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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)

#51

Post by JoshuaDavid »

dfox wrote: post #47

There is well defined name for "useful reals": Algebraic numbers. Of course the well-definedness necessitates some limit on how the symbolic description looks like (ie. algebraic numbers are roots of polynomials with rational coefficients) because every real number can be described by some arbitrarily complex symbolic notation. Edit: I vaguely remember that there used to be some name for the intersection of algebraic…

> every real number can be described by some arbitrarily complex symbolic notation

This seems like it would have to be false, because otherwise the reals would be countable (iterate through every possible 1-character string, then every possible 2 character string, then 3 chars, etc and in a finite (but potentially very very large) amount of time you would come across the description of any real number that can be described).

Re: The field of “useful reals” between rational and real numbers (2019)

#52

Post by superjan »

this reminds me of unit testing, where the tests come up with arbitrarily defined numbers, and the function you test tries to come up with a consistent way to count them. If you can change your function each time a test is added, the tester never wins. Isn’t this similar? It seems like cherrypicking to include simple formulas with e and pi in your numbering system.

Re: The field of “useful reals” between rational and real numbers (2019)

#53

Post by OscarCunningham »

ramshorns wrote: post #42
jepler wrote: post #5

Not "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.

Yes, though "slightly expanded" is probably a misnomer too, in maybe a Lebesgue measure sense, since 100% of useful reals are transcendental.

Technically you can't compare two countable sets using the Lebesgue measure, because they will both have measure 0.

EDIT: You could instead say something like 'the useful reals are an infinite-degree field extension of the rationals'. (Although as I mentioned elsewhere it's actually impossible to define the useful reals.)

Re: The field of “useful reals” between rational and real numbers (2019)

#54

Post by tromp »

emacdona wrote: post #16
emacdona wrote: post #14

Earlier quoted context omitted.

I say "it" when I mention Chaitin's Constant, but really I believe it's an entire set of constants. Is that set countable? So many questions... :-)

Looks like the wikipedia page says there's a Chaitin's constant for each Computable Function, so yeah, countable. That's if I'm reading it correctly. Even if the constant differs for every program that computes a given Computable Function... still countable, though (if I'm doing my math right).

Yep, it's countable. I have defined a relatively simple one in https://tromp.github.io/cl/Binary_lambda_calculus.html#Halti...

Re: The field of “useful reals” between rational and real numbers (2019)

#55

Post by evanb »

dfox wrote: post #47

There is well defined name for "useful reals": Algebraic numbers. Of course the well-definedness necessitates some limit on how the symbolic description looks like (ie. algebraic numbers are roots of polynomials with rational coefficients) because every real number can be described by some arbitrarily complex symbolic notation. Edit: I vaguely remember that there used to be some name for the intersection of algebraic…

π and e are not algebraic numbers. So the set of "useful" (read: definable) reals is larger than the algebraic numbers.

Re: The field of “useful reals” between rational and real numbers (2019)

#56

Post by [deleted] »

JoshuaDavid wrote: post #51
dfox wrote: post #47

There is well defined name for "useful reals": Algebraic numbers. Of course the well-definedness necessitates some limit on how the symbolic description looks like (ie. algebraic numbers are roots of polynomials with rational coefficients) because every real number can be described by some arbitrarily complex symbolic notation. Edit: I vaguely remember that there used to be some name for the intersection of algebraic…

> every real number can be described by some arbitrarily complex symbolic notation This seems like it would have to be false, because otherwise the reals would be countable (iterate through every possible 1-character string, then every possible 2 character string, then 3 chars, etc and in a finite (but potentially very very large) amount of time you would come across the description of any real number that can be des…

[deleted]

Re: The field of “useful reals” between rational and real numbers (2019)

#57

Post by thaumasiotes »

OscarCunningham wrote: post #24
PaulHoule wrote: post #20

This is one of my favorite obscure math topics. I think of the "useful reals" being the "reals that have names". Alan Turing developed the Turing machine to get a handle on the "useful reals" since you can make a Turing machine write them out one digit at a time. Given that, I don't like the term "real numbers" at all because they are phony compared to the "useful reals" -- if you reject the axiom of choice then the…

> if you reject the axiom of choice then the construction that Cantor does to construct a real isn't valid Are you talking about Cantor's argument that the reals are uncountable? That doesn't need choice.

Elaborating, the hypothesis that Cantor disproves is "The real numbers are countable -- that is to say, the real numbers can be put into one-to-one correspondence with the natural numbers".

You never have to use the axiom of choice, because the hypothesis tells you there is a one-to-one function between the reals and the naturals. You can then order the reals in the order suggested by their image in the naturals: f(0), f(1), f(2), ...

Re: The field of “useful reals” between rational and real numbers (2019)

#58

Post by perl4ever »

thaumasiotes wrote: post #49
perl4ever wrote: post #9

Earlier quoted context omitted.

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.

> that was the question I had - clearly it's smaller than reals, but how and why is this field larger than rational numbers? As pdonis points out sidethread, this isn't really a valid question. (Or rather, the question is fine, but the answer to all questions of this form is already well-known, so there's no point in asking this specific question.) It is not possible to prove that a set is both smaller than the reals…

"It is not possible to prove that a set is both smaller than the reals and larger than the rationals, because such a set would disprove the continuum hypothesis."

Sure, even without much of a mathematical background, people generally take it for granted. Which is why it's disappointing that a suggestion of overturning it isn't fulfilled.

Re: The field of “useful reals” between rational and real numbers (2019)

#59

Post by stephencanon »

Aside from all the other issues people have raised, equality is not decidable for the “useful reals”. While they form a field, they do not form a computably-ordered field, which makes them quite a bit less useful than many other number systems.

Re: The field of “useful reals” between rational and real numbers (2019)

#60

Post by thaumasiotes »

perl4ever wrote: post #58
thaumasiotes wrote: post #49

Earlier quoted context omitted.

> that was the question I had - clearly it's smaller than reals, but how and why is this field larger than rational numbers? As pdonis points out sidethread, this isn't really a valid question. (Or rather, the question is fine, but the answer to all questions of this form is already well-known, so there's no point in asking this specific question.) It is not possible to prove that a set is both smaller than the reals…

"It is not possible to prove that a set is both smaller than the reals and larger than the rationals, because such a set would disprove the continuum hypothesis." Sure, even without much of a mathematical background, people generally take it for granted. Which is why it's disappointing that a suggestion of overturning it isn't fulfilled.

Suggestion of overturning it? It's a proof. That article would be headlined "disproving the independence of the continuum hypothesis" or some such; it would be huge news, not somebody's fun blog post.
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