Re: How do We Know that ZF is the Axiomatization that Proves everything Provable?

On Feb 5, 3:25 pm, Jan Burse <janbu...@xxxxxxxxxxx> wrote:
Aatu Koskensilta schrieb:



On 2008-02-05, in sci.logic, Jan Burse wrote:
With the problem that ZFC is not a "logical" theory,
but rather a "set" theory. So it doesn't really
refer to modes of reasoning, rather to classical
reasoning inside set theory.

Here modes of reasoning does not refer to logical principles, but
rather to mathematical principles employed routinely in set theoretic
arguments, e.g. applying transfinite induction, defining operations on
sets by transfinite recursion, constructions involving the axiom of
choice and so on. It has been shown that using such principles all the
usual theorems of mathematics can be proved, and from this observation
it follows, by the completeness theorem and the conceptual analysis
due to Kreisel, that their formalisations are formally derivable in
ZFC which, directly or indirectly, contains the formlisations of these
principles. That is, on basis of the work of logicians, set theorists,
analysts and who not, we have a reduction of ordinary mathematics to
set theory, and, as ZFC is a formalisation of set theory, implicitly
to ZFC.

I think there are some competing approach to
formalize math.

And this is clearly a fallacy:

     math_is_formalizable_in(ZFC) ->
        (forall X (math_is_formalizable_in(X) -> X=ZFC))

How do you define formalizing in ZFC? Are you allowed to add axioms?

If not, I'd like to see how the Pythagorean Theorem is proven. If you
do have to add axioms, then I would still like to see it, and
especially how the ZFC axioms played any actual role in that
particular proof (as opposed to saying "You need ZFC to state that
there is an infinite set." in response to every theorem, which is not
the point.)

The ZFC axioms have nothing to do with anything but an attempt to
avoid Russell's Paradox. The Peano Axioms are another matter, as they
guarantee that Addition, Multiplication and the Universal Set are
representable, which has many implications. But that's all you have -
some theorems of Set Theory from ZFC proper, and much of simple
Arithmetic from the Peano Axioms that are added to that.

The fact that there are no references to anything outside of sets and
integers (not e.g. Geometry or Trigonometry) ever being carried out in
ZFC shows that.

And it's not the ability to "represent" the objects of the system that
is at issue. You can pair up any two r.e. sets of expressions, and
call the first set or graph or list or wff or program zero. And the
next one you call one.

The problem is that you need to know specific relationships among
those objects, and the ZFC axioms state only certain relationships.
You could probably prove the following interesting question: For any
r.e. set, use Peano's Axioms to prove that it is r.e. But this
doesn't give you things like the Pythagorean Theorem and the rest of
common Mathematics.

C-B

And this is also a fallacy:

     math_is_formalizable_in(X) ->
         can_be_reduced_to(X,ZFC)

Isn't it?

Check out for example:
Appendix 4
Descriptive Set Theoryhttp://linas.org/mirrors/www.ltn.lv/2001.03.27/~podnieks/gtaa.html

If you think that trying of "experimental" axioms of set theory
(constructibility, determinateness, inaccessible cardinals etc.) is a
business not serious enough for a true mathematician, you may follow
prominent personalities like as H.Lebesgue, E.Borel, R.Baire,
P.S.Aleksandrov, N.N.Luzin and many others. Instead of quick postulating
new axioms that would allow solving of unsolvable problems (for example,
the continuum problem), these people prefer working in the classical set
theory (i.e. ZFC). If the axioms of ZFC do not allow proving the
continuum hypothesis, a true mathematician should not search for
additional axioms. Instead, he should ask: if I cannot prove the
continuum hypothesis, i.e. that there are no infinite sets of real
numbers with cardinality between (the countable) aleph0 and the
cardinality of the entire continuum, then, perhaps, I can prove that
there are no "simple" or "definable" sets of this kind?

[...]

The axiom of projective determinateness (PD)

Best Regards- Hide quoted text -

- Show quoted text -

.

Relevant Pages

  • Re: Well Ordering the Reals
    ... most of the standard axioms would get scrapped ... you claim that set theory is ... theory in which to express virtually all of mathematics. ... S (call this function 'omega pre S'). ...
    (sci.math)
  • Re: question about axiomatic set theory
    ... the other which depends on set theory. ... by Hilbert in his Hilbert program: a game played with symbols ... the axioms of ZFC. ...
    (sci.math)
  • Re: Skolems Paradox and why is math the way it is?
    ... > This is not a job the axioms were ever meant to do. ... other person's interpretation require a winning strategy, no more, no ... I'm pretty sure than any model of set theory is intuitively ... figuring out how I tell what is real in mathematics. ...
    (sci.math)
  • Re: Set Theory: Should You Believe
    ... Why, in your opinion, is the orthodoxy in set theory ... mathematics, ignoring the misgivings of geniuses like Poincare, Weyl, ... When NW said that "You don't need axioms", ... NAFL theories (but infinite proper classes, ...
    (sci.logic)
  • Re: Countable models of ZFC
    ... Might not some set be the domain of a model of ZFC? ... I thought you were talking about the intended model. ... underpinnings of your mathematics. ... think they can understand the first-order language of set theory. ...
    (sci.logic)
Loading