[FOM] An intuitionistic query

[Robert Lubarsky]

Yes. If |A|=2 then there are x and y such that, for all z in A, z=x or z=y.
Instantiate z with each of a, b, and c. With an "or", one can do a case
analysis. That is, if you have "phi or psi", you can assume first phi and
then psi to come up with your theorem. (I.e. phi -> theta and psi -> theta
implies (phi or psi) -> theta.) Doing this case analysis with the given
equalities yields the desired result.

Bob Lubarsky


-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
Arnold Neumaier
Sent: Thursday, September 03, 2009 5:50 AM
To: Foundations of Mathematics
Subject: [FOM] An intuitionistic query


Alice tells Bob that she has a set A={a,b,c} of cardinality 2, but she
is silent about any further detail. Intuition tells Bob that a, b, c
are names for two distinct elements, so that a=b or a=c or b=c.
Can intuitionistic logic prove this argument correct?

In other words, is
    A={a,b,c}, |A|=2    ==>   a=b v a=c v b=c          (*)
intuitionistically provable with generic interpretations of
the symbols on the left hand side of (*)?


Arnold Neumaier





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