|
|
vfilipch ha scritto:
> I can think up about an element which may have a "property" that
> invalidates axioms of Set Theory. Would it be correct to say that Set
> Therory CANNOT be applied to such element?
>
> If "yes", does it mean that there are some restrictions on elements in
> the Set Theory?
Essentially yes. Set Theory is an AXIOMATIC system set-up to
encode in a compact form what is called mathematical logic or
Aristotelian logic (there are people trying to propose/develop other
logic systems). As such, given that the axioms are conflict-free,
Set Theory can be neither proved nor disproved.
It defines sets and set elements indirectly by means of the
axioms and, in principle, does not care whether out there in the
real world exists anything compatible with them (in practice,
of course, the real-world rules of logic were used as a MODEL
to construct it).
It requires very little of the set elements - essentially just that
they
can be uniquely referenced (labelled), distinguished from each other,
and, if need be, associated into n-tuples which then can be
treated as new elements.
It does not care about the elements having other properties. That may
be a downer but it is also why Set Theory is so NEARLY universally
applicable. Additional properties Math cares about are only those
which are added a-posteriori by mathematicians themselves under
the form of assumptions.
All math theorem have the form "assuming *this*, it follows *that*" and
do not give a damn whether "this" is true or not, much less how often;
those are problems typical of applications, not of math. That, I feel,
may be your problem - you tend to think that math is about real objects
and thus confuse math with its applications. Math is really just a
large
collection of pre-packaged logic deduction chains, ready to be used by
whomever [else] might find a situation matching the starting
assumptions.
So should you find an ontological entity which for some reason does
not fit well the Set Theory axioms, Set Theory would simply not apply
to it. Nobody pretends that it MUST be applicable to everything.
It might be hard to use it to handle things like "universal love"
or "magnanimity" (though it might handle references to them).
That is another thing - math does NEVER handle ANY actual objects
(say planets) but just references to objects ("planets" p1, p2, p3,
...).
When mathematician says "let us take a planet p15 ..." nobody is
fooled into thinking that he actually handles the planet, right ?
So that is what I meant. If you can think about something, you have
already given it a name in your head, and that name (reference, label)
is all that math needs to get hold of it.
|
|