[FOM] Ultrafilters and consistency
José Manuel Rodríguez Caballero
josephcmac at gmail.com
Wed Jul 10 18:45:34 EDT 2019
Dear FOM members,
One of the most popular quotes from Andre Weil is the following one: We
know that God exists because mathematics is consistent and we know that the
devil exists because we cannot prove the consistency [1].
Godel's axiomatic treatment of theology (section 7 in [3]) is based on the
notion of ultrafilter. The proof that the existence of God (according to
Godel) implies the consistency of mathematics can be found in [2]. I quote
from [2] one of the main ideas:
It occurred to us that if we take God out of the class of all objects,
> treating God as exceptional, but keeping the positivity ultrafilter, pruned
> to be over the class of objects excluding God, then the positivity
> ultrafilter is no longer trivial. In fact, it is a nontrivial ultrafilter
> over the class of all objects without God. It is well known that in the
> context of set theory, certain kinds of ultrafilters are enough to prove
> the consistency of mathematics, as formalized by the usual ZFC axioms. In
> particular, a nontrivial countably complete ultrafilter serves this
> purpose.
Nevertheless, Weil's quotation is about the converse statement, i.e., the
consistency of mathematics should imply the existence of God (in the
above-mentioned formal sense). Is this converse statement true?
A counterexample may be to show a model in which God, in Godel's sense,
does not exist, provided that ZFC is consistent. Of course, even in the
hypothetical case such a counterexample exists, this does not mean that God
does not exist, but just that Weil's quote cannot be formalized as a
theorem.
Kind Regards,
Jose M.
References.
[1] As quoted in Mathematical Circles Adieu (Boston 1977) by H Eves
[2] Friedman, H. M. (2012). A divine consistency proof for mathematics. *The
Ohio State*. URL =
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.379.3330&rep=rep1&type=pdf
[3] Oppy, Graham, Ontological Arguments, The Stanford Encyclopedia of
Philosophy (Spring 2019 Edition), Edward N. Zalta (ed.), URL =
https://plato.stanford.edu/cgi-bin/encyclopedia/archinfo.cgi?entry=ontological-arguments
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