FOM: infallibility and inexpressibility

Neil Tennant neilt at
Thu Jan 21 10:00:24 EST 1999

Joe Shipman's concept of an infallible cardinal is interesting,
because it enables one to reach certain philosophical conclusions
about the limitations of set theory without making use of the heavy 
armour of G"odel's second incompleteness theorem.

Definition (Joe Shipman): alpha is an infallible cardinal just in case 
all sentences true in V are true in V_alpha.

Theorem: If the concept "x is an infallible cardinal" is expressible
in the language of set theory, then there is no infallible cardinal.

Proof: Suppose that the concept  "x is an infallible cardinal" is expressible
in the language of set theory. Let I(x) be the expressing formula.
Suppose an infallible cardinal exists. 
Then by the well-ordering of the ordinals, there is a least one. Call it i.
It is true (informally) that there is an infallible (for our second
supposition guarantees it). So "ExI(x)" is true in V.
But "ExI(x)" is false in V_i, since i itself is not in V_i.
So V_i falsifies a sentence that is true in V.
Hence i is not infallible.
Thus there is no infallible.

Philosophical corollaries:

The concept of infallibility is either inexpressible in the language of set
theory, or empty.

If there is an infallible cardinal, then one cannot, in ordinary set
theory, say that there is one.

If the truths of V are made true at any rank, then the language of set
theory is expressively incomplete.

Note that no appeal is made to the G"odel phenomena in reasoning to
these conclusions.

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