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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)

#91

Post by dchyrdvh »

PureParadigm wrote: post #88
dchyrdvh wrote: post #86

The premise of this idea - that anything describable can be written in a binary firm and is thus countable - seems wrong. It's wrong because we easily invent new concepts and put them into a symbolic form. We could invent a new concept, agree on a new symbol for it and add it to our alphabet. The set of ideas isn't countable and so our alphabet isn't countable. This alphabet can't be translated into some binary form…

Why is the alphabet not countable? If each time you think of a new idea and make a symbol for it, I can also assign it to an integer (because there is always a next integer like there is always a new symbol you can come up with). When you come up with a new concept, it should also be possible to write out a definition of it. If you can write down your definition (in English, math notation, etc.), then it comes from a…

We don't "come up" with ideas from other ideas using some closed form rules of logic, like in Coq or some Turing machine. Instead, we discover new ideas.

There is a world of ideas and the real world. People live in both worlds. When they discover a new idea, often by accident, they label it with a symbol and use it in the real world. Other people can see the same idea and since they can't fully describe it with words, they agree to use the new symbol.

We describe new concepts with words, but those definitions are underspecified: they refer to things with vague or non existent descriptions, or just common sense. What is "set" for example? The same words often mean different things in different contexts. This extra meaning that's always attached to words is what makes these definitions non countable.

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

#92

Post by dchyrdvh »

lisper wrote: post #87
dchyrdvh wrote: post #86

The premise of this idea - that anything describable can be written in a binary firm and is thus countable - seems wrong. It's wrong because we easily invent new concepts and put them into a symbolic form. We could invent a new concept, agree on a new symbol for it and add it to our alphabet. The set of ideas isn't countable and so our alphabet isn't countable. This alphabet can't be translated into some binary form…

What makes you think that the set of ideas isn't countable?

So long as every real number exists, has properties and so on. Every such number is a separate idea. They exist, no matter whether we know about them or not.

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

#93

Post by impendia »

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.

Slight correction. Your first bullet point, and hence your proof that ℚ cannot contain π, is correct.

A number x is algebraic over ℚ if and only if it generates a finite field extension, i.e. if x, x^2, x^3, etc. have a linear dependence relation.

However, as jopolous pointed out, you can get infinite dimensional algebraic field extensions by adjoining infinitely many algebraic numbers. For example, the set of all numbers which are algebraic over ℚ is a field, and this field is an infinite degree extension of ℚ.

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

#94

Post by Someone »

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.

”π, π², π³, … which are linearly independent.”

Do we know that? My search doesn’t get more than https://www.encyclopediaofmath.org/index.php/Lindemann_theor..., which proves it for “𝑒, 𝑒², 𝑒³, …“.

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

#95

Post by circlefavshape »

D_Alex wrote: post #84

If you look at the integers between say Graham's number ( https://en.wikipedia.org/wiki/Graham%27s_number ) and TREE3( https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem ) you can observe that practically all of these integers, while "computable", cannot be defined within the known constraints of this universe. Which raises an interesting question: In what meaningful sense do these numbers exist ? They are just o…

In what meaningful sense do any numbers exist? This comes up with my kids sometimes ... are numbers real?

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

#96

Post by steerablesafe »

lonelappde wrote: post #80
H8crilA wrote: post #32

Earlier quoted context omitted.

It has sqrt(2), for starters? Not sure what do you mean by useful. It is not "useful" in the sense that reals are most "famous" for: it is not complete. Cauchy sequences can diverge in the useful reals field.

Completeness in the "full" reals is a useless feature, though. All is gives you is an emotional crutch to pretend your cauchy sequences can be mapped to regular numbers. But it doesn't give you anything you didn't already have in the cauchy sequences and useful reals.

You are of course right, reals are isomorphic to equivalence classes of Cauchy sequences on Q. But once you are dealing with equivalence classes of Cauchy sequences on Q you might as well give it a name. Maybe call it R.

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

#97

Post by lisper »

dchyrdvh wrote: post #92
lisper wrote: post #87

Earlier quoted context omitted.

What makes you think that the set of ideas isn't countable?

So long as every real number exists, has properties and so on. Every such number is a separate idea. They exist, no matter whether we know about them or not.

Ah. Personally, I distinguish between potential ideas and actual ideas. To be an actual idea, it has to reside in someone's brain (or a computer, or some other data-processing system). The reals correspond to the set of potential ideas, but the set of actual ideas is not only countable, but almost certainly finite.

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

#98

Post by thaumasiotes »

lonelappde wrote: post #72
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…

Aside, it's a shape that hypothesis in math (axiom) is the opposite of hypothesis in science (claim to test)

Not really. The words are used in closely analogous ways. But the "hypothesis" in "continuum hypothesis" is part of the name of the continuum hypothesis, carried over from a time when we didn't know the answer.

Names are just names. Euclid's Algorithm is an algorithm. The Division Algorithm is a theorem.

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

#99

Post by ummonk »

klodolph wrote: post #2

Note 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”.

Name one such sequence of "useful reals" that is Cauchy but doesn't converge to a "useful real". You can't, can you? "Useful" Cauchy sequences of "useful reals" (i.e. those you can define) all converge to a "useful real".

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

#100

Post by joppy »

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.

When talking about subsets of an infinite set, and in particular fields, the common understanding of the word “between” means in terms of subsets, not cardinality. For instance, the field Q(sqrt 2) lies between the fields Q and Q(sqrt 2, sqrt 3).
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