On 05 Nov 2012, at 15:08, Stephen P. King wrote:

On 11/5/2012 7:43 AM, Roger Clough wrote:
Hi Bruno Marchal

OK, you say propositions might have a contradiction but you might not
yet have found the contradictions. That's a profound point.
In other words, one can't ever be sure if a proposition is
necessarily true, because, as Woody Allen says, forever
is a long time. And the variety and number of possible copntradictions
is possibly vast. Shades of Nietzsche ! Tell me it isn't so !

I guess that's the same as saying that you can never be sure
of contingency either. I need to lie down for a while. This
is beginning to look like existentialism.


Roger Clough, rclo...@verizon.net
11/5/2012
"Forever is a long time, especially near the end." -Woody Allen
Hi Roger,

Great question! If we are allowed to take forever to pay back a debt, then we have an effective free lunch!

I don't see this. The debt remains. Many countries have such "free lunch", which of course are not free at all.



What you are thinking about with the concept of "propositions might have a contradiction but you might not yet have found the contradictions" is what is known as omega-inconsistent logical systems.

Not really. Even if we can look at all the proofs possible, they might all not get the falsity. The omega-inconsistent theories keep saying that they are inconsistent, and they remain consistent as we cannot exclude the existence of non standard infinite proofs in the system. But the "proof of inconsistency" will have a non standard length, and is not a proof in the usual sense of the word.




;-) Theories that are consistent right up until they produce a statement that is not consistent.

No, that's an inconsistent theory. omega-inconsistent theories never produce a contradiction. But they just disbelieves this.


By the way, the usual rules of logical inference in math assumes that truth theories are never inconsistent.

It is not an assumption. It is provable. Soundness implies consistency, but the reverse is false. An omega-inconsistent theory is consistent but not sound. They assert arithmetical falsity, like the fact that they are inconsistent.

Bruno


What about theories that are only 'almost' never inconsistent? This might help us think about the shade of Nietzche a bit more.

--
Onward!

Stephen

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