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    A Gravitational Lens need not produce an Odd Number of Images

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    Given any space-time MM without singularities and any event OO, there is a natural continuous mapping ff of a two dimensional sphere into any space-like slice TT not containing OO. The set of future null geodesics (or the set of past null geodesics) forms a 2-sphere S2S^2 and the map ff sends a point in S2S^2 to the point in TT which is the intersection of the corresponding geodesic with TT. To require that ff, which maps a two dimensional space into a three dimensional space, satisfy the condition that any point in the image of ff has an odd number of preimages, is to place a very strong condition on ff. This is exactly what happens in any case where the odd image theorem holds for a transparent gravitational lens. It is argued here that this condition on ff is probably too restrictive to occur in general; and if it appears to hold in a specific example, then some ff should be calculated either analytically or numerically to provide either an illustrative example or counterexample.Comment: 8 pages, amste

    Berkeley on God's Knowledge of Pain

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    Since nothing about God is passive, and the perception of pain is inherently passive, then it seems that God does not know what it is like to experience pain. Nor would he be able to cause us to experience pain, for his experience would then be a sensation (which would require God to have senses, which he does not). My suggestion is that Berkeley avoids this situation by describing how God knows about pain “among other things” (i.e. as something whose identity is intelligible in terms of the integrated network of things). This avoids having to assume that God has ideas (including pain) apart from his willing that there be perceivers who have specific ideas that are in harmony or not in harmony with one another
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