[FOM] CFR and Programme: SoTFoM III and The Hyperuniverse Programme, Vienna, 21-23 September 2015.
Harvey Friedman
hmflogic at gmail.com
Fri Aug 28 00:50:38 EDT 2015
I know about this result http://arxiv.org/pdf/math/0311165.pdf
and should have mentioned it. But this is a very brutal and totally
unsatisfactory kind of definability for use in such a fundamental
issue, and I said "none in various senses". One sense I had in mind is
this.
THEOREM. There is no formula of ZFC such that, provably, in ZFC,
defines a proper elementary extension (R*,Z*,+*,dot*) of (R,Z,+,dot)
together with an element of Z*\Z.
Here (R,Z,+,dot) is the field of reals with a predicate for the integers.
The above means roughly that you cannot name ANY new integer. They are
all ephemeral. In every mathematical situation I know of that is not
deeply pathological in the sense under discussion, when you extend
some beloved structure you have motivating examples of new elements
being introduced.
This doesn't mean that nonstandard analysis is a bad idea. It just
highlights some sharp differences from a lot of celebrated
mathematics.
By the way, I think the impressive paper above, which concerns only
definable the global structure, does raise some interesting issues.
For example,
THEOREM. There is no formula of ZFC such that, provably in ZFC,
defines a proper elementary extension (R*,Z*,+*,dot*) of (R,Z,+,dot)
lying in HOD(R).
Harvey Friedman
On Thu, Aug 27, 2015 at 12:47 PM, <katzmik at macs.biu.ac.il> wrote:
> This is an error, Harvey. A definable model of the hyperreals was constructed
> by Kanovei and Shelah in
>
> Kanovei, Vladimir; Shelah, Saharon A definable nonstandard model of the reals.
> J. Symbolic Logic 69 (2004), no. 1, 159–164.
>
> MK
>
> On Thu, August 27, 2015 08:12, Harvey Friedman wrote:
>> 4. `Non-standard analysis as a computational foundation.'
>>
>> Alternative foundational schemes with some systematic robust idea is
>> always red meat. An interesting feature of non standardism is both its
>> drawbacks and its successes. On the drawback side, we know that there
>> cannot be any preferred or even describable model of nonstandard
>> analysis in various senses. There is probably more to do along this
>> negative vein than has been done. The most well known is that there is
>> no formula which, provably in ZFC, defines a particular example of
>> various particular kinds of nonstandard models.
>
>
>
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