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Math Help - The causality function, Kakutani's fixed-point theorem, and an event causing itself

  1. #1
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    The causality function, Kakutani's fixed-point theorem, and an event causing itself

    When we attempt to define a causality function, that projects any past event unto the collection ("set") of its causing events, in my impression, a reasonable definition of such causality function, will satisfy Kakutani's fixed-point theorem, and therefore project at least one event unto itself; meaning somehow that the event is a fixed point and therefore causes itself.

    This result would be very similar to the result described by Aristotle, in his work "Physics" (350 BC), in which he says that there only one unmoved mover, and that the cause for the unmoved mover is the unmoved mover itself. I somehow suspect that Aristotle would have used Kakutani's fixed-point theorem, if it had been available to him at that time.

    I would also like to say that Kant's criticism in his "Critique of Pure Reason" on Aristotle's finding, saying that causality is a property of timespace, and therefore does not exist prior to its existence, is not valid, because during Aritotle's limit approach of the initial timespace point from the right, timespace exists at every point. Kant's criticism is only valid for the left limit approach, which is however, never needed for Aristotle's result.

    A full verification of why such causality function would satisfy Kakutani's fixed point theorem:here.

    Could anybody verify if this makes sense?
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  2. #2
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    You assert that the time-space continuum is compact but I see no proof of that. In fact, since unbounded sets are NOT compact, I don't see how you could prove this. Are you maintining that "time-space" is bounded? If so, what physical evidence do you have of that?
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    Quote Originally Posted by HallsofIvy View Post
    You assert that the time-space continuum is compact but I see no proof of that. In fact, since unbounded sets are NOT compact, I don't see how you could prove this. Are you maintining that "time-space" is bounded? If so, what physical evidence do you have of that?
    Thanks for raising this point.

    We could proceed by investigating if timespace is closed and bounded and therefore compact.

    Timespace is simply the history of the universe. The boundary for the history of the universe is the present universe, which is entirely contained in the collection of all its versions, that is, timespace. Therefore, timespace is closed; assuming that the future does not (yet) exist at no point in time.

    The Big Bang model conjectures that the universe expanded from an initial timespace singularity. Since the expansion process is continuous and proceeds by addition, it cannot produce an actual infinite. Consequently, under the Big Bang conjecture, the universe is bounded.

    Contemporary calculations intimate that the universe would be 13.7 billion years old and approximately 150 billion lightyears across.

    If we accept such age-size calculations for the universe as an assumption for the underlying model, timespace is closed and bounded, and therefore compact.
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    Quote Originally Posted by HallsofIvy View Post
    Are you maintining that "time-space" is bounded? If so, what physical evidence do you have of that?
    To cut a long story short, under the model conjectured by the Big Bang, the present universe is the boundary of timespace; so timespace is closed; and the entire history of timespace is also bounded by it: 13.7 billion years old and approximately 150 billion lightyears across. Assuming the Big Bang model, timespace can therefore be deemed compact.
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