Re: I am confused by the diagonal argument

David C. Ullrich wrote:
|Honest. If you assume (1) and (2) then both of the following
|statements follow:
|
|(i) I am the king of France
|(ii) I am not the king of France.
|
|Really. Both (i) and (ii) _do_ follow from (1) and (2). The
|fact that (ii) follows does not show that (i) does not follow.

Indeed. The reasoning we use in a mathematical proof by
contradiction is different from the kind of reasoning that
one uses in a lot of ordinary situations in a few key ways.

For one thing, it has to be more rigorous. A lot of ordinary
reasoning only shows that the conclusion is very likely to
follow.

A lot of ordinary reasoning also begins with premises that
you can be pretty sure aren't absolutely inconsistent with
each other. There may be flawed premises, and you might find
that it's not credible that the premises are all true. But
there tends to be some consistent story that one can make
based on the premises.

As a result, in ordinary reasoning it may make sense to take
back conclusions. If you seem to deduce (i), but then have
another line of reasoning that shows (ii) actually follows,
it might make sense to say, "well, it seemed like (i) was
going to be a valid conclusion here, but evidently it was
actually (ii) that follows" (sorry David). In a mathematical
argument, however, when a contradiction arises, you can't
conclude that some of the inferences that have been drawn
are inappropriate because they conflict with others.

This means that one has to be more careful with reasoning
in a proof by contradiction, because you no longer have
possible inconsistencies as a guide for spotting errors of
reasoning. It's a fairly common problem people have when
they start doing proofs by contradiction, that they make
some kind of mistake in their reasoning, which makes some
of what they're doing inconsistent with some of the rest
of what they're doing, and they conclude they're finished
with the proof because they've reached a contradiction.

One has to be sure of the reasoning being sound, in spite
of its being reasoning about a situation that doesn't ever
occur.

These difficulties are partly why people presenting the
diagonal proof often don't present it as a proof by
contradiction. You really aren't using the assumption
that the list is a complete list until you get to the
point where it leads to a contradiction. Instead, you
can just leave that assumption out, and infer in a
direct way that any list, however it's defined, is an
incomplete list.

Keith Ramsay

.

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