Spiritus seems to think so. At least, instead of discussing “sufficiently powerful” systems long after we’ve established what “sufficiently powerful” means, he keeps referring to a specific axiomatic system instead.
I realize that showing something is true in the most basic axiomatic system we’re discussing shows it in all of them…
But he’s ignoring a principle that’s been demonstrated by mathematicians far more competent than either of us.
I’ve shown examples where a statement can be proven true relative to a system yet unprovable in that system. That’s all I need to do to show that’s possible.
The important point is that truth does not depend on provability… yet Spiritus doesn’t seem to grasp this. All things that are provable within a consistent system are true, but not all true things are provable. That point is is dispute… or at least, SM keeps disputing it.
I’m sorry but you must completely ignore everything he posts if you think so. For ease of speech in terms of the topic I recall him hypothesizing the soundness of the axioms in a sidenote to ultrafilter, in which case, as you say, all provable things are true, but not all true things are provable, in a sufficiently powerful system. He emphasized, repeatedly, and pointed out to you very early on in fact, that GIT was completely unconcerned with soundness.
Spiritus Mundi continues to dispute the idea that a statement can be true within a PA without being demonstrable. I’ve pointed out that mathematicians know this is true in general… but the demands keep coming.
If he does understand, why does he keep asking? If he doesn’t understand, why isn’t that enough reason to continue?
SM did. I’m fairly certain he was the first to use the word. My OP just states the consequences of GIT… and, as you’ve pointed out, GIT doesn’t change or contradict the definition of soundness one bit.
I could check to make sure that SM first brought up the issue of soundness, but I don’t think it’s worth the effort at this point.
I think you should just say what the weakness in your argument is. That way, you will either demonstrate that Spiritus missed it, and that he is incompetent at mathematics. Or he will be forced to prove you were wrong about the weakness in your argument, and therefore end up supporting your original position. Either way, you can’t lose.
Otherwise, keep the truth a secret if you like – and this whole thread will be nothing more than a waving of prosthetic dick implants to show who’s is bigger.
He neglected to pick up on the point that it’s not the case for every type of system that a contradiction leads to all possible statements becoming valid.
That was one of the first things my sources noticed and complained about…
Oh yeah, Spiritus is an expert on mathematical logic, all right…
Well, I just read the first three pages and thought, to begin with, TVVA was parodying Legomancer’s ‘Ask the Dalek’ thread. But then I realised TVVA is actually the Iraqi Information Minister with a new gig; Hi dude, how’s the beret ?
You folks sure were a little slow on the uptake there …
And do you actually have any understanding of what we’re talking about? If you can logically support either side with a reasoned argument, you’re more than welcome to do so… but otherwise, kindly shut your pie hole.
I rather enjoyed the more creative insults that were thrown about in the earlier portion of this thread, but they’re starting to lose their savor.
And in any case, Spiritus, we can’t rule out the possibility that we could show it to be true in regard to the original axioms set by adding more axioms. I believe I’ve linked an example earlier where this is demonstrated?
But of course you won’t actually refute that example. You’ll probably just call me an idiot again.
NO ! We do have savor. We have very much savor. It is Spiritus Mundi who has none because he is a liar and an infidel. I will show you he is a lying dog, not a lying down dog. IN ONE HOUR!