Re: continuum hypothesis and 0=1

MoeBlee says...

I think I accept your argument here. Perhaps I would put it this way
(even if not exactly your own argument): Regularity does not even come
up in ordinary mathematics, but IF it did, then it's a safe bet that
ordinary mathematics wouldn't hesitate to use it if needed, thus we
may as well regard it as within the scope of acceptable principles of
ordinary mathematics.

I think it's interesting to ask *why* regularity plays no
role in ordinary mathematics. In ordinary mathematics, we
usually think in terms of *types*, rather than (pure) sets.
There is generally has some base type that we're interested in.
The base type could be naturals, or integers, or rationals,
or reals, or complex numbers, etc. Then we may also deal with
higher-order objects defined in terms of these base objects:
functions, sets, sequences, etc. But the set-theoretic structure
of the base objects is usually completely irrelevant. Asking about
the elements of the natural number 2 seems almost a category error.
Numbers have successors and predecessors, but they don't have
members.

So in ordinary mathematics, if X is a set of interest, it is
always a set of objects of a particular type T. If x is an
element of X, then asking whether x and X have any elements
in common is almost a category error. X is a set of Ts, while
x is a T, (not a set of T).

Regularity only is of interest if you are in a type-free universe
where everything is a pure set. In that setting, it is always
meaningful to ask whether X and x have any elements in common.

--
Daryl McCullough
Ithaca, NY

.

Relevant Pages

  • Re: continuum hypothesis and 0=1
    ... role in ordinary mathematics. ... usually think in terms of *types*, rather than (pure) sets. ... There is generally has some base type that we're interested in. ... of the base objects is usually completely irrelevant. ...
    (sci.math)
  • Re: Russell paradox solved by infinite multi-layering
    ... derived from ZFC or an EVEN WEAKER set of axioms. ... I have no idea what theorems of ordinary mathematics you think require ... replacement or regularity. ... This set theory also shows a similar thing. ...
    (sci.logic)
  • Re: Russell paradox solved by infinite multi-layering
    ... derived from ZFC or an EVEN WEAKER set of axioms. ... I have no idea what theorems of ordinary mathematics you think require ... replacement or regularity. ... This set theory also shows a similar thing. ...
    (sci.logic)
  • Re: continuum hypothesis and 0=1
    ... within the deductive power of ordinary mathematics. ... Though replacement is rarely needed - a few results about Borel sets come to ... mind - regularity is thoroughly innocent. ...
    (sci.math)
  • Re: continuum hypothesis and 0=1
    ... whether adding the deductive power of regularity and replacement stays ... within the deductive power of ordinary mathematics. ... Though replacement is rarely needed - a few results about Borel sets come> to ...
    (sci.math)