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Originally posted by royalchickenHere are two DP references.
I know essentially nothing about OOP or DP
www.onlydp.com
www.herfirstdp.com
I don't have any OOP links in my bookmarks.
The existence of this post 24 hours from now will constitute a nonconstructive proof for the unique existence of at most 2 people still reading this thread, a postulate that I have long been entertaining.
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Originally posted by DoctorScribblesMeh. As I see it, this random bit of pontification is less relevant to most RHPers than my last lengthy bit of random pontification, the NG incident, which I considered frightfully important at the time. Only about 4 people read that (you, me, Bennett and Hrothgar), so while you're probably right, I'm not bothered.
Here are two DP references.
www.onlydp.com
www.herfirstdp.com
I don't have any OOP links in my bookmarks.
The existence of this post 24 hours from now will constitute a nonconstructive proof for the unique existence of at most 2 people still reading this thread, a postulate that I have long been entertaining.
It's up; I was doing real work, laundry, packing, and haggling with virgins all afternoon and evening, so I didn't start this until quite late. Consequently, you'll have less meat to chew than you might have expected. Then again, I inspire that overestimate in a lot of people 😛.
Originally posted by DoctorScribblesActually, there is a third person reading this thread (so far), who cannot understand anything you two are saying (being a mathematical dunderhead, who has forgotten what very little differential calculus he once knew), but—
Here are two DP references.
www.onlydp.com
www.herfirstdp.com
I don't have any OOP links in my bookmarks.
The existence of this post 24 hours from now will constitute a nonconstructive proof for the unique existence of at most 2 people still reading this thread, a postulate that I have long been entertaining.
(a) simply enjoys eavesdropping on two intelligent people who know what their talking about;
(b) occasionally might find some comment or phrase that he rip totally out of context to stimulate his thinking in some wholly different area; and
(c) is waiting for an existence theorem, employing a non-constructive proof, for satan which guarantees his (its) existence “but leaves us no way of knowing what it might be.” 😉
Originally posted by vistesdIf you accept that "the devil is in the details," then, as royalchicken pointed out, you already have such a proof, as I have supplied the details.
(c) is waiting for an existence theorem, employing a non-constructive proof, for satan which guarantees his (its) existence “but leaves us no way of knowing what it might be.” 😉
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Originally posted by vistesdRoyalchicken and I are both happy to explain anything you don't understand. Just ask. You don't need anything so mathematically advanced as differential calculus to follow along. And we won't say, "Until you become a true mathematician, you simply won't be able to understand," or God forbid, "Just believe." Anybody who has good reading comprehension can understand mathematical proof after a little practice.
Actually, there is a third person reading this thread (so far), who cannot understand anything you two are saying (being a mathematical dunderhead, who has forgotten what very little differential calculus he once knew), but—
Additionally, seeing as you're 54, if you have heart problems, I'd caution you not to follow my references to Design Patterns.
Originally posted by DoctorScribblesThank you. You are a gentleman as well as a scholar.
Royalchicken and I are both happy to explain anything you don't understand. Just ask. You don't need anything so mathematically advanced as differential calculus to follow along. And we won't say, "Until you become a true mathematician, you simply won't be able to understand," or God forbid, "Just believe." Anybody who has good reading co ...[text shortened]... 54, if you have heart problems, I'd caution you not to follow my references to Design Patterns.
What is a diagonalization proof?
BTW, I’m younger at 54 that I was at 40 (maybe even 34)!
2nd BTW: I picked up the Battle for God book today; will start it soon...
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Originally posted by vistesdOf anything pertaining to this discussion that you could have asked, that question is the most difficult, but one of the most interesting. I'll take a shot at it, but I bet royalchicken can provide a better explanation.
What is a diagonalization proof?
The abstract answer is that diagonalization proofs are used to demonstrate claims about sets of things being countable or listable.
One standard example of a diagonalization proof is that one which demonstrates that there exist more real numbers between 0 and 1 than there are integers. Intuitively, when considering the truth of this claim, you'd note that there are infinitely many integers, and infinitely many real numbers between 0 and 1, so you might suspect the claim to be false.
Suppose you then claim it is false.
You would then be claiming that there exist at least as many integers as there are real numbers between 0 and 1.
If I think the claim is true, then I would formulate this challenge: put those real numbers in a list to correspond with the list of integers.
Maybe you'd start like:
1 --- 0.340000...
2 --- 0.894420...
3 --- 0.600000...
4 --- 0.999300...
and so on, with each real on the right ending in an infinity of digits.
Since there are an infinite number of integers, our list will be infinite.
However, if I can demonstrate that there is a real number that cannot occur in the right-hand side of the list (that is, that for any way you order the reals in correspondence with the integers, there will be a real missing from your list), I have shown that there are in fact more reals between 0 and 1 than there are integers.
I will construct such a real number right now.
Let x be any real number 0.abcdefg..., where
a is not equal to the first digit of the first real in the list,
b is not equal to the second digit of the second real in the list,
c is not equal to the third digit in the third real in the list,
and so on ad infinitum.
It must be the case that x is nowhere in the list. (It can't be the first real in the list, since its first digit differs from that; it can't be the 2nd, 3rd, etc. one either, for the similar reason.) However, it is also the case that x is a real number. Thus, there must exist more real numbers between 0 and 1 than there are integers, since we couldn't fit at least one real number into a one-to-one correspondence with the integers.
So, why the name diagonalization? Look at how the a, b, c, ... are defined. They correspond to the diagonal of the digits of the reals aligned in a list; i.e., the first from the first, the second from the second, and so on.
This may be a boring explanation, but the proof technique is clever and has been used to demonstrate some difficult, profound and counter-intuitive theorems.
Originally posted by DoctorScribblesThank you. I actually think I understand it, but I want to work through it a bit, so I'm going to print it out...
Of anything pertaining to this discussion that you could have asked, that question is the most difficult, but one of the most interesting. I'll take a shot at it, but I bet royalchicken can provide a better explanation.
The abstract answer is that diagonalization proofs are used to demonstrate claims about sets of things being countable or lista ...[text shortened]... clever and has been used to demonstrate some difficult, profound and counter-intuitive theorems.
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Originally posted by DoctorScribblesThanks again. Cantor, huh? I read a book sometime back called The Mystery of Aleph: Mathematics, the Kabbalah, and the Search for Infinity that was mostly about Cantor and set theory. Maybe it's worth a re-read...
Here is a more succint illustration of the proof.
http://scidiv.bcc.ctc.edu/Math/diag.html
Originally posted by vistesdAh, yes. Perhaps you recall from that reading that the level of infinity corresponding to the number of integers is referred to as aleph-zero, and that correspoding to the number of reals is called aleph-one, and so on.
Thanks again. Cantor, huh? I read a book sometime back called The Mystery of Aleph: Mathematics, the Kabbalah, and the Search for Infinity that was mostly about Cantor and set theory. Maybe it's worth a re-read...
Originally posted by DoctorScribblesVaguely, yes. (I just pulled the book off the shelf again.)
Ah, yes. Perhaps you recall from that reading that the level of infinity corresponding to the number of integers is referred to as aleph-zero, and that correspoding to the number of reals is called aleph-one, and so on.