A trivial question about the space of Euclidean

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The discussion centers on the implications of defining space as Q^3 x Q_t instead of the classical E^3 or R^3 x R_t in classical mechanics. Questions arise about the continuity of particle movement between coordinates, particularly whether a particle would transition continuously or "hop" between rational points. The concept of solving Newton's laws in a space with "holes" raises concerns about potential logical paradoxes. Participants explore whether this alternative framework could fundamentally alter our understanding of motion and mechanics. The conversation highlights the complexities of defining space and its impact on physical laws.
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Classical Mechanics define our space is E^3
that's also assumed to be R^3x R_t
I just wondering what if Q^3x Q_t?
will it make any significant difference? will it cause any logical paradox?

thanks for reading!
 
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The first thing I wonder about is how a particle in one dimension will get from coordinate a to coordinate b. In my mind, it should always do so in a continuous fashion. But you suppose it would "hop" from one rational to the "next"? You would solve Newton's laws (like F = m x'') in an interval with "holes" (e.g. [a, b] \cap \mathbb{Q})?
 
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