"This mechanism is fundamentally different from that of an actual greenhouse, which works by isolating warm air inside the structure so that heat is not lost by convection."
Trying to conflate the mechanism of a greenhouse with a different phenomenon that happens to share a similar name is not helpful.
'In the abstract' is probably key here. Maybe 'hypothetical number' would be a better tag for Omega? Hypothetical number might aptly describe the other uncomputables too.
There is a valid distinction between being undefinable and merely uncomputable. If a number is definable (even if uncomputable), there is a finite string that specifies exactly what conditions that number uniquely satisfies (e.g., "Pi is the ratio of a circle's circumference to its diameter, in Euclidean space"). Given this string, you could "talk about" the number in the ways you describe (assuming the string is short enough for you to comprehend within your lifetime). But there are only aleph-null definable numbers, and there are at least aleph-one undefinable numbers. For the undefinables, there is no way to describe them in a finite string, so you can't talk about them. That's the distinction I think you are looking for.
So a better statement in your original post would be "That we can talk about it at all means pi is a definable number."
I'm not an expert.
Nor am I. But I think the article I just linked backs me up.
The article you linked mentions that there are uncomputable numbers that are nevertheless definable (and so can be talked about, in the abstract). It give Chaitin's number as an example.
I just thought of a counterexample that shows that a proof following my outline won't work. In binary:
Let's say you want to embed an infinite pattern of binary bits a0, a1, a2, a3, a4,... in an irrational number.
First, construct the sequence 01_001011_... as follows: The initial 01 contains bits 0 and 1, and the 2-bit sequence 01. The next 6 bits contain the other possible 2-bit patterns, as well as some 3-bit, 4-bit, and 5-bit patterns, and is itself a 6-bit pattern. There are 63 other 6-bit patterns, so after the second "_", insert the concatenation of those patterns (378 bits). Now you've got all 3-bit, 4-bit, 5-bit, and 6-bit patterns. Add another underscore and the concatenation of all not-yet-covered 378-bit patterns (this will be quite long!), and another underscore, and so on ad infinitum. Now you have an infinite sequence that contains infinite copies of every finite series of bits. Nothing repeats regularly so it is at least irrational.
All that's left is to insert the bits of our desired infinite pattern (a0, a1, a2) in place of the underscores, one bit (consecutively) at each underscore.
This does *not* mean that inclusion of an infinite pattern in a series that contains all finite patterns of bits is possible if the infinite pattern's bits to be found at regular intervals.
Moral: If we want to find "meaningful" patterns (whatever that means) in pi, we may need to sample the sequence at ever-expanding (hyperexponential) intervals.
I suspect it may be possible to prove that no pattern in pi can extend forever.
If any arbitrary sequence of digits can be found in pi (I think this has been shown), and if a purported pattern is found starting at digit N, then there are 2^N - N patterns of length N, each of which must occur following digit N. I think forcing these patterns into the series for all values of N would disrupt any such pattern.
Obviously that's not a formal proof, but it suggests a possible outline of one.
I place it at 1 in 6.779 billion...There will always be somebody, somewhere in the world, who is connected to the crime and has a DNA match to the evidence (assuming the DNA in the evidence really came from someone associated with the crime). But in most cases, there will just be one such person.
As another person replied, a series hybrid will never be more efficient than a straight engine, but that's ignoring the charging of the batteries through third party options like regenerative braking, solar collection and wall sockets.
It's also ignoring that the losses through conversion are only in the 5% to 10% range. If the gain from running the engine at a constant speed is enough to offset this loss, the hybrid *will* be more efficient.
Perhaps this was once a binary star system. The planet was orbiting the *other* star. Something (???) happened to the other star and the planet was recaptured by the remaining star.
Slashdot Mirror
Pen and paper can't be used to record athletic records?
And that both involve the Dutch.
I've been on /. too long...I first read that last word as something different.
I have free will because as far as I can tell I exercise it.
Free from what? That's the real question.
When and where was that done?
From wikipedia's article on the greenhouse effect:
"This mechanism is fundamentally different from that of an actual greenhouse, which works by isolating warm air inside the structure so that heat is not lost by convection."
Trying to conflate the mechanism of a greenhouse with a different phenomenon that happens to share a similar name is not helpful.
and that testing either challenges your beliefs or affirms them. Would you rather his quote had said "have had their beliefs affirmed"?
...and between honesty and cherry-picking.
the infamous spinning beach ball?
