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Post by Eugene 2.0 on Mar 6, 2021 12:20:31 GMT
There's a line from 0 to 1. On the line the row of real numbers like 0.000001, 0,0201309012... etc.
So how to calculate all of them?
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antor
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Post by antor on Mar 6, 2021 19:48:21 GMT
Calculate is not the same as count and write down. In my vocabulary.
But anyway. The real numbers are all fractions, all irrational numbers and all transcentental numbers.
So the method for fractions would be to do x/y for x and y from 0 to inf, all combinations
Same goes for irrational and transcendental numbers where you take the above fractions and multiply by new fractions of each irrational and transcendental nummber
Now you have written down alot of numbers, hypothetically. Then you select those between 0 and 1. Hypothetically of course.
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Post by Eugene 2.0 on Mar 6, 2021 20:47:15 GMT
Calculate is not the same as count and write down. In my vocabulary. But anyway. The real numbers are all fractions, all irrational numbers and all transcentental numbers. So the method for fractions would be to do x/y for x and y from 0 to inf, all combinations Same goes for irrational and transcendental numbers where you take the above fractions and multiply by new fractions of each irrational and transcendental nummber Now you have written down alot of numbers, hypothetically. Then you select those between 0 and 1. Hypothetically of course. Well, yes taking fractions would be really helpful, yet I can't say that each fraction x/y realizes in some real one number z. Besides, the transcendental numbers are seemed to be out of the line; I mean only the real number has projections on a line, the transcendental numbers are being realized in a square, the Cartesian field, for instance. |
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Post by karl on Mar 6, 2021 21:02:22 GMT
Every irrational number can be approximated arithmetically, which means that a computer with unlimited capacity could express each irrational number by the algorithm that approximates it. Problem is, it's theoretically impossible to program a computer to identify which algorithm approximates an irrational number, and which one doesn't. So identifying irrational numbers have to be done by conscious intelligence. It can't be an automated process.
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Post by Eugene 2.0 on Mar 6, 2021 21:10:04 GMT
Every irrational number can be approximated arithmetically, which means that a computer with unlimited capacity could express each irrational number by the algorithm that approximates it. Problem is, it's theoretically impossible to program a computer to identify which algorithm approximates an irrational number, and which one doesn't. So identifying irrational numbers have to be done by conscious intelligence. It can't be an automated process. Oh, it's really interesting. A sentinel person can be deceived also. And it's also curious why among the other algorithms the one that implies irrational numbers cannot be picked up? I mean what barriers a computer to do this? |
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Post by karl on Mar 6, 2021 21:28:31 GMT
Every irrational number can be approximated arithmetically, which means that a computer with unlimited capacity could express each irrational number by the algorithm that approximates it. Problem is, it's theoretically impossible to program a computer to identify which algorithm approximates an irrational number, and which one doesn't. So identifying irrational numbers have to be done by conscious intelligence. It can't be an automated process. Oh, it's really interesting. A sentinel person can be deceived also. And it's also curious why among the other algorithms the one that implies irrational numbers cannot be picked up? I mean what barriers a computer to do this?
Because the set of all irrational numbers is unenumerable. Not that this is limited to irrational numbers. It's the same if one attempts to identify every algorithm that produces a never-ending sequence of digits.
Image you show a computer an algorithm and ask: "Does this algorithm have property A?". The computer may say, "yes", "no", or "I don't know.". If property A is the property of calculating a rational number, then if only the computer is programmed correctly, it can be trusted to always give the right answer. For the set of all rational numbers is enumerable. If property A is the property of approximating an irrational number, then one has two choices.
1. The computer will have to state "I don't know" for an infinite number of algorithms that do have property A.
2. The computer will state "yes" for every algorithm that has property A, but it will also state "yes" for an infinite number of algorithms that do not have property A.
The computer will either have to be incomplete (not being able to answer every meaningful question), or inconsistent (give false answers).
Same thing if property A is the property of producing a never-ending sequence of digits.
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antor
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Post by antor on Mar 6, 2021 22:21:49 GMT
Well I wrote from a mathematics point of view. Still I think calculate is almost an overkill term since its just multiplication in my method. No need to implicitly assume its done by a computer. Could be done by any entity capable of multiplication/division. The method/formula/whatever is there in any case. You are right in that undiscovered irrational and transcendental numbers maybe exist. The method I described deals only with the known. "I mean only the real number has projections on a line, the transcendental numbers are being realized in a square, the Cartesian field, for instance." The transcendetal numbers are real by definition. Real numbers is a definition made by mathematicians. Not something that for them is up for debate. I dont mean to be rude but if you want to group number types differently please name your group something else  There are no perfect circles in the world so you can argue pi is not "real" or cant be realized but then youre talking about a different realism than the mathematical definition of real numbers. Anyway pi can be proven to be between some two values on the real number line therefore its real. In contrast to being an imaginary number which is another topic. (I'm not a mathematician so I may be wrong on some details) |
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Post by Eugene 2.0 on Mar 6, 2021 22:49:46 GMT
Well I wrote from a mathematics point of view. Still I think calculate is almost an overkill term since its just multiplication in my method. No need to implicitly assume its done by a computer. Could be done by any entity capable of multiplication/division. The method/formula/whatever is there in any case. You are right in that undiscovered irrational and transcendental numbers maybe exist. The method I described deals only with the known. "I mean only the real number has projections on a line, the transcendental numbers are being realized in a square, the Cartesian field, for instance." The transcendetal numbers are real by definition. Real numbers is a definition made by mathematicians. Not something that for them is up for debate. I dont mean to be rude but if you want to group number types differently please name your group something else ;) There are no perfect circles in the world so you can argue pi is not "real" or cant be realized but then youre talking about a different realism than the mathematical definition of real numbers. Anyway pi can be proven to be between some two values on the real number line therefore its real. In contrast to being an imaginary number which is another topic. (I'm not a mathematician so I may be wrong on some details) I'm not a mathematician either, but what I find fascinating is that you and karl both agree at that there's no perfect circles in the world, and at the same time karl has said that no irrational numbers are numerable, while you've said that the transcendental numbers can be found on a line. Yes, your objection is correct toward mine the transcendental number line. It was my fault to confuse them with imaginary ones. I had made a rough mistake. It seems obvious, especially considering the karl 's answer given, that knowledge of this sum is technically limited. |
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Post by karl on Mar 6, 2021 23:06:29 GMT
Eugene 2.0 I wrote that the set of all irrational numbers is not enumerable. Some subsets of irrational numbers are enumerable.
