Re: Question about proof by contradiction

On 2008-03-10, in sci.math, goodchild.trevor@xxxxxxxxx wrote:
Perhaps my question should be restated as: how do I determine whether
the law of the excluded middle can be violated within a given
axiomatic system?

That a given axiomatic system is consistent is a combinatorial
mathematical statement. As with any mathematical statement, we
establish the truth or falsity of such statements by means of a
mathematical proof. Sometimes such a proof is forthcoming -- e.g. in
case of first-order logic, Peano arithmetic, Kripke-Platek set theory
--, sometimes not -- e.g. in case of Quine's New Foundations or ZFC +
there exists mind-bogglingly large cardinals.

Of course, some axiomatic systems are codifications of mathematical
principles we accept as correct, in the sense that in ordinary
mathematical contexts we accept a mathematical statement as true if we
are presented with a proof from these principles. If it should turn
out that an axiomatic system of this sort was inconsistent it would of
course indicate something is wrong in our ordinary mathematical
reasoning, and we'd need to react appropriately, trying to work out
what goes wrong, restricting the applicability of this or that
principle and so on. Reflecting on such purely theoretical
possibilities might be great philosophical fun the same way as
pondering whether other people exist at all is, but as a matter of
practical reality there are no realistic doubts of the correctness of
the principles codified in the commonly considered axiomatic systems,
and consequently no realistic doubts of the consistency of these
systems.

It is a perennial philosophical question why we accept these and those
mathematical principles as correct, and whether this or that
mathematical principle is or is not admissible. Without venturing into
deep philosophical waters, we may say that some principles are, in
fact, unreservedly accepted in ordinary mathematics, and that most
people find these principles -- the principle of mathematical
induction, the basic principles of set theory, ... -- compelling based
on the conceptual grasp of the notions involved.

To whatever extent one does accept some bunch of principles as correct
one should also be equally willing to accept the combinatorial
statement that no contradiction follows from the formalisation of
these statements as an axiomatic system, as it is a trivial
mathematical consequence of the principles themselves -- as are indeed
much stronger correctness properties of the axiomatisation. This
trivial mathematical consistency proof of course carries no
epistemological weight, in the sense that if someone doubts the
principles in question he will find the argument singularly
unconvincing, but the same applies to any mathematical proof using the
principles and the consistency assertions are not special in this
regard in any way.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

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

Relevant Pages

  • Re: Question about proof by contradiction
    ... That a given axiomatic system is consistent is a combinatorial ... As with any mathematical statement, ... are presented with a proof from these principles. ... fact, unreservedly accepted in ordinary mathematics, and that most ...
    (sci.math)
  • Re: Question about proof by contradiction
    ... As with any mathematical statement, ... are presented with a proof from these principles. ... out that an axiomatic system of this sort was inconsistent it would of ... fact, unreservedly accepted in ordinary mathematics, and that most ...
    (sci.math)
  • Re: Godel proved maths inconsistent not incompleteness theorem
    ... It's the *axioms* ... Otherwise it's not an axiomatic system. ... generator to compare to a proof checker in the first place. ... The program was to be able to rewrite all mathematics starting using ...
    (sci.logic)
  • Re: accruate "sundials"
    ... >> All experiments done with Foucault's pendulum and the Earth's ... The Equation of Time principles for the equable 24 hour day were ... intuition and mathematics were in balance until ... their eyes to the principles of mathematics to which they are unused. ...
    (sci.astro.amateur)
  • Re: How does ZFC proof that PA is consistent?
    ... which it follows that a that theory is inconsistent be inconsistent? ... irrelevant to our understanding of mathematics. ... theory T as a formalization of mathematical and logical principles ... kind of minimal check on the validity of our principles and intuitive ...
    (sci.logic)
Quantcast