# [FOM] An intuitionistic query

Frank Waaldijk frank.waaldijk at hetnet.nl
Mon Sep 7 15:22:32 EDT 2009

```Arnold Neumaier wrote:

> 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 (*)?

I think so, if I understand you correctly.

In my view, the statement |A|=2 should be interpreted as `there is a
bijection from A to the set {0,1}'. Let's take h to be such a bijection.
Without loss of generality:

h(a)=0, h(b)=0, h(c)=1

Therefore, in this case a=b. The other cases are similar.

[However, this already touches on the question `what is a function'. I would
advocate the normal set-theoretic approach: a subset of the Cartesian
product such that if (a, x) and (a,y) are in the subset, then x=y. (this can
be strengthened a little using apartness relations rather than
equality/equivalence).
Then the appropriate subset corresponding to h above would be: {(a,0),
(b,0), (c,1)} and the inverse h^-1 would be {(0,a), (0,b), (1,c)}].

frank waaldijk
http://home.hetnet.nl/~sufra/mathematics.html

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