Re: Cantor's circular "proof" that evens = integers

On 2007-05-19, in sci.logic, herbzet wrote:
I'll take your word for this, since I am struggling with the
concept of a consistent theory proving a "false" proposition.
A consistent theory has models, and what it proves is true in
all of them. If a consistent theory proves a false proposition,
it must be that the structure in which we are interpreting the
theory is not a model of the theory.

Indeed. If a theory proves a false arithmetical statement, for example, then
the naturals (possibly with some relevant additional structure, sometimes
embedded in a larger structure) are not a model of the theory. If the theory
is intended as a formalization of some body of knowledge about naturals, so
that we might speak e.g. of a number theoretical theorem being provable or
unprovable in it, proving false arithmetical statements is just as bad as
being inconsistent. Another point I wished to make was that people usually
obsess over consistency simply because they think a theory being consistent
or inconsistent is somehow more concrete or objective a state of affairs
than it being unsound for some class of sentences. We can counter this by
noting that a theory T proving "T is incosistent" is equally "objective". Of
course, people who have sentiments like this are usually not consistent in
applying their doubts, and do not similarly doubt the objectivity of
statements of form "the algorithm A terminates on all inputs" and such like.

It is also important to note that that a consistent theory might prove false
statements is a mathematical observation no more problematic than, say, the
existence of non-Archimedean fields or non-Abelian groups.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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