[FOM] Arithmetical soundness of ZFC

steel@math.berkeley.edu steel at math.berkeley.edu
Sun May 24 22:24:12 EDT 2009

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> BUT WHAT WOULD A PROOF OF THE NON ARITHMETICAL SOUNDNESS OF ZFC LOOK
LIKE?
>
> More vaguely,
>
> WHAT WOULD EVIDENCE OF THE NON ARITHMETICAL SOUNDNESS OF ZFC LOOK LIKE?
>
> We might get somewhere if we ponder this question.
>
> Harvey Friedman

You could have evidence of arithmetic unsoundness which is not
evidence/proof of inconsistency.

I don't see any difference in this context between first and second
order arithmetic. Evidence could be either "top down" or "bottom up".

Bottom-up evidence would be consequences of ZFC which do not fit
well with existing number theory. Like the consequences of ZFC + V=L in
the language of 2nd order arithmetic which do no fit well with
the basic Analysis of low-level projective sets. E.g. what if
ZFC refutes the Riemann Hypo.?

Top-down would be some alternate foundation T for mathematics
which is preferable to ZFC, can talk about the natural numbers,
and proves the negation of some arithmetic consequence of ZFC.
Replace T by ZFC + PD, and ZFC by ZFC + V=L, and 1st order arithmetic by
2nd order, and you get an example of such top-down evidence.

As far as we know, NF (New Foundations) could be consistent,
but arithmetically incompatible with ZFC, right? One could imagine a
history in which NF was accepted, then later the cumulative hierarchy and
ZFC were isolated, and ZFC was shown to have arithmetic consequences
incompatible with NF. In that alternate history, one would have
come up with pretty good evidence that one's foundational theory
was arithmetically unsound, without finding an inconsistency.

The alternate theory doesn't have to be one we understand.
Here's a science-fiction scenario: we make contact with aliens
who are much further advanced than we are. Their mathematical
theory T is much more powerful than ours. Its language  has a part which
seems to be about natural numbers, e.g. T proves translations of all the
arithmetic consequences of 2nd order arithmetic. The language
of T also has a part we just don't understand, being too weak-minded, and
T refutes the translation of some arithmetic consequence of ZFC.

This points out that there is an issue of translation in the
"better alternate theory" scenario. Perhaps we have just mis-translated
talk about natural numbers from one language to the other.

At the moment, finding out that ZFC is arithmetically unsound
does seem like science fiction. No one has a foundational idea that
diverges from ZFC at the arithmetical level.

John Steel

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