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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)

#41

Post by effie »

klodolph wrote: post #11
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.

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…

> “closeness” does not necessarily exist in a vector space.

The asker will give a definition. For example, two vectors are close if sqrt of dot product of difference of the two vectors is smaller than some number delta.

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

#42

Post by ramshorns »

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.

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

#43

Post by solinent »

OscarCunningham wrote: post #38

> A “useful real” is just a real number that can be precisely described (not just approximated!) by some symbolic notation. Obviously, this definition is loose and depends greatly on your choice of symbols and their definitions. In fact, the definition is necessarily loose. If you could make it precise then you could carry out Cantor's diagonalisation procedure to produce a precise description of a real which couldn'…

Another argument that's not completely non-constructive.

The real numbers have to be constructed. Typically, a number is represented by a Cauchy sequence or a Dedekind cut.

To determine if a real number is representable symbolically, we simply need a finite sequence of symbols which stands for this Cauchy sequence, lets say.

Theroem: The real numbers and definable numbers are the same set.

Assume a real number exists but is not definable. This means at the very least we have a mathematical statement saying there exists a number such that some logical predicate is valid (we may not even have a construction in ZFC), which can also be constructed using a Cauchy sequence. This mathematical statement is embedded in ZFC, and since we are humans it must be finite. In fact, you could come up with a binary representation for such a statement using methods from Godel, by mapping each symbol to some binary representation. Therefore, this number can be represented as a sequence of zeros or ones, a contradiction.

QED

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

#44

Post by pdonis »

wolfgke wrote: post #33
klodolph wrote: post #19

Earlier quoted context omitted.

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 s…

> 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.” Rather: If countability is important to you, you should change the rules so that the property that the field is closed w.r.t limits of Cauchy sequences does not make your set uncountable. Redefining the rules if something does not wo…

> If countability is important to you, you should change the rules so that the property that the field is closed w.r.t limits of Cauchy sequences does not make your set uncountable.

Ok, fine: I hereby declare that the set Q of rationals is a closed field, because I define "closed" to mean "closed under Cauchy sequences whose limit points are rational numbers".

Does that seem OK to you?

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

#46

Post by thaumasiotes »

OscarCunningham wrote: post #38

> A “useful real” is just a real number that can be precisely described (not just approximated!) by some symbolic notation. Obviously, this definition is loose and depends greatly on your choice of symbols and their definitions. In fact, the definition is necessarily loose. If you could make it precise then you could carry out Cantor's diagonalisation procedure to produce a precise description of a real which couldn'…

> the definition is necessarily loose. If you could make it precise then you could carry out Cantor's diagonalisation procedure to produce a precise description of a real which couldn't be precisely described

Is this true without the Axiom of Choice? Don't you need a choice function to order the numbers before you can diagonalize them?

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

#47

Post by dfox »

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 and real numbers, but I neither can remember it nor can find it on wikipedia.

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

#48

Post by OscarCunningham »

solinent wrote: post #43
OscarCunningham wrote: post #38

> A “useful real” is just a real number that can be precisely described (not just approximated!) by some symbolic notation. Obviously, this definition is loose and depends greatly on your choice of symbols and their definitions. In fact, the definition is necessarily loose. If you could make it precise then you could carry out Cantor's diagonalisation procedure to produce a precise description of a real which couldn'…

Another argument that's not completely non-constructive. The real numbers have to be constructed. Typically, a number is represented by a Cauchy sequence or a Dedekind cut. To determine if a real number is representable symbolically, we simply need a finite sequence of symbols which stands for this Cauchy sequence, lets say. Theroem: The real numbers and definable numbers are the same set. Assume a real number exists…

I don't understand that argument. But in any case, Cantor's argument is very constructive. It literally gives you the decimal expansion of the new number not in your set.

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

#49

Post by thaumasiotes »

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.

> 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 and larger than the rationals, because such a set would disprove the continuum hypothesis. (And symmetrically, it isn't possible to prove that no such set exists, because that would be a proof of the continuum hypothesis.)

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

#50

Post by OscarCunningham »

thaumasiotes wrote: post #46
OscarCunningham wrote: post #38

> A “useful real” is just a real number that can be precisely described (not just approximated!) by some symbolic notation. Obviously, this definition is loose and depends greatly on your choice of symbols and their definitions. In fact, the definition is necessarily loose. If you could make it precise then you could carry out Cantor's diagonalisation procedure to produce a precise description of a real which couldn'…

> the definition is necessarily loose. If you could make it precise then you could carry out Cantor's diagonalisation procedure to produce a precise description of a real which couldn't be precisely described Is this true without the Axiom of Choice? Don't you need a choice function to order the numbers before you can diagonalize them?

In this case you can define an order without using Choice. By definition of 'useful number' each useful number has some finite string that describes it. The finite strings can be put into lexicographic order, and then the useful numbers can be ordered according to the position of the lexicographically first string that describes them.
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