Re: Non-Standard Analysis & Other Infinities
- From: hrubin@xxxxxxxxxxxxxxxxxxxx (Herman Rubin)
- Date: 28 Dec 2005 11:34:49 -0500
In article <dostdb$fp7$1@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
Dave Seaman <dseaman@xxxxxxxxxxxx> wrote:
>On 27 Dec 2005 16:06:33 -0800, Daryl McCullough wrote:
>> Dave Seaman says...
>>>On 27 Dec 2005 14:07:16 -0800, markwh04@xxxxxxxxx wrote:
>>>> Non-standard analysis is simply defined as the free extension of
>>>> ordinary arithmetic, modulo the addition of a new distinguished
>>>> constant (w) and a set of axioms (0<w; 1<w; 2<w; 3<w; ...).
>>>What other properties does this "w" have? Surely it is not the same as
>>>the ordinal number omega. I take it you agree that w is infinite. Is it
>>>the smallest infinite number? Note that in Robinson's theory there is no
>>>such thing as a smallest infinite number, nor is there a largest.
>> No, w is an arbitrary hyperfinite natural. It's not the largest
>> hyperfinite (because w+1 is larger) and it's not the smallest
>> hyperfinite (because w-1 is smaller).
>I agree that this should be the case if the theory is Robinson's, but I
>don't see how to derive the existence of w-1 or w+1 from only the
>assumptions stated above, and no others. Are we assuming that "ordinary
>arithmetic" satisfies the axioms of an integral domain, or perhaps a
>field?
It is a first-order extension, so all first-order theorems
of arithmetic are satisfied. If one has the completeness
axiom of the reals, it is still true that it holds for all
bounded sets defined by first-order statements.
>>>Since (Ex)(x>w) is true in Robinson's theory, I don't see why you claim
>>>that the theory is omega-inconsistent.
>> The definition of omega-consistency is that if
>> Phi(0)
>> Phi(1)
>> Phi(2)
>> ...
>> are all theorems, then
>> exists x, not Phi(x)
>> is not a theorem. This is false for the specific Phi(x)
>> Phi(x) == x=w
>Phi(0) is not a theorem in that case. Perhaps you meant
> Phi(x) == x<w
>But I still wonder whether your statement actually holds, even for this
>Phi. I suppose it depends on what "..." means.
For first-order results, it is absolutely meaningless.
Ordinary sequences do not exist in any model of the
axioms given; this is what confuses people. The sum
of 1/x(x+1) from x=1, ..., N = 1 - 1/(N+1), even if
N is bigger than w, and from OUTSIDE the model, there
are at least c, the cardinality of the continuum,
integers in the model less than N. But INSIDE the
model, N is finite.
Also, all things like the smallest set which contains 0,
and whenever it contains n, it contains n+1, is meaningless
in first-order mathematics. One cannot form the
intersection of all sets of a given type.
>> Being omega-inconsistent isn't a *bad* thing, it just
>> means that the theory is nonstandard.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.
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