As I said, we know it from an indirect proof. We can prove that not-X is false, hence we assume that mathematics is consistent, hence X is true (as either X or not-X is true).If there is no possible way to know that expression X is true then we can't possibly know
that expression X is true. AKA when X lacks a truth-maker then X is not true.
We must get through this key point first because it is the core foundation of everything
that I am saying. I oversimplified this a little bit so that you can get the gist of what I am saying. — PL Olcott
Let's debate this from another angle.
In mathematics, do you think there can be true, but unprovable statements?
Do you think that all true statements are also provable?
And finally, can an indirect reductio ad absurdum proof prove something?
Yes on No answers would be appreciated (of course with reasoning too). — ssu
is in my opinion one of the folks on TPF whose voice carries weight in his subject matter. I am not. But I think I can answer these, and maybe tones will correct errors I may make.TonesInDeepFreeze — ssu
Nice I wouldn't know, but otherwise I think you're exactly right, because I suffer that a lot myself - and could wish for even just a fewer trees here on TPF! So I feel gratitude to the mods for saving me from myself while I learn to do it for myself!But sometimes a tree looks nice to bang your head into. — ssu
Lol, well, if math is consistent, you'll get the proof. :razz:Indirect reductio ad absurdum proofs. Not sure what those are. Do you have an example? (In trust we both get the joke, here.) — tim wood
It's typical to underestimate the people on this Forum. But the issue here is that there's still a lot of debate on exactly what the undecidability results mean. What exactly is the realm of "true, but unprovable mathematical statements" or the role of non-computable mathematics. Obviously something that at first mathematicians don't want to find their answers to be in the realm of.As I read our now-banned member, It seemed to me he was making a limited and ultimately trivial claim, that he had unfortunately persuaded himself was somehow general and significant. All I could do was ask him for clarity, which he could not provide. Tones on the other hand was setting him straight, which (in my opinion) he was too disturbed to accept, appreciate, or follow. — tim wood
This clearly shows the person simply doesn't grasp the Undecidability results. And just to repeat again and again that Gödel is referring to the Liar paradox (and hence you can disregard it) is the typical error here.And the substance of the claim, as I read it, was that if you have a closed system/listing of propositions that are proved true (whatever that is), and your standard for inclusion in this listing is that the proposition be provably true and everything else false, then - and here insert his claims. The main difficulty being that while his claims may have been true for some closed system in theory, he wanted it to apply across-the-board - and never mind that his closed system was (I think) not even theoretically possible. — tim wood
Especially in logic, mathematics, but also in philosophy I think there's a good guideline. If someone says that you are wrong or misunderstand something, that isn't yet worrysome. But if two or more say that, you really have to consider going over what were saying. Now if people especially after some discussion simply fall silent, then you perhaps can have a point. Either nothing to add, or it goes totally over their head (which is also a possibility).Nice I wouldn't know, but otherwise I think you're exactly right, because I suffer that a lot myself - and could wish for even just a fewer trees here on TPF! So I feel gratitude to the mods for saving me from myself while I learn to do it for myself! — tim wood
Well, there is what they are. What they mean is just that which they exactly say. Significance a different question.that there's still a lot of debate on exactly what the undecidability results mean. — ssu
Exactly. And referring to the significance of Gödel's results usually gets a response of someone questioning you exactly how the difficult proof goes and repeating it (and sidelining the part that you were talking about: the significance of the results).What they mean is just that which they exactly say. Significance a different question. — tim wood
That's an absolutely great question.The trick is self-reference. But are there propositions in ω - or arithmetic - that are true but unprovable that do not involve self-reference? — tim wood
And in something as logical and rigorous as mathematics, the last thing is for us to accept that we have feelings about how it should be. Or that they would matter. It's like the person who declares: "Philosophy is meaningless to me, I do just science" has a quite specific philosophical view about science.Or another way: that there is the subject matter, and also how we feel about it. — tim wood
In economics one way to model self reference or basic interaction of the variables is to use dynamic models. And there you have to make quite careful models that stay in some equilibrium. The amount of premises just increase.If open and easy, good. If closed, and difficult, then either you freak out, blow up, or simply function in increasing error. — tim wood
Hm, I'm not sure what you mean by this.The trick not to goggle at them. — tim wood
But I applaud The Philosophy Forum for its toleration, allowing even the most incorrigible cranks to start thread after redundant thread, spewing disinformation like a crudely written bot. — TonesInDeepFreeze
I wondered if it wasn't my ignorance of the topic that didn't allow me to follow the conversation. — Lionino
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