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I've given notes of where in Turing's paper any type use triviality could be found, with direct page citations of the relevant original math, along with the mat

I've given notes of where in Turing's paper any type use triviality could be found, with direct page citations of the relevant original math, along with the mathematical reasons it doesn't have anything to do with the plain English statements you are making. This effort was a kindness done in good faith that I might find said mathematical definition of non-triviality for the halting problem somewhere in the paper, not something needed to show your claim devoid of a shown basis in mathematical reasoning so far. That much is apparent by the lack of a single mathematically defined claim specified by any of your messages. That said, if you'd like me to formally show what e.g. the triviality in the rules of implication Turing was talking about are in a purely mathematical form I'd be glad to state with no English commentary out of good faith as well. This would seem a waste of time unless that's the part of the paper you think the definition of non-triviality can be sourced, but at least I'd at last have a clear mathematical claim from you to discuss. It is now your opportunity to make a mathematical claim, of which proof by assertion this paper should apply because it proves something in general is not.

lutusp 5d ago
> That much is apparent by the lack of a single mathematically defined claim specified by any of your messages. I posted the Turing Halting Problem Wikipedia page. It describes a theorem, not a conjecture that I need to prove, which shows the original poster's claim is contradicted by established facts. Let me put it this way. If I say, "There are an infinity of primes," will you reply, saying, "I disagree"? That position would be equally appropriate -- that is to say, not appropriate at all. Am I obliged to prove the infinity of primes by generating an infinity of candidate integers and prove that some of them are prime? No, and by the same token, I'm not obliged to reply to your naive demands and teach you why the Halting Problem falsifies the claim made by the original poster. That is not my responsibility, it is yours. In mathematics, some things are conjectures -- the Millennium Challenge problems, for example. Others are theorems, meaning established truths, beyond dispute. Turing's Halting problem is a theorem. Here is another authoritative reference to the fact I originally posted: "Did Turing prove the undecidability of the halting problem?" from the Oxford University Press -- https://academic.oup.com/logcom/article/36/1/exaf075/8417148 https://academic.oup.com/logcom/article/36/1/exaf075/8417148 . The question in the title is rhetorical, as you will discover if you read and understand the article I just linked. > It is now your opportunity to make a mathematical claim, of which proof by assertion this paper should apply because it proves something in general is not. How many more mathematical literature references will you require before you realize I have already met any burden of proof? I could post the entire technical proof here, but (a) the editors of this forum would kick me out, and (b) you would still refuse to accept the evidence. How do I know this? Because you keep refusing to learn what you need to know to engage in this conversation. There are an infinity of primes -- your turn.
- zamadatix 5d ago
  The answer to what I'd say if this were about primes and a program which blocks outputting them in pure mathematics, despite there being infinitely many, would be (using a similar type of argument I gave early on for the current discussion): Define: A_n = n\*n n ∈ Z_>=2 ∴ A_n ∉ ℙ ⇒ ∀n ∴ |ℙ| = ∞ ∧ ∀n ∈ ℤ_>=2, A_n ∈ ℕ ∖ ℙ Q.E.D. (Apologies for the formatting, HN does not allow latex) The thing blocking me from being able to do the same here is I defined a similarly styled argument about it being possible to construct a program (glad to give it in pure math if tou'd like) which blocks all non-halting programs by only allowing ones showable to be halting (n.b. much as the above example with primes, this result did not require listing every non-halting program and so does not disagree with the general halting problem) and you said Turing's paper contains some definition of non-triviality which says why that cannot be and then refuse to say what and where in this paper you believe this definition of non-triviality to be even though I could find no such thing. If you said Euclid showed the above proof on primes does not apply because he gave some definition of non-triviality, telling me it's somewhere in his book Elements, and I said book IX proposition 20 does not say anything like that, so you say "it's on Wikipedia"... you could see why I have doubts you have any mathematical claim to produce defining said definition. At the very least, you could see how I'd still be awaiting to hear the actual mathematics you're using.
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  - lutusp 5d ago
    > The answer to what I'd say if this were about primes and a program which blocks outputting them in pure mathematics, despite there being infinitely many, would be ... [ snip ] ... Q.E.D. Now I get it. I should have realized at the outset that this outcome lay in the future. You are here to pointlessly argue, not discuss nor debate. > At the very least, you could see how I'd still be awaiting to hear the actual mathematics you're using. Yes -- notwithstanding that I have posted links to that exact evidence from multiple sources. The problem is not a lack of evidence, the problem is that you refuse to read it. On that topic, here is a link to Alan Turing's 1936 paper -- 36 pages long: https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf Andrew Wiles conclusively (dis)proved Fermat's Last Theorem. If you disagree, and since you have demonstrated a willingness to disagree with anything, I would have to post a link to the evidence, not the evidence itself, because the (dis)proof is 129 pages long. As I post this, I'm trying to imagine the corpus of mathematical knowledge you're unwilling to accept, because it won't fit into a finite-sized Hacker News post -- relativity, both special and general, quantum mechanics, dozens of others. I'm also trying to imagine someone whose literacy filter consists of "Tl;DR!"

