Timeline for What are the advantages of the more abstract approaches to nonstandard analysis?
Current License: CC BY-SA 3.0
11 events
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| Feb 15, 2016 at 9:00 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Dec 25, 2015 at 15:10 | history | bounty ended | CommunityBot | ||
| Dec 23, 2015 at 16:47 | vote | accept | Mikhail Katz | ||
| Dec 23, 2015 at 14:54 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
Improved exposition; added sample consequence of saturation
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| Dec 20, 2015 at 11:36 | comment | added | Mikhail Katz | Certainly Nelson's take on the story is that the real world has the real numbers, definitely. Also, this real number line contains elements that test negative for the standardness predicate. The standard points does not form a set so one can't speak of a pair of worlds one being embedded inside the other. Your expression "the real numbers" involves a rather confident use of the definite article that seems to be at variance with the philosophy of the multiverse, but I am certainly not an expert. | |
| Dec 20, 2015 at 10:48 | comment | added | Joel David Hamkins | I find nonstandard analysis to be fundamentally about the interaction between two different worlds -- the standard world and the nonstandard counterpart -- and therefore fundamentally it involves multiverse issues. Nevertheless, when I used the phrase, "thinking of the nonstandard world as the real world," I had meant to refer to something a bit milder. Namely, which world has the real numbers? On the third perspective, what others might call the nonstandard reals are effectively the real reals. One can consider this difference merely semantics or terminology rather than philosophy. | |
| Dec 20, 2015 at 8:38 | comment | added | Mikhail Katz | ...even in the total time allotted to our civilisation. In other words, inhomogeneity as captured in Leibniz's distinction between assignable and inassignable numbers should be a starting point rather than the second story. To summarize, this way of describing Nelson's framework seems based on philosophical assumptions that go counter to the ideas of mutliverse and switches. | |
| Dec 20, 2015 at 8:37 | comment | added | Mikhail Katz | In the context of your idea of multiverse and switches I wonder whether it is accurate to describe Nelson's approach as "thinking of the nonstandard world as the real world". This would seem to presuppose that what is described as the "intended model" is the appropriate primary point of reference. The said intended model assumes a certain homogeneity about numbers that could be contested. Nelson would argue that there is a vast difference between the ordinary counting numbers on the one hand and an integer that's so big that it cannot be expressed by a computer the size of the universe... | |
| Dec 18, 2015 at 1:29 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Dec 18, 2015 at 0:34 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Dec 17, 2015 at 22:55 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |