The field of “useful reals” between rational and real numbers (2019)
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Re: The field of “useful reals” between rational and real numbers (2019)
#2Note that like the rational numbers, the field of “useful reals” is not complete.
So if you have a sequence of “useful reals” that is Cauchy, it will converge to a real number but it may or may not converge to a “useful real”.
Re: The field of “useful reals” between rational and real numbers (2019)
#3Note that like the rational numbers, the field of “useful reals” is not complete. So if you have a sequence of “useful reals” that is Cauchy, it will converge to a real number but it may or may not converge to a “useful real”.
Sorry if I misunderstand you. If I have a Cauchy sequence of "useful reals", wouldn't the convergence be, by definition, a "useful real"? That is, I can write down the Cauchy sequence, so it's now symbolically noted, right?
Or are you referring to a Cauchy sequence that exists, but can't be defined using our symbology?
Re: The field of “useful reals” between rational and real numbers (2019)
#4A nit:
"reals are a field extension of ℚ. They could be considered an algebraic number field..."
This is not an algebraic extension. Pi is a "useful real number" and it is not algebraic over Q.
Re: The field of “useful reals” between rational and real numbers (2019)
#5Not "between" in the sense of having an intermediate cardinality between rationals and reals, since they are exactly the numbers available from strings in some symbolic system or other. Seems to be a slightly expanded case of algebraic numbers, since additional forms (like infinite definite integrals) are allowed.
Re: The field of “useful reals” between rational and real numbers (2019)
#6Not "between" in the sense of having an intermediate cardinality between rationals and reals, since they are exactly the numbers available from strings in some symbolic system or other. Seems to be a slightly expanded case of algebraic numbers, since additional forms (like infinite definite integrals) are allowed.
Yep, that's right. Its cardinality is the same as rationals, since it's countable.
Re: The field of “useful reals” between rational and real numbers (2019)
#7Note that like the rational numbers, the field of “useful reals” is not complete. So if you have a sequence of “useful reals” that is Cauchy, it will converge to a real number but it may or may not converge to a “useful real”.
Sorry if I misunderstand you. If I have a Cauchy sequence of "useful reals", wouldn't the convergence be, by definition, a "useful real"? That is, I can write down the Cauchy sequence, so it's now symbolically noted, right? Or are you referring to a Cauchy sequence that exists, but can't be defined using our symbology?
You cannot write a general Cauchy sequence of useful reals with a finite number of symbols. Hence you cannot in general express its limit with a finite number of symbols.
Re: The field of “useful reals” between rational and real numbers (2019)
#8Note that like the rational numbers, the field of “useful reals” is not complete. So if you have a sequence of “useful reals” that is Cauchy, it will converge to a real number but it may or may not converge to a “useful real”.
Sorry if I misunderstand you. If I have a Cauchy sequence of "useful reals", wouldn't the convergence be, by definition, a "useful real"? That is, I can write down the Cauchy sequence, so it's now symbolically noted, right? Or are you referring to a Cauchy sequence that exists, but can't be defined using our symbology?
There are uncountably many Cauchy sequences of useful reals. You can’t write them all down. So now you have to also restrict yourself to “useful Cauchy sequences of useful reals”.
This is a rabbit hole with no end.
Re: The field of “useful reals” between rational and real numbers (2019)
#9Not "between" in the sense of having an intermediate cardinality between rationals and reals, since they are exactly the numbers available from strings in some symbolic system or other. Seems to be a slightly expanded case of algebraic numbers, since additional forms (like infinite definite integrals) are allowed.
Yep, that's right. Its cardinality is the same as rationals, since it's countable.
How disappointing.
Unlike most of the time, I read the article first and now that I'm here, that was the question I had - clearly it's smaller than reals, but how and why is this field larger than rational numbers? Guess it's not.
Re: The field of “useful reals” between rational and real numbers (2019)
#10The author claims in the notes that "The useful reals are similar, but not quite equivalent to other ideas in mathematics, such as [...] computable numbers."
Is that correct? What is the complement of the Computable Numbers in the Useful Reals? What is the complement of the Useful Reals in the Computable Numbers?
I've always thought of Computable Numbers as all numbers able to be represented by a finite string, ie: a computer program that would generate the number to any desired precision. How does that differ from the set of numbers with a finite symbolic representation?
Hmmmm... maybe by asking that question I've led myself to the answer. Chaitin's Constant has symbolic representations, one of which being the Wikipedia page that describes it: https://en.wikipedia.org/wiki/Chaitin%27s_constant. Does that mean it's included in the complement of the Computable Numbers in the Useful Reals? Are the Computable numbers a subset of the Useful Reals?