Another problem I see is “Clearly if R is provable but there is no proof of !R, then R is false, by definition of the oracle H …”.

Perhaps I glossed over this point: This is because in this case the oracle H differs only finitely from the oracle B which Baker, Gill, and Solovay demonstrated such that P^{B} != NP^{B}, and therefore it must also be true that P^{H} != NP^{H}.

Similarly if !R is provable, but R is not provable, then H differs only finitely from Baker, Gill, and Solovay’s oracle A such that P^{A} == NP^{A}, and therefore P^{H} == NP^{H}.

The finite number of differences between the sets B and H, or A and H, respectively, can be managed as a finite number of exceptions by the Turing machine or primitive recursive algorithm that queries the oracle.

]]>I’m not going to waste my time on someone whose interest is in finding details which he can bend and abuse to attack someone personally.

I have made no comment on your person. (See the sentence quoted above for contrast.)

As for content you quite realiably don’t care and fail. I asked you twice to tell us the html code for ==> and you offered but → and a unicode symbol which is or is not visible in a users browser depending on his code settings and installed fonts.

How much anyone cares is irrelevant.

As for “failing”: I gave you **⇒** (in bold even) as a way of writing ⇒ in response to your question.

The question of availability of glyphs is independent of what we’re talking about.

“HTML codes” (character references) are a notational device only. A browser will produce the same tokens for character references as for the characters they reference. (See e.g. tokenizing character references in HTML spec for the details.)

]]>I once more tried to reasonably talk with you. Now I’m stopping that non-discussion. I’m not going to waste my time on someone whose interest is in finding details which he can bend and abuse to attack someone personally.

As for content you quite realiably don’t care and fail. I asked you twice to tell us the html code for ==> and you offered but → and a unicode symbol which is or is not visible in a users browser depending on his code settings and installed fonts.

]]>“Moving the goal posts” – That’s as much a self-accusation of you as it is an accusation of myself. You started it by arbitrarily focussing on an example rather than on the issue at hand.

No, it is not.

You made a claim about a recursively defined factorial function as an example of the supposed difference between the mathematical realm and the system realm.

I pointed out that your claim was bogus.

After implicitly acknowledging this, you immediately start talking about other functions that… [Goalposts are moved here.]

FYI, this time you used tu quoque.

Unfortunately things with Unicode are complicated and ugly. That even starts with entering Unicode symbols.

That’s not Unicode. That’s your keyboard and software. So use decent software and find out what’s possible. Entering code points as hexadecimal numbers isn’t the only option.

In my tools I do, of course, have everything needed (usually more or less Latex like) and I have special fonts, too. But as soon as I have to put things in HTML it gets ugly and painful and poor.

If your tools can produce characters like → directly, then why not use those characters directly in HTML?

We get ever new loads of BS like emoticons and whatnot but we still have no reasonable set of math symbols in HTML. Yuck!

(I think it’s about adding glyphs for more emoticons/emojis, but whatever.)

Which math symbols are missing?

]]>In provability logic we have to consider that R might be …

And indeed I did discern between R and PR (the latter being “R is provable”). However, I regard that as a niche case because: If R isn’t true I need not propose it; “true” meaning “provably true”. Pardon me if I don’t care too much about exotics.

Suppose that either Q or !Q is provable …

That’s why I wrote the above. If Q isn’t provable any statement with Q in it is a conjecture in the best of cases. Please keep in mind that while diverse niches of mathematics are certainly interesting and promising, it still holds that “provable” can not contradict well established math. Accordingly while I’m willing to discern and to write R and PR, I still consider “Q is provable” as a proposition, in this case one related to another one, namely R.

Back to your proposition:

First I think that your undertaking and attempt is interesting. Compliments. Way better than the usual delivery of bits and pieces found somewhere on the internet.

But still I do see at least some instability and loose ends in it. It is, for instance, not very useful to make a statement about an oracle without saying much more about it than that it’s primitive recursive. Moreover, again, the very definition of an oracle isn’t about its inner working but about its magic capability tp provide an answer to a problem of any complexity in a single step (nor is it about the Turing machine it provides with “answers”). Otherwise we know little; we merely consider it to be a mysterious part of (Turing) state machine because it’s a useful assumption (or image) for input series that belong to a (however defined) set.

Another problem I see is “Clearly if R is provable but there is no proof of !R, then R is false, by definition of the oracle H …”. You should make clear how a complexity statement is defined by an oracle; an oracle is assumed to properly work over a certain set of values in a certain problem constellation. An oracle for “Is Hm a valid SHA512 hash value?” is capable to decide that question for any member of the set of all strings that meet the condition for an output of SHA512. The fact that the problem of determining that is of complexity (say) NP is completely independent of the oracle. You, however, seem to posit a (even well definable) relation.

