FOM: infinitesimals and AST

[Vladimir Sazonov]

Note, that P. Vopenka used a motivation of his Alternative Set 
Theory in terms of an idea of "horizon". In a universe for 
AST it *should be* true that some concrete and even not too big 
number (like 1000, - V.S.) must be "after horizon" as all of us 
can see in the real world.  He called such a universe "witnessed". 
Unfortunately, there was not "constructed" in that book such a 
witnessed universe.

In this respect P.Vopenka writes: "We will not investigate these
problems in this book mainly because they are still insufficiently
understood."

Nevertheless, "... we will sometimes motivate various notions
introduced in our theory with the help of such situations".



Let me preasent some other interesting citations from (the Russian 
version) of the book of P.Vopenka related to f.o.m:

"Convincingness of proofs according to the lows of logic decreases
with their length."

"At present time the existence of actual infinite sets became 
a dogma in which the most of mathematicians believe; moreover, 
mathematicians try to inspire other peoples with this dogma. 
...
the statements on infinite sets loose their phenomenological 
contents. As a result, the further development of set theory 
completely depends on formal considerations which prove to be 
the only reliable leader in the darkness gloaming around sets."

According to a recent posting by Solomon Feferman this also looks 

"like a work of fiction: what's true in "Hamlet" or "Don Quixote" 
or "War and Peace"?  
...
We do feel in mathematics that the stories, if that's what they are, 
are less arbitrary than works of fiction.  That's because the kinds 
of objects these are supposed to be about are refined to have a 
minimum few characteristics, and then one has significantly fewer 
options as to what to tell about them."

Again Vopenka:

"[Cantorian] Set theory brought to mathematics the whole scale of
partial cases of actual infinity. However, the most of them cannot 
be reasonably interpreted in the real world."

"Mathematics based on Cantorian set theory have been reduced to
mathematics of Cantorian set theory.

Cantorian set theory is responsible for this defective [I am not sure
in the correctness of the translation from Russian - V.S.] development
of mathematics; on the other hand, it puts on mathematics restrictions
which are not so easy to overcome. ... That is why mathematicians are
so helpless in comprehending such non-exact in their essence notions
as realizability, relationship between continuous and discrete, etc.

Thus, contemporary mathematics learns a construction whose relation
to the real world is at least problematic. Moreover, this construction
is not the only possible one and, actually, not the best appropriate
from the point of view of the mathematics itself. This makes
questionable the role of mathematics as a scientific and useful
method. This is not a danger for mathematics in the future but an
immediate crisis of contemporary mathematics. It is revealed also in the
fact that quite often deep and witty mathematical results provoke
no interest to people which are not professional mathematicians and
even also to mathematicians currently working on the problems with a
different disposition of pieces on the chess board."

"We do not reject a logic as a tool of inference from axioms, however
we will present a number of critical considerations about it."

"... the phenomenon of continuity is connected with the 
indistinguishability of the separate elements of an observable 
class."

"Crisis capturing Cantorian set theory begins progressively to catch
also other mathematical disciplines. This should be expected because
these disciplines leaned on Cantorian set theory with such a faith
that quite often they replaced their own problems by problems
induced by set theory."

"Immortality of Cantorian set theory consists not in its reality
but, quite contrary, in its fantastic nature..."

"... it cannot be excluded that there may be more than one standard
set of natural numbers..."

"Mathematicians assumed the right to decide which assertions on
infinite sets are true, as if they are staying on the place from
which these sets can be contemplated, i.e. on the place accessible 
only to the God."

"In spite of our endeavours we cannot contemplate infinite sets
by "eyes of God". Looking on such a set, we see only what we see,
not what we would like to see."


Vladimir Sazonov