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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)

#11

Post by klodolph »

perl4ever wrote: post #9
xtacy wrote: post #6

Earlier quoted context omitted.

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.

It’s larger than the rational numbers in the sense that it is a strict superset. Cardinality is what a lot of people reach for when they are talking about “larger” or “smaller”, but there are lots of other useful concepts which we can translate to “larger” and “smaller”.

So when someone says “larger” or “smaller”, your first step might be to try and translate that relationship into a more precise mathematical concept, like cardinality or measure.

Casual terminology also leads to weird discussions. Like when someone asks whether some function is “close” to another, and these functions are defined in terms of vector spaces. Unfortunately, “closeness” does not necessarily exist in a vector space. So the answer may be that the question does not make sense.

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

#14

Post by emacdona »

emacdona wrote: post #10

The 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…

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... :-)

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

#15

Post by wolfgke »

klodolph wrote: post #8
function_seven wrote: post #3

Earlier quoted context omitted.

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.

> There are uncountably many Cauchy sequences of useful reals. You can’t write them all down.

This does not hold if you demand that, for example, the map

k -> a_k

that represents the Cauchy sequence, is a computable function.

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

#16

Post by emacdona »

emacdona wrote: post #14
emacdona wrote: post #10

The 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…

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).

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

#17

Post by ur-whale »

perl4ever wrote: post #9
xtacy wrote: post #6

Earlier quoted context omitted.

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.

> Guess it's not.

Isn't there a theorem that speaks of the existence or non-existence of a set whose cardinality is strictly larger than Q and strictly smaller than R.

And a conjecture that says this theorem might well be unprovable?

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

#18

Post by pdonis »

wolfgke wrote: post #15
klodolph wrote: post #8

Earlier quoted context omitted.

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.

> There are uncountably many Cauchy sequences of useful reals. You can’t write them all down. This does not hold if you demand that, for example, the map k -> a_k that represents the Cauchy sequence, is a computable function.

To prove that a field is complete, your proof must hold for any Cauchy sequence, not just the ones that meet some constraint you impose.

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

#19

Post by klodolph »

wolfgke wrote: post #15
klodolph wrote: post #8

Earlier quoted context omitted.

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.

> There are uncountably many Cauchy sequences of useful reals. You can’t write them all down. This does not hold if you demand that, for example, the map k -> a_k that represents the Cauchy sequence, is a computable function.

I am a bit shocked, because that seems like a really dishonest way of doing things. “Surprise! Actually, I am not talking about Cauchy sequences, but only computable Cauchy sequences.”

If you change the rules you had better be up front about it.

What you are describing is a completely different definition for “complete metric space” than what is commonly accepted by the mathematical community at large. So do not be surprised that by using different definitions, you come to different conclusions.

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

#20

Post by PaulHoule »

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 construction that Cantor does to construct a real isn't valid.

Despite calling for a rebuild of math and science based on computation, Steve Wolfram has yet to take the critical step of rejecting the axiom of choice. I wish he would man up.

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