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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)

#61

Post by benkuhn »

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'…

You can make it precise if the language in which you define a "useful real" is richer than the language in which individual useful reals must be defined. For instance, model theorists will talk about definable reals in a model of set theory: https://en.wikipedia.org/wiki/Definable_real_number#Definabi...

> A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds (see Kunen 1980, p. 153). This notion cannot be expressed as a formula in the language of set theory.

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

#62

Post by perl4ever »

thaumasiotes wrote: post #60
perl4ever wrote: post #58

Earlier quoted context omitted.

"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.

So, you know better than to be fooled by this clickbait. Noted. All I'm saying is that it was very effective on me, unlike a lot of things.

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

#63

Post by perl4ever »

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…

"It’s larger than the rational numbers in the sense that it is a strict superset"

Ok, did I miss the explanation of that? Or is it something in "part 2" which I didn't see a link to?

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

#65

Post by BeetleB »

ur-whale wrote: post #17
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.

> 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?

In this particular case, though, that is irrelevant. This set is strictly of the same cardinality as the rationals.

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

#66

Post by shawnz »

perl4ever wrote: post #63
klodolph wrote: post #11

Earlier quoted context omitted.

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…

"It’s larger than the rational numbers in the sense that it is a strict superset" Ok, did I miss the explanation of that? Or is it something in "part 2" which I didn't see a link to?

Clearly every single rational number is "useful", plus others that are not rational

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

#67

Post by jopolous »

arberavdullahu wrote: post #27
klodolph wrote: post #23

Earlier quoted context omitted.

Yes—and to elaborate, the reason why an algebraic field extension of ℚ cannot contain π is because: - If it is a field, it contains π, π², π³, … which are linearly independent. - By definition, an algebraic field extension is finite dimensional.

You are wrong! The algebraic field extension ℚ[π] contains π.

I guarantee that is not an algebraic extension.

It's not even a finite extension

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

#68

Post by jopolous »

klodolph wrote: post #23
doomrobo wrote: post #4

A 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.

Yes—and to elaborate, the reason why an algebraic field extension of ℚ cannot contain π is because: - If it is a field, it contains π, π², π³, … which are linearly independent. - By definition, an algebraic field extension is finite dimensional.

Another nit, not all algebraic extensions are finite dimensional. Just adjoin an infinite number of linearly independent roots (square roots, cube roots, etc). Algebraic, but not finite

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

#69

Post by PeterisP »

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'…

It seems to me that Cantor's diagonalization fails here because of the very different nature of descriptions vs (for example) decimal notation. Every possible string of digits is a valid, unique number. That does not apply to descriptions.

I'd assume that every number that can be precisely described by some symbolic notation can be described in that notation in multiple ways, and likely in an infinite number of multiple different ways. E.g. the number 2 can be described as 1+1, 1+1+1-1, 1+1+1+1-1-1, ad infinitum.

Furthermore, I'd assume that not every string in that symbolic notation constitutes a valid, precise description of some real.

So Cantor's diagonalization produces some unique description of a number that differs from all of the descriptions - but it's possible and plausible that the description refers to a number that is in the list but has been described differently; and it's possible and plausible that the constructed description does not describe any real whatsoever.

Or am I completely misunderstanding you and you did not intend to apply Cantor's diagonalization to the descriptions?

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

#70

Post by NelsonMinar »

Another related topic of interest is constructivism in mathematics. Unfortunately the wikipedia article is pretty abstruse, anyone have a more down to earth one? https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...

(Note this is different from constructible numbers, which the author mentions. That has to do with classical geometry.)

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