Dunn has proved that an inconsistent field with (a) a pair of classically distinct real numbers x,y identified x=y, and (b) the resulting theory closed under the laws of classical fields, then we have that every real number is provably identical to every other (r,s)(r=s). The proof is simple: from x=y we have 0=(y-x), then both sides can be multiplied by any factor we like to get 0=r and 0=s for any r,s. Hence by Leibniz Law r=s. This is avoidable only by restricting functional substitution. How then is it prevented in the present construction? Especially in light of the (unproved) result of the preservation of function values P8? Is it because the only inconsistency contemplated in this construction is simply adding ~(x=y) for some classically identical x=y? If so, doesn’t this restrict its usefulness? There are of course inconsistent constructions which prevent Dunn’s argument, such as Mortensen’s.
Inconsistent Mathematics
C.E. Mortensen
https://books.google.co.il/books?id=KYDrCAAAQBAJ&dq=Dunn+has+proved+that+an+inconsistent+field&source=gbs_navlinks_s
i (for one) am not very impressed by the proof. however, just to ring an old bell let me mention Nietzsche's "identity of the eternal return." indeed, in an infinite universe, computing the probabilities that you have a duplicate and therefore your fate will return with the exact same path is possible, yet improbable. and for the very reason that the expansion deems the stage to expand into emptiness, back to a vacuum etc. as in Olbers' paradox, we see again how an expanding universe saves us from the fatal repetition of our tragicomical lives -- "This life, as you live it now and have lived it, you will have to live again and again, times without number; and there will be nothing new in it, but every pain and every joy and every thought and every sigh and all the unspeakably small and great in your life must return to you, and everything in the same series and sequence—and in the same way"
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