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

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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, foreveris a long time. And the variety and number of possiblecopntradictionsis 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 AllenHi Roger,Great question! If we are allowed to take forever to pay back adebt, 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 mighthave a contradiction but you might not yet have found thecontradictions" is what is known as omega-inconsistent logicalsystems.

`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 astatement 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 assumesthat 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? Thismight help us think about the shade of Nietzche a bit more.-- Onward! Stephen --You received this message because you are subscribed to the GoogleGroups "Everything List" group.To post to this group, send email to everything-list@googlegroups.com.To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com.For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.

http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.