Matheology § 091

[WM]

Matheology § 091

No set-theoretically definable well-ordering of the continuum can be
proved to exist from the Zermelo-Fraenkel axioms together with the
axiom of choice and the generalized continuum hypothesis.
[S. Feferman: "Some applications of the notions of forcing and generic
Sets", Talk at the International Symposium on the Theory of Models,
Berkeley (1963)]

It is well known, that no well-ordering of the reals can be
accomplished by mortal humans. But usually matheology can at least
prove that things, that cannot be done, can be done. Now even this
proof fails! What a pity!

On the other hand, well-ordering of the reals must be done. At all
costs! Otherwise the hierarchy of infinities would break down and
research on inaccessible cardinals would appear like nonsense. So let
us pray to the Gods of matheology that they do what no mortal human
can do: Well-ordering the real numbers. Perhaps they can even write a
list of all real numbers? But that must be kept secret! Because
otherwise the research on inaccessible cardinals ...

Regards, WM

[Virgil]

In article
<7583a1bc-73b7-4371...@d6g2000vbe.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> Matheology � 091
>
> No set-theoretically definable well-ordering of the continuum can be
> proved to exist from the Zermelo-Fraenkel axioms together with the
> axiom of choice and the generalized continuum hypothesis.
> [S. Feferman: "Some applications of the notions of forcing and generic
> Sets", Talk at the International Symposium on the Theory of Models,
> Berkeley (1963)]
>
> It is well known, that no well-ordering of the reals can be
> accomplished by mortal humans. But usually matheology can at least
> prove that things, that cannot be done, can be done.

In WM's WMatheology, it seems as if anything WM wants to have proved can
be proved, but standard mathematics is much fussier about what
constitutes a valid proof.
>
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