Also note that “R is provable but ¬R is not” does not imply that R is false. Let’s for a moment use standard logic and formulate:

Let P(R) be the fact that R is provable. Then we can simplify to R, because if R (whatever R may happen to be) is not provable it is not true. If it is provable, however, then P(R) is equivalent to R. Also note that “is provable” is a statement with a mathematical meaning.

If ¬P(¬R), i.e. if ¬R is not provable then it may or may not be true and can anyway not be used for the purpose of proving no matter what. R is (provably) true isn’t any less true because we can not prove that ¬R is true.

No! In provability logic we have to consider that R might be proven but not true, true but not proven, both true and proven, or neither proven nor true. And then we can talk about whether a proof that a proof of R exists implies that such a proof really does exist…

Anyways, back to the proof…of my last unproven claim above.

Suppose that either Q or !Q is provable and furthermore we can prove that if Q and R are equivalent, then our system of logic is inconsistent. This is equivalent to proving that either R or !R, respectively, is provable. By Lob’s theorem, this will not help us prove R or !R if we cannot already do so without the supposition that Q and R are equivalent.

I will have to think through the implications, but my intention here is to have proven what I set out to do in my first post in this thread.

]]>Moving the goalposts. You claimed, with a smidgen of hyperbole…

No, no hyperbole, just “loosely speaking” to make my point clear, a point that we encounter painfully often, namely the difference between the mathematical realm and the physical system realm.

“Moving the goal posts” – That’s as much a self-accusation of you as it is an accusation of myself. You started it by arbitrarily focussing on an *example* rather than on the issue at hand.

*HTML and math*

There is a reason why I want HTML codes: smallest common demoninator. I can reasonably assume that anyone with a properly configured OS and browser will be able to see what I write.

Unfortunately things with Unicode are complicated and ugly. That even starts with entering Unicode symbols. On Linux (seemingly at least gtk based) one presses and holds Ctrl-Shift, then types ‘u’ and the unicode in hex while still having Ctrl-Shift pressed; finally a single ‘space’ ends the sequence. I have read that KDE works differently (or not at all; don’t know, don’t use it). Staying within unix but switching to, say, FreeBSD changes things. Let alone apple which, as I understand, does not even have a Ctrl key (but presumably other ways to achieve the same), and so on.

In my tools I do, of course, have everything needed (usually more or less Latex like) and I have special fonts, too. But as soon as I have to put things in HTML it gets ugly and painful and poor.

As for the HTML “rarr” code, I know that and in fact I used it (“PR and ¬¬PR → ¬ PH = NPH”, which btw. contains an error; I should have put parantheses around “PH = NHP” but was too distressed by the HMTL sh*t) but after encountering problems like these often enough with HTML I’m unnerved. We get ever new loads of BS like emoticons and whatnot but we still have no reasonable set of math symbols in HTML. Yuck!

In case you really know more about Unicode and all the problems around it than I do (which is quite easily possible), congratulations.

]]>Unfortunately we also found out that not every recursive function is implementable in a tail-recursive fashion.

Moving the goalposts. You claimed, with a smidgen of hyperbole, that if factorial is implemented in a recursive manner the world production of RAM isn’t sufficient for the needed stack.

As for your immensely smart remark re. HTML and math symbols just give me the proper HTML code for what in ASCII is typically written as “==>” (implication). Good luck.

Who needs an “HTML code” when you can input any character you please using a keyboard? (You don’t use an “HTML code” to input the letter “a”, do you? Or maybe you do. But then, how do you enter the ampersand without using the ampersand?)

Have a ⇒. If you feel your really need a special “HTML code”, you can use **⇒** instead. Right single arrow is **→** as you can see: →. Or you just use your keyboard and type →.

Congrats to your yellow belt.

I don’t do belts.

]]>Oh, in fact, they invented even loops replacing recursive functions on my planet. Unfortunately we also found out that not every recursive function is implementable in a tail-recursive fashion. Moreover we found out that those functions may need local variables which are used both before and after the subroutine call.

As for your immensely smart remark re. HTML and math symbols just give me the proper HTML code for what in ASCII is typically written as “==>” (implication). Good luck.

My karate teacher once explained to a worried local politician “I don’t worry about my black belts. While they certainly could kill someone they also have the quality not to do so. What I’m worried about is idiotic wannabees with a fresh yellow belt who lack both the capability to survice what they might start and the wisdom to not start it in the first place” – Congrats to your yellow belt.

]]>[recursion] In math I can find it elegant and simple to make statements about 3000000 digits factorials; On a system, however, the world production of RAM isn’t sufficient for the needed stack.

I see, implementing proper tail calls has yet to happen on your planet.

Sidenote: F*ck [HT|x]ML! We’re in the 21. century and still have to either fight with unicode intricacies or use distastingly poor HTML symbols like a single arrow for implication.

Yeah. let’s blame HTML and Unicode for the difficulties you have with your own keyboard and software. (How’s it go? A poor something blames his something?)

Have a double turnstile: ⊨

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