Re: Definition of real number ℝ --infinitesimal--

Keith Thompson <Keith.S.Thompson+u@gmail.com> writes:

Ross Finlayson <ross.a.finlayson@gmail.com> writes:
[...]

Fred Katz wrote an M.I.T. Ph.D. dissertation to the effect
that there is a size relation in sets wherein that proper
subsets are smaller than their supersets.
>
Even "only one less...".
>
So, maybe you should bend your efforts to getting M.I.T. disaccredited.

>
https://arxiv.org/abs/math/0106100
https://arxiv.org/pdf/math/0106100.pdf
>
The PDF is 119 pages.  There's no point in my attempting to read the
whole thing, but others here might be interested.

Yes, thanks.  I found it interesting.

The author claims to refute what he says are two principles from Cantor:
>
    ONE-ONE. Two sets are the same size just in case there is a one-one
    correspondence between them.
>
    CANTOR<. A set, x, is smaller than a set, y, just in case x is the
    same size as some subset of y, but not the same size as y itself.
>
If he's able to construct an internally consistent model in which those
principles do not apply, that could be useful and interesting.  If he's
asserting that he's right and Cantor was wrong, he's got a lot of work
to do to convince anyone.

The author is not a crank, so he is not saying that Cantor is wrong.  He
is trying to salvage a "common sense" notion of size in some theories of
sets, though he skates close to crankoloy by introducing the problem as
one of size whilst almost immediately accepting that cardinality is not
really about size in the usual sense.  Since no one has any common sense
notions of cardinality, there is nothing to salvage unless you think of
it in terms of size.

His alternative theory is very limited and he knows that as soon as it
grows to accommodate bijections, cardinality will re-appear as a
consistent propriety of sets.  His notion of size will then be a
refinement of the ordering by cardinality.

    This paper proposes a theory of set size which is based
    on intuitions, naive and otherwise. The theory goes beyond
    intuitions, as theories will, so it needs both justification
    and defense. I spend very little time justifying the theory; it
    is so clearly true that anyone who comes to the matter without
    prejudice will accept it. I spend a lot of time defending the
    theory because no one who comes to the matter comes without
    prejudice.
>
Sounds familiar.

He puts it in a rather combative way, but it's true that any new theory
has a up-hill struggle.  And the hill will be steep if the theory is
motivated by common sense since most mathematicians follow Russell and
are sceptical of common sense.  After all, we (as animals) have no
physical experience of the infinite so what value can our common sense
expectations of it have?

The PDF is a little odd as a PhD thesis: the bibliography has only a
dozen entries, some quotations appear not to be cited property and there
is no section giving conclusions.  But the author is clearly not a
crank; he has done the work to propose (the beginnings of) an
alternative.  Usenet maths cranks only do the part involving telling
everyone else they are wrong.  They never do to the part the involves
proposing a clear and consistent alternative.

--
Ben.