state" of the system, and how that state evolves with time. When we say that a model is incomplete, it means that we're missing some assembler instructions, a register, or something else that doesn't allow us to faithfully simulate reality. Discovering a new particle is like discovering that there are a bunch of new registers and associated instructions that we need to account for.
cscheid 10y ago
(Minor nitpick. Presburger arithmetic can represent integers and is decidable: https://en.wikipedia.org/wiki/Presburger_arithmetic https://en.wikipedia.org/wiki/Presburger_arithmetic You were probably looking for Peano arithmetic.)
- jsprogrammer 10y ago
  Does that page say Presburger arithmetic requires infinitely many axioms ? How do you make an actual decision when you need to take into account an infinite number of axioms?
GFK_of_xmaspast 10y ago
What does that even mean?
- jsprogrammer 10y ago
  Gödel showed that any system meeting a certain level of arithmetical complexity will have the property of either being incomplete or inconsistent. The person I responded to said that SM was incomplete, but that must be the case, unless the model is self-contradictory (assuming it's not, as it would not qualify as a scientific theory if it were), or, the model must not have enough arithmetical complexity to fall under Gödel's rubric.

> Musn't SM be either incomplete or inconsistent, per Gödel's results? No. The point of physical models is not to be able to answer all possible questions that you can ask about the models, that as you point out, is not possible. However, the point of a physical model is to simulate reality. This does not entail being able to answer arbitrary questions (in finite time) about it. Consider building something like a Nintendo Emulator. As a first pass, you might simply say "I just need to specify the registers and assembly instructions." You might miss out some assembly instructions, and when you pass in a program that uses them, your emulator would crash. You then add that in, and might find out that in order to do faithful reproduction, you need to model instruction execution time and so forth. In short, your building an emulator requires you to figure out 1) What the state of the Nintendo system needs to be (this may include a time variable). 2) How to correctly handle state transitions. However, even after you get a complete model of the Nintendo, this obviously doesn't let you answer questions like "If I were to pass in an arbitrary program to my emulator, would it halt on arbitrary inputs?" Physics is the same. We want to figure out how to specify the "state" of the system, and how that state evolves with time. When we say that a model is incomplete, it means that we're missing some assembler instructions, a register, or something else that doesn't allow us to faithfully simulate reality. Discovering a new particle is like discovering that there are a bunch of new registers and associated instructions that we need to account for.

cscheid 10y ago

(Minor nitpick. Presburger arithmetic can represent integers and is decidable: https://en.wikipedia.org/wiki/Presburger_arithmetic https://en.wikipedia.org/wiki/Presburger_arithmetic You were probably looking for Peano arithmetic.)

jsprogrammer 10y ago
Does that page say Presburger arithmetic requires infinitely many axioms ? How do you make an actual decision when you need to take into account an infinite number of axioms?

GFK_of_xmaspast 10y ago

What does that even mean?

jsprogrammer 10y ago
Gödel showed that any system meeting a certain level of arithmetical complexity will have the property of either being incomplete or inconsistent. The person I responded to said that SM was incomplete, but that must be the case, unless the model is self-contradictory (assuming it's not, as it would not qualify as a scientific theory if it were), or, the model must not have enough arithmetical complexity to fall under Gödel's rubric.