[FOM] FOM Digest, Vol 65, Issue 8, Nonconservative law of excluded middle (Daniel M?hkeri)

rathjen@maths.leeds.ac.uk rathjen at maths.leeds.ac.uk
Mon May 12 06:19:32 EDT 2008

>But then PA + transfinite induction along it could not
>prove consistency of HA + transfinite induction along
>it, could it?

If it's a "natural" well-ordering then PA + transfinite induction
is of the same strength as HA + transfinite induction, so
could not prove the consistency of the latter.

> I expect I have misunderstood your
>example, or else my original question was badly
>worded: I was really wondering about HA + X + LEM
>proving Con(HA + X).

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.

>As to the even larger set axioms you mentioned
>(measurable, supercompact, huge etc) may I also ask
>what you think the range of plausible strengths would
>be for such extensions? I ask because someone else
>gave a URL to the paper of yours with the suggestive
>title "Inaccessible Set Axioms May Have Little
>Consistency Strength". Do you expect them all to be
>below, say, second-order arithmetic?

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 also notice that paper uses something weaker than
>CZF (namely it drops well-foundation), and with a
>large set axiom it only goes up to the
>Feferman-Schütte ordinal, but with LEM, it goes to the
>existence of strong inaccessibles.

As proof theorists know, dropping Foundation (i.e. elementhood induction)
has a considerable
impact on proof-theoretic strength when it comes to theories without full
separation. The paper with the title   "Inaccessible Set Axioms May Have
Little Consistency Strength" is concerned with the inaccessible sets on
the basis of CZF minus elementhood induction.

Best Michael Rathjen

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