Re: Looking for Undecidable Propositions in Systems without a certainamountofarthimetic.

Nam Nguyen wrote:


Now that has settled, we can take a note that the definitions mentioned
right above ignore words like "tautology/validity" and "contradiction"!

Ignore the words do they. And my (good) intention is to move definitions
of tautology and contradiction to syntactical-provability base.

In their places, we'd have logical-theorems and non logical-theorems
that are logical formulas. In other words, we've translated tautological
truth and contradiction falsehood to provabilities!

However, my replacing them was a bit too strong, to the point of being
incorrect: it would accidentally weed out tautologies such as p <-> p,
or p \/ ~p. So let me re-do the replacements:

- A contradiction is a formula F which is equivalent to (F' /\ ~F'),
for some F'.

- A tautology is a formula F such that ~F is a contradiction.


The point being is: truth is always expendable; provability is not!


Which is still the point.

--
"To discover the proper approach to mathematical logic,
we must therefore examine the methods of the mathematician."
(Shoenfield, "Mathematical Logic")
.

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