lutusp 5d ago

> That much is apparent by the lack of a single mathematically defined claim specified by any of your messages. I posted the Turing Halting Problem Wikipedia page. It describes a theorem, not a conjecture that I need to prove, which shows the original poster's claim is contradicted by established facts. Let me put it this way. If I say, "There are an infinity of primes," will you reply, saying, "I disagree"? That position would be equally appropriate -- that is to say, not appropriate at all. Am I obliged to prove the infinity of primes by generating an infinity of candidate integers and prove that some of them are prime? No, and by the same token, I'm not obliged to reply to your naive demands and teach you why the Halting Problem falsifies the claim made by the original poster. That is not my responsibility, it is yours. In mathematics, some things are conjectures -- the Millennium Challenge problems, for example. Others are theorems, meaning established truths, beyond dispute. Turing's Halting problem is a theorem. Here is another authoritative reference to the fact I originally posted: "Did Turing prove the undecidability of the halting problem?" from the Oxford University Press -- https://academic.oup.com/logcom/article/36/1/exaf075/8417148 https://academic.oup.com/logcom/article/36/1/exaf075/8417148 . The question in the title is rhetorical, as you will discover if you read and understand the article I just linked. > It is now your opportunity to make a mathematical claim, of which proof by assertion this paper should apply because it proves something in general is not. How many more mathematical literature references will you require before you realize I have already met any burden of proof? I could post the entire technical proof here, but (a) the editors of this forum would kick me out, and (b) you would still refuse to accept the evidence. How do I know this? Because you keep refusing to learn what you need to know to engage in this conversation. There are an infinity of primes -- your turn.

zamadatix 5d ago
The answer to what I'd say if this were about primes and a program which blocks outputting them in pure mathematics, despite there being infinitely many, would be (using a similar type of argument I gave early on for the current discussion): Define: A_n = n\*n n ∈ Z_>=2 ∴ A_n ∉ ℙ ⇒ ∀n ∴ |ℙ| = ∞ ∧ ∀n ∈ ℤ_>=2, A_n ∈ ℕ ∖ ℙ Q.E.D. (Apologies for the formatting, HN does not allow latex) The thing blocking me from being able to do the same here is I defined a similarly styled argument about it being possible to construct a program (glad to give it in pure math if tou'd like) which blocks all non-halting programs by only allowing ones showable to be halting (n.b. much as the above example with primes, this result did not require listing every non-halting program and so does not disagree with the general halting problem) and you said Turing's paper contains some definition of non-triviality which says why that cannot be and then refuse to say what and where in this paper you believe this definition of non-triviality to be even though I could find no such thing. If you said Euclid showed the above proof on primes does not apply because he gave some definition of non-triviality, telling me it's somewhere in his book Elements, and I said book IX proposition 20 does not say anything like that, so you say "it's on Wikipedia"... you could see why I have doubts you have any mathematical claim to produce defining said definition. At the very least, you could see how I'd still be awaiting to hear the actual mathematics you're using.
- deleted 5d ago
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- lutusp 5d ago
  > The answer to what I'd say if this were about primes and a program which blocks outputting them in pure mathematics, despite there being infinitely many, would be ... [ snip ] ... Q.E.D. Now I get it. I should have realized at the outset that this outcome lay in the future. You are here to pointlessly argue, not discuss nor debate. > At the very least, you could see how I'd still be awaiting to hear the actual mathematics you're using. Yes -- notwithstanding that I have posted links to that exact evidence from multiple sources. The problem is not a lack of evidence, the problem is that you refuse to read it. On that topic, here is a link to Alan Turing's 1936 paper -- 36 pages long: https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf Andrew Wiles conclusively (dis)proved Fermat's Last Theorem. If you disagree, and since you have demonstrated a willingness to disagree with anything, I would have to post a link to the evidence, not the evidence itself, because the (dis)proof is 129 pages long. As I post this, I'm trying to imagine the corpus of mathematical knowledge you're unwilling to accept, because it won't fit into a finite-sized Hacker News post -- relativity, both special and general, quantum mechanics, dozens of others. I'm also trying to imagine someone whose literacy filter consists of "Tl;DR!"