In a Consistent System, Can a True Sentence Imply a False One?
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From: Aatu Koskensilta on 31 Jul 2010 04:43 MoeBlee <jazzmobe(a)hotmail.com> writes: > UNBELIEVABLE! The guy just won't relent! It's totally believable. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 31 Jul 2010 04:54 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Shouldn't mathematical reasoning be technically clear where it needs > to (such as definitions that would communicate a fact or its negation > be the case)? What are "definitions that would communicate a fact or its negation to be the case"? > For example, if you merely defined a prime as one having only 1 and > itself as divisors, then you shouldn't get angry when I or anybody > confront you with the question like: Does your "prime" definition > make sense in the case of 1? There's no need for you to get upset. The definition makes perfect sense in case of 1, and on that definition of prime 1 is a prime. As it happens, it's not the standard definition these days. Similarly, on the standard definition of provable and disprovable, every formula is both provable and disprovable in an inconsistent theory. This is a straightforward and trivial observation. If you dislike standard terminology there's not much you can do but grind your teeth in impotent terminological discomfort. If you have some philosophical argument, observation, complaint about the way these purely technical notions relate to this and that in our mathematical experience you need to spell it out. Bizarre tirades about what makes or does not make "technical sense" are pointless. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 31 Jul 2010 04:55 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Now then, as I asked before, if e.g. I tell you I have a T that has a > disprovable formula in it, would you be able to tell if that T is > consistent, or not? No. What of it? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 31 Jul 2010 05:06 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > But let me cut the chase here and say that a genuine technical > definition has a few purposes but one important of which is it should > not lead to contradictions, even if accidentally (via definition) by > ambiguities. An important purpose of a genuine technical definition is not to lead to contradictions? Your babbling is getting more and more incoherent. > For example if I defined a theory as T "para-consistent" iff there's a > disprovable formula F in it, would anybody here really know if T is > consistent or not, given that it must be one way or another, and that > that's a _clear cut_ technical definition hence can't be wrong? This is incoherent babbling. Read charitably, your definition of para-consistent theory is fine, if a bit pointless since on that definition every theory is para-consistent, but your question simply makes no sense. > I mean: what is a useful definition if it's actually useless? A useless useful definition? > Of course, as I alluded to before, what would be _the technical > points_ to define something like "Bush-provable", "Obama-disprovable" > or the like even when we technically could? We will of course introduce such definitions as we think will prove useful or convenient in our mathematical pursuits. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 31 Jul 2010 05:14
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Be honest, straight forward, to the points, logical, conforming to the > 4 Principles (Consistency, Compatibility, Symmetry, and Humility). But > most important of all, be kind in wordings and not attacking thy > opponents just because thou are about to loose thy arguments. Setting aside the four principles, about which I'm entirely ignorant, this is swell advice. Do take heed! -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |