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Math Help - Is universal timespace compact?

  1. #1
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    Is universal timespace compact?

    In order to apply Kakutani's fixed-point theorem successfully on the causality function that projects each past event onto its causing events, I need to demonstrate that timespace is a non-empty, compact and convex Euclidean n-dimensional space.

    If the conjecture is acceptable, it would provide support for Aristotle's theorem about the unmoved mover: The causality function would have a fixed point; in other words, there would be at least one event that causes itself.

    Under the model conjectured by the Big Bang, the present universe is the boundary of timespace -- which is simply the history of the universe up till present -- and entirely contained in timespace; so timespace is closed; given the expansion of the universe, timespace is also bounded by the present universe, because the future does not exist yet. The largest universe ever is always the present universe. The universe's age-size would be estimated at 13.7 billion years old and approximately 150 billion lightyears across. Assuming the Big Bang model, timespace would be closed and bounded, and therefore compact.

    Does this make sense?
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  2. #2
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    It's not clear to me why this is posted in a mathematics forum. Yes, every closed and bounded subset of a Euclidean space is compact but your argument that the universe is closed, as well as your argument that it is bounded, is physics, not mathematics. Indeed, I am not sure whether Kakutani's space is Euclidean but I am sure that the question of whether it is or not is one of physics, not mathematics.
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  3. #3
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    The problem is that physicists do not deal with Kakutani's fixed-point theorem, while mathematicians do not deal with the properties of timespace, so, indeed, you are right, is it physics or mathematics? Maybe it's even philosophy, since the issue eventually revolves around Aristotle's theorem. Maybe it should be enough for a problem to have a substantial mathematical component, such as Kakutani's fixed-point theorem, for mathematicians to deal with it? In the end, the real world does not seem to lend itself that well to chopping it up neatly into distinct disciplines ...
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  4. #4
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    ??? Physicists certainly do use mathematics and I see no reason not to apply Kakutani's theorem to physics- if you state clearly what physical objects the mathematical symbols represent and show that the hypotheses of the theorem are met (to some reasonable accuracy) with those representations. That is what you have not yet done.
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