IIRC, ToE not possible, per Goedel

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In summary, the conversation discusses Goedel's incompleteness theorem and its implications for proof systems that are rich enough to prove arithmetic. It is mentioned that any system containing second order propositional calculus, such as the popular ZF axioms of set theory, is either undecidable or incomplete. However, Tarski's work in geometry and recent developments in operations with real numbers show that not all mathematical systems are undecidable or incomplete. It is also noted that physics does not aim to prove the completeness of mathematics and may not necessarily require a complete math system for its own completeness.
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mister studebaker
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If one is seeking to explain everything, doesn't one run up against Goedel's incompleteness theorem?
 
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Goedel's theorem concerns proof systems that are rich enough to prove arithmetic. Second order propositional calculus is such a proof system, and any system that contains it, such as the popular ZF axioms of set theory, is either undecidable or incomplete, by Goedel's theorem or its successors.

But any mathematical system than does not base itself on such a proof system is not necessarily undecidable or incomplete. Tarski showed that geometry is complete back in 1948, and work in the last ten years or so has shown that a broad class of operations with real numbers is complete, too. So it's not a foregone conclusion that the math of physics will run into Goedel problems.
 
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Physics in not an attempt to prove the completeness of mathematics. Nor has it been proven that you even need a complete math system to prove the completeness of physics. Physics is deliberately using only a subset of math since it's an effort to reduce all things to a brief formula, it does not require every valid mathematical formula for its completeness.
 

Related to IIRC, ToE not possible, per Goedel

1. What is IIRC and ToE?

IIRC stands for "if I remember correctly" and ToE stands for "Theory of Everything." IIRC is often used in informal conversations and online discussions, while ToE refers to the hypothetical ultimate theory that explains all physical phenomena in the universe.

2. Who is Goedel and why is his work relevant to the possibility of a ToE?

Kurt Goedel was a mathematician and logician who is best known for his incompleteness theorems. These theorems showed that in any axiomatic system, there will always be true statements that cannot be proven within that system. This has implications for the possibility of a ToE, as it suggests that there may always be fundamental aspects of the universe that cannot be explained by a single theory.

3. Does Goedel's work definitively prove that a ToE is not possible?

No, Goedel's work does not definitively prove that a ToE is not possible. It raises important questions and challenges for the development of a ToE, but it does not necessarily rule out the possibility entirely. It is still an ongoing area of research and debate in the scientific community.

4. Are there any alternative theories to a ToE that have been proposed?

Yes, there are alternative theories to a ToE that have been proposed. Some scientists suggest that instead of a single theory that explains everything, there may be multiple theories that each explain different aspects of the universe. Others propose that a ToE may not be possible due to limitations in our understanding and perception of the universe.

5. How does the question of a ToE impact current scientific research?

The question of a ToE has a significant impact on current scientific research, particularly in fields such as physics and cosmology. Many scientists are actively working towards finding a unifying theory that can explain all physical phenomena, while others are exploring alternative approaches. The pursuit of a ToE drives much of the cutting-edge research in these fields and continues to be a hot topic of discussion and debate in the scientific community.

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