[FOM] Con(Z) vs Con(ZF), nine years before Intrade.com

Vaughan Pratt pratt at cs.stanford.edu
Fri May 27 13:26:18 EDT 2011

I was wrong about Halmos's "Naive Set Theory" omitting Replacement 
(pointed out to me by Brian White).  Hence appeals to "naive set theory" 
as defined by Halmos can be considered to assume not just Con(Z) but 

How much worse is that?  In 1992 I wanted to know how the denizens of 
sci.math and sci.logic felt about Con(Z) vs. Con(ZF) (Harvey and Steve 
had not yet started FOM).  In what must have been one of the earliest 
uses of an online prediction market for mathematical logic questions 
(Intrade.com was not founded until 2001) I offered two bets, the details 
of which can be seen at
http://boole.stanford.edu/ConZconZF.html .  (During a lunch break at a 
conference in 1975 I had bet $5 with each of two CS theorists that P = 
NP, but those were not placed online.)

The bottom line was that I drew two lines in the sand, with 100:1 odds 
against ZF being found inconsistent in the next 20 years, extended to 
1000:1 against in the case of Z.  I was willing to pay $100 if ZF 
survived to Sept. 19, 2012, in return for $10,000 if it were found 
inconsistent by then.  And I was willing to pay $10,000 if Z were found 
inconsistent by then, in return for $10 if it weren't.  (Obviously I'm 
not offering these today as 2012 is now only one year away.)

In 1992 ZF had only been in existence for around three-quarters of a 
century.  If failure to find an inconsistency in that period were the 
only basis for confidence in Con(ZF), odds of 20:75 for finding an 
inconsistency in the next 20 years might seem about right.  Hence anyone 
willing to risk $10,000 for a mere $100 would be demonstrating a degree 
of confidence in the consistency of ZF that went more than an order of 
magnitude beyond the simple fact of no inconsistency to date.

On the other hand, anyone willing to risk $10 in return for $10,000 if Z 
were found inconsistent would be demonstrating a lack of confidence in 
the consistency of Z that could arguably be compared to that shown much 
more recently for PA by Voevodsky, McIntyre, etc.

(I chose numbers on the large side to increase the likelihood that the 
bets were made in good faith, and declined scaled-down bets because 
their meaning would be less clear.  The much earlier $5 bets on P=NP 
gave little indication of anyone's strength of conviction.)

As it turned out I had one taker for each of the two bets.  Three other 
parties interested in the first bet declined eventually, two because I 
wouldn't scale down the bets, one because I might be hard to find in 20 
years time to collect the $100.

With only a year to go it looks like I will owe one taker $100 and be 
owed $10 by the other.

The most valuable outcome for me however was an extended correspondence 
with Bob Solovay on some of the issues raised by such questions, 
centering on possible size of proofs of inconsistency, role of Lob's 
theorem in that connection, etc.

Vaughan Pratt

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