# [FOM] RE : Nonconservative law of excluded middle

Daniel Méhkeri dmehkeri at yahoo.ca
Wed May 14 21:25:28 EDT 2008

> If X asserts transfinite induction for a "natural"
> well-ordering then
> this is not possible. One might be able to cook up
> an artificial
> well-ordering though where HA + X +LEM proves
> Con(HA+ X).
>  Also HA is consistent with lots of principles X
> which classically yield
> an inconsistency. So trivially such an X would work.
>

Ah yes :-)

For instance so-called "Church's Thesis". CT and LEM
are incompatible - as I recently read, HA + CT proves
statements that are "false but realisable" and doesn't
prove statements that are "true but unrealisable",
whereas straight HA only proves "true and realisable"
statements. Though I suppose we might as well say that
PA proves statements that are "idealisable but false"
and doesn't prove statements that are "unidealisable
but true", with HA only proving "idealisable and true"
statements. However HA, HA + CT and HA + LEM are
equiconsistent, and these points of view can coexist
via appropriate translation.

Is CT also still conservative over HA with transfinite
induction over a natural well-ordering? And what is it
about naturality that ensures conservativity of LEM
(and of CT, if it is conservative) which is absent for
an arbitrary well-order?

Now CZF, like HA, contains no LEM-incompatible axioms
like CT. But to say that it proves no LEM-incompatible
theorems is to assert the consistency not just of CZF
but of ZFC! And the possibility of translation does
not exist in this case. So what about CT, then? Is CT
conservative over (or for that matter, is it even
compatible with) CZF? CZF + large set axioms so far
studied?

> No I don't expect the strengths of these very large
> set axioms plus CZF to
> be below the strength of second order arithmetic.
> But they may turn out to
> be of the strength of ZF plus smaller large
> cardinals, which would be
> quite interesting. It would also be interesting to
> look at large cardinal
> "axioms" that classically yield inconistencies, e.g.
> elementary
> embeddings of the universe into itself.
>
I have only the foggiest notion of what an elementary
embedding of the universe into itself is. I do recall
that it is a large cardinal axiom incompatible with
the axiom of choice, but so far not with ZF (or am I
thinking of something else?)

Are there other large cardinal axiom candidates which
were found incompatible even with ZF, but which might
not be incompatible with CZF??

Best regards,
Dan Mehkeri, ever curious.

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