'In the abstract' is probably key here. Maybe 'hypothetical number' would be a better tag for Omega? Hypothetical number might aptly describe the other uncomputables too.
There is a valid distinction between being undefinable and merely uncomputable. If a number is definable (even if uncomputable), there is a finite string that specifies exactly what conditions that number uniquely satisfies (e.g., "Pi is the ratio of a circle's circumference to its diameter, in Euclidean space"). Given this string, you could "talk about" the number in the ways you describe (assuming the string is short enough for you to comprehend within your lifetime). But there are only aleph-null definable numbers, and there are at least aleph-one undefinable numbers. For the undefinables, there is no way to describe them in a finite string, so you can't talk about them. That's the distinction I think you are looking for.
So a better statement in your original post would be "That we can talk about it at all means pi is a definable number."
I'm not an expert.
Nor am I. But I think the article I just linked backs me up.
The first rule of undefinable numbers is, you do not talk about undefinable numbers!
Wow. You're the king of "whoosh"!
Even today, they still used to cost $6.
When you say "base 10", what base is the "10" using?
I guess every base is really base 10...but not necessarily base ten.
The article you linked mentions that there are uncomputable numbers that are nevertheless definable (and so can be talked about, in the abstract). It give Chaitin's number as an example.
I just thought of a counterexample that shows that a proof following my outline won't work. In binary:
Let's say you want to embed an infinite pattern of binary bits a0, a1, a2, a3, a4, ... in an irrational number.
First, construct the sequence 01_001011_... as follows: The initial 01 contains bits 0 and 1, and the 2-bit sequence 01. The next 6 bits contain the other possible 2-bit patterns, as well as some 3-bit, 4-bit, and 5-bit patterns, and is itself a 6-bit pattern. There are 63 other 6-bit patterns, so after the second "_", insert the concatenation of those patterns (378 bits). Now you've got all 3-bit, 4-bit, 5-bit, and 6-bit patterns. Add another underscore and the concatenation of all not-yet-covered 378-bit patterns (this will be quite long!), and another underscore, and so on ad infinitum. Now you have an infinite sequence that contains infinite copies of every finite series of bits. Nothing repeats regularly so it is at least irrational.
All that's left is to insert the bits of our desired infinite pattern (a0, a1, a2) in place of the underscores, one bit (consecutively) at each underscore.
This does *not* mean that inclusion of an infinite pattern in a series that contains all finite patterns of bits is possible if the infinite pattern's bits to be found at regular intervals.
Moral: If we want to find "meaningful" patterns (whatever that means) in pi, we may need to sample the sequence at ever-expanding (hyperexponential) intervals.
Hey, this is Slashdot. Stuff like that belongs on Backslashdot, not here.
If N is a prime, I can factor it *very* quickly.
I suspect it may be possible to prove that no pattern in pi can extend forever.
If any arbitrary sequence of digits can be found in pi (I think this has been shown), and if a purported pattern is found starting at digit N, then there are 2^N - N patterns of length N, each of which must occur following digit N. I think forcing these patterns into the series for all values of N would disrupt any such pattern.
Obviously that's not a formal proof, but it suggests a possible outline of one.
I'd be even more impressed if it said "Sure thing, I'll get back to you when I'm done!" and then pretended to work while surfing the web.
Improved it for you.
I place it at 1 in 6.779 billion...There will always be somebody, somewhere in the world, who is connected to the crime and has a DNA match to the evidence (assuming the DNA in the evidence really came from someone associated with the crime). But in most cases, there will just be one such person.
As a upcoming freshmen let me make a few points.
27. Christopher Columbus has always been getting a bad rap. Not in my school(s).I think it means "Christopher Columbus has always been getting a bad rap somewhere."
walking to class backwards in six feet of snow for at least ten miles
You forgot "uphill both ways".
Also, I actually still *have* a record player (working) and a portable tape player (broken).
As another person replied, a series hybrid will never be more efficient than a straight engine, but that's ignoring the charging of the batteries through third party options like regenerative braking, solar collection and wall sockets.
It's also ignoring that the losses through conversion are only in the 5% to 10% range. If the gain from running the engine at a constant speed is enough to offset this loss, the hybrid *will* be more efficient.
Perhaps this was once a binary star system. The planet was orbiting the *other* star. Something (???) happened to the other star and the planet was recaptured by the remaining star.
Just a thought.