[FOM] Infinitesimal calculus

[Harvey Friedman]

> On May 26, 2009, at 7:07 PM, Brian Hart wrote:
>
> Gödel sent a letter to Robinson commending him on his fine work on
> non-standard analysis and he stated that he thought that it would
> become much more important than it has become since.  Why is this?  Is
> it because it just isn't as useful as Gödel thought it would be or
> that it could become much more useful in the future but we just don't
> know yet?  Interestingly enough from the perspective of the history of
> math, Robinson was good enough of a mathematician that he was
> seriously considered to replace Gödel at the IAS but tragically died
> shortly thereafter.

Here is a speculation about the enthusiasm of Goedel.

Perhaps Goedel thought that nonstandard analysis, or the ideas of  
nonstandard analysis, suitably extended, would be "logically  
powerful", perhaps interpreting ZFC and beyond.

The way A. Robinson treated infinitesimals through ultrapowers, limits  
is logical power: it can be handled perfectly well within weak  
fragments of set theory.

However, there is a much more logically powerful form of nonstandard  
ideas present in my Concept Calculus. The first installment of this  
theory uses

better than
much better than

with the "infinitely large" idea present in the "much" is "much better  
than". See http://www.math.ohio-state.edu/%7Efriedman/manuscripts.html  
#63, Concept Calculus: Much Better Than. Actually, an improved version  
will be put up shortly.

There will be a book on Concept Calculus with many other results of  
this kind, even where informal concepts are used to interpret systems  
well beyond ZFC, that include a variety of large cardinal axioms.

Harvey Friedman