And to clarify, in regards to the original question; If the set of all irrational numbers is not enumerable, then neither can the set of all irrational numbers between N and N+1, N being an integer.
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Post by Eugene 2.0 on Mar 7, 2021 7:43:26 GMT
Eugene 2.0 I wrote that the set of all irrational numbers is not enumerable. Some subsets of irrational numbers are enumerable.
And to clarify, in regards to the original question; If the set of all irrational numbers is not enumerable, then neither can the set of all irrational numbers between N and N+1, N being an integer.
But you said: "the set of all irrational numbers is unenumerable". Saying "no rational number are numerable" I meant exactly this Shrug1 And you're right. Any real numbers from N to N+1 isn't numerable either. Anyway I don't want to stop thinking about it. I don't understand why can't we find the very first number of the real ones? Maybe the logic of their calculation is different to the other numbers?.. :( |
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Post by karl on Mar 7, 2021 19:02:25 GMT
Eugene 2.0 I wrote that the set of all irrational numbers is not enumerable. Some subsets of irrational numbers are enumerable.
And to clarify, in regards to the original question; If the set of all irrational numbers is not enumerable, then neither can the set of all irrational numbers between N and N+1, N being an integer.
But you said: "the set of all irrational numbers is unenumerable". Saying "no rational number are numerable" I meant exactly this Shrug1 And you're right. Any real numbers from N to N+1 isn't numerable either. Anyway I don't want to stop thinking about it. I don't understand why can't we find the very first number of the real ones? Maybe the logic of their calculation is different to the other numbers?.. :(
And what does it mean that it's the very first number? That it's the first irrational number larger than 0? Whatever irrational number one constructs that's larger than zero, one can always use to create a smaller one. The very approximation of an irrational number is what I wish to refer to as a restless mathematical process that isn't really going anywhere. Imagine you mark a distance with the length of 1 km. First you move towards 1/2 km, then you turn around and move backward, 1/2 of that 1/2 km (so you're at 1/4 km), then you turn around again and move 1/2 of 1/2 of 1/2 (1/8 km). Now you're at 3/8 km. Then you turn around once more, and move 1/16 km. And so on. What value does this approximate? None in the rational number system. One must either state that it approximates nothing, and that you only move back and forth on an ever decreasing interval, or one can just define this approximation as that of an irrational number. -And this is basically what irrational numbers are; Mathematical processes that confines themselves to a smaller and smaller interval, but do not approximate a rational number. This makes the very concept of an irrational number allusive, which is the real reason why no AI could evaluate an algorithm and always know whether or not it approximates an irrational number.
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antor
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Post by antor on Mar 7, 2021 19:42:19 GMT
Karl:
Alot of mathematics is just discovered and then not going anywhere.
A mathematician would want to see a proof of that second sentence. And no you cant simply "define it as an irrational number either, there are strict methods of that. Which I dont know by heart.
In my opinion you are saying for example we cant take the square root of 2. But, why not? What separates 2 from 4 really? And about AI well for all practical purposes we dont actually need them to calculate pi for example. We just feed them a good enough value.
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Post by karl on Mar 7, 2021 20:51:31 GMT
antor 1. I have no quarrel with the first statement. 2. An irrational number is defined as a real number that isn't rational. A real number is any value along a line that has either a finite number of decimals or infinite decimal representation. A rational number can be written as a fraction. The example I gave can be be written as follows: 1/2-1/4+1/8-1/16+1/32-1/64+1/128.... Or: 1/(4a-2)-1/(4a) Summed up from a=1 to infinity. The temporary sums will get an ever larger denominator, and hence don't converge toward a fraction. Instead it may be written as an infinite decimal representation. 3. I do not see how what I wrote implies that you can't do a square root of 2.
A more abstract definition of real numbers: "A quantity that can be represented as an infinite decimal expansion."
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Post by fschmidt on Mar 7, 2021 21:43:45 GMT
You can't. You can't even calculate one irrational number, say pi-3. But why even asked this question? Aren't there more practical questions worth asking?
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Post by Eugene 2.0 on Mar 8, 2021 12:57:08 GMT
You can't. You can't even calculate one irrational number, say pi-3. But why even asked this question? Aren't there more practical questions worth asking? ;) Thanks! You're truly right. |
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There are no perfect circles in the world so you can argue pi is not "real" or cant be realized but then youre talking about a different realism than the mathematical definition of real numbers. Anyway pi can be proven to be between some two values on the real number line therefore its real. In contrast to being an imaginary number which is another topic.