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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)

#21

Post by pdonis »

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?

A "theorem" is a proved proposition, so there is no such thing as a theorem that might well be unprovable.

The proposition you refer to (which is not a theorem since no proof is known, and in fact it has been shown that this proposition is logically independent of the usual foundations of set theory, so it cannot be proved in that framework) is called the Continuum Hypothesis:

https://en.wikipedia.org/wiki/Continuum_hypothesis

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

#22

Post by ultrafilter »

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.

Set theorists have located the "end" for all practical and most impractical purposes. Let M be the minimal countable transitive model of ZFC. Declare a real to be useful if and only if it is in M.

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

#23

Post by klodolph »

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.

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

#24

Post by OscarCunningham »

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.

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

#26

Post by OscarCunningham »

ultrafilter wrote: post #22
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.

Set theorists have located the "end" for all practical and most impractical purposes. Let M be the minimal countable transitive model of ZFC. Declare a real to be useful if and only if it is in M.

This needs axioms beyond ZFC though. Even assuming ZFC is consistent isn't enough to know that there's a minimal countable transitive model.

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

#27

Post by arberavdullahu »

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.

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

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

#28

Post by OscarCunningham »

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.

I don't think that's the usual definition of algebric extension. Wikipedia (https://en.wikipedia.org/wiki/Algebraic_extension) says that an algebraic extension is one where every element is the root of some nonzero polynomial over the base field. So for example the algebraic numbers would be algebraic over the rationals, even though they're infinite dimensional over the rationals.

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

#29

Post by avmich »

klodolph wrote: post #19
wolfgke wrote: post #15

Earlier quoted context omitted.

> 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 s…

Interesting that for Cauchy sequences in question they necessarily have to be non-constructive, i.e. one can't name any element in the sequence.

Edit: oops, not that. The sequence itself - not the "useful reals" element of the sequence - has to be non-constructive...

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

#30

Post by OscarCunningham »

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 think people would normally call that a transcendental extension and not an algebraic one.
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