Twin Paradox and the Logical Foundation of Relativity Theory

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SUMMARY

The discussion centers on the paper by Judit X. Madárasz, István Németi, and Gergely Székely (2005) regarding the Twin Paradox and its implications for the logical foundation of relativity theory. The authors argue that while inertial frames can be addressed using algebraic numbers for a first-order axiomatization, accelerated frames necessitate additional axioms, potentially involving real numbers and higher-order axioms. This distinction is crucial as acceleration introduces differential equations, extending beyond the scope of algebraic numbers. The conversation highlights the limitations of first-order theories as discussed by Herb Enderton.

PREREQUISITES
  • Understanding of first-order logic and axiomatization
  • Familiarity with algebraic numbers and their properties
  • Knowledge of differential equations and their role in physics
  • Basic comprehension of the Twin Paradox in relativity theory
NEXT STEPS
  • Research the implications of algebraic numbers in physics
  • Study the role of differential equations in accelerated frames
  • Explore higher-order axioms in mathematical logic
  • Examine the limitations of first-order theories in formal mathematics
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Physicists, mathematicians, and logicians interested in the foundations of relativity theory, particularly those exploring the mathematical frameworks underlying inertial and accelerated frames.

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I just noticed the following on the Philsci archive.

Judit, X. Madárasz and István, Németi and Gergely, Székely (2005) Twin Paradox and the Logical Foundation of Relativity Theory.
http://philsci-archive.pitt.edu/archive/00002358/

They seem to be saying that inertial frames can be dealt with using algebraic numbers, which allows a 1st order axiomatization. However accelerated frames require further axioms, equivalent to moving to the real numbers (I'm not sure whether this means higher order axioms). This seems reasonable to me, since acceleration means differential equations, which means going beyond the algebraic numbers.

I'd be interested to hear other people's opinions on this.
 
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