Re: Solving the Tony Orlow mystery (part 1 of ?)

In article <437d7782$0$25970$7a628cd7@xxxxxxxxxxxxxxxxxxxxx>, denis
feldmann <denis.feldmann.sansspam@xxxxxxxxxxxxxxxx> wrote:

> In trying to understand and criticise TO's vision of integers, set
> theory and all that, I had to synthetize his position, giving to it much
> more consistency it ever had. But when your opponent is so much lower
> than yourself, it is often necessary first to raise him before being
> able to punch him...
>
> First, the fact that TO believes that [0,1,2,... "here a miracle
> happens", ..., aleph0-1,aleph0] is a valid sequence of integers implies
> (besides the part about miracles ("human intellect is too limited" was
> one of his recent replies) that TO's integers are non-standard ones, and
> the "miracle" is the elusive point where naïve integers leave place to
> non-standard ones (in more classical words, the naîve integers (NI for
> now on) don't form a (standard) set, and non-naive =infinite ones (in
> non-standard analysis, they are for instance the inverse of
> infinitesimals) don't form a (standard) set either) To avoid apparent
> contradictions, I will for now on set the discussion in the IST (Nelson)
> version of non-standard analysis, admit that TO's integers are the real
> ones (of ZF), and that ours are the standard ones *and only them*, ie
> the NI,( x is a NI iff x is in IN and st(x), in Nelson notation). Now,
> it seems clear that TO (despite his protests) doesn't admit the axiom of
> infinite, so I will restrict the analysis to (hereditary) finite sets in
> ZF(C) (but I will use, of course, infinite sets myself)
>
>
> In spite of this restriction, we still have infinites (of all sizes) :
> the non-standard integers. We fix one (called a0) ; this is the TO's
> version of aleph-null, which he indulges at times to use for making us
> believes he speaks of the same thing as we do, but betrays himeslf at
> once by speaking of a0-1 and the likes. Never mind. Let's try now to
> evaluate the cardinal of the set NS (non-standard, or, more precisely,
> external) of the NI. It is clear that this set is "finite" (it is a
> subset of {x in N / x<a0} ), so its cardinal w should be an integer,
> which is absurd (as then w-1, and so w and w+1, would be naive). So w
> (this elusive cardinal) is greater that all naive (or standard)
> integers, smaller that all infinite ones (and especially much smaller
> than a0, as , for instance, it is smaller than log(a0)) and so not an
> integer at all, even if it is situated in the miracle (or twilight)
> zone. At this stage, the reader should perhaps get a good book on IST,
> to check that the concept of w is *not* in contradiction to the axioms
> of ZFC (but probably useless :-))
>
>
> What do you think so far? For my next essay, I will try to examine TO's
> sequences, and reals....
I confess I have not been following the Tony Orlow bundle of threads at
all well. (Still resting my palate, after JSH.) But your summary sounds
reasonable -that is, as a precis of the thought of someone who is
making very heavy weather of the concept of "infinity."

Perhaps "surreal numbers" are a model of what TO is trying to get at.
In nonstandard analysis, the infinite integers (which are the same as
the non-standard ones, as I understand it) are anonymous objects; they
appear during the working of proofs, but take them selves off-stage for
the results. There is nothing much to say about an t one particular
nonstandard integer.

In surreal numbers, \omega is just as well-defined an individual as 2
or 17. I like Gonshor's book about them, noting that it does require a
good grounding in set theory.

--
Chris Henrich
http://www.mathinteract.com
God just doesn't fit inside a single religion.
.

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