Re: Nilpotent Infinitesimals & the Law of the Excluded Middle

On Jan 11, 11:49 pm, Math1723 <anonym1...@xxxxxxx> wrote:
I have been looking at Bell's Smooth Infinitesimal Analysis and find
it very interesting, particularly the concept of nilpotent
infinitesimals (higher powers of an infinitesimal is 0). It however
has the rather unfortunate attribute of contradicting the Law of the
Excluded Middle. That is, given a statement ~~A, we cannot conclude
A. This seems a terrible defect to me.

Are there any other nil-potent infinitesimal axiomatic systems which
preserves this Law (and does not give up any other basic logical law)?

To quote from page 104 of Bell's book 'In connection with (8.4) it is
interesting to note that, in most models of smooth infinitesimal
analysis, the law of excluded middle is true in a certain restricted
sense, namely, if a is any closed sentence, i.e. having no free
variable, then a or not a holds. (Notice that the sentence in (8.3) is
not of this form because its quantifier appears 'outside'.) Thus, in
smooth infinitesimal analysis, the law of excluded middle fails 'just
enough' for variables so as to ensure that all maps on R are
continuous, but not so much as to affect the propositional logic of
closed sentences.'

And on page 6 'Now at first sight the failure of the law of excluded
middle in smooth worlds may seem to constitute a major drawback.
However, it is precisely this failure which allows nonpunctiform
infinitesimals to be present. To get some idea of why this is so, we
observe that since the law of excluded middle fails in any smooth
world S, so does its logical equivalent the law of double negation:
for any statement A, not not A implies A.' The argument goes on from
here.

It is important to bear in mind that the LEM is equivalent to the so-
called axiom of choice. An interesting insight on this subject can be
found on page 256 of The Foundations of Mathematics by Ian Stewart and
David Tall: 'The status of this axiom is intuitively hard to grasp,
but is now well understood. Neither its truth nor its falsity
contradict axioms (S1)-(S13)... For this reason it is customary in
mathematics to point out whenever the axiom of choice is being used,
whereas the ordinary axioms (S1)-(S13) are not normally mentioned...
Assuming the axiom of choice allows us to tidy up one loose end. It
implies that for any sets x, y, we have x >= y or y >= x.'

This 'loose end' is what allows nilsquare infinitesimals to exist
since without it an entity can in theory be in the vicinity of zero
without being distinct from it. The axiom of choice is only introduced
in Stewart and Tall's book at the beginning of their discussion of
cardinal (infinite) numbers. Thus one can believe in infinite numbers
or infinitesimals but not both, or at least not both at the same time.
This may explain why Cantor was so hostile to infinitesimals.
.

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