31–40 of 121 posts
In order for the set of symbols to be finite field it must grow therefore since rational is infinitely countable from Cantor it must hold that "useful reals" is uncountable.
Does this field behave differently from Q in some 'useful' way?
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.
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…
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 work is how you do mathematics works all the time:
- A PDE does not have a solution in a classical sense and you hate this? No problem: You invent the theory of weak solutions and distributions and simply change the concept what is to be considered a solution of the PDE.
- The concept of algebraic varieties turns out to be to limiting to obey the rules that you would love them to have? No problem: You define the concept of algebraic schemes and now talk about algebraic schemes instead of varieties (https://en.wikipedia.org/w/index.php?title=Scheme_(mathemati...).
TLDR: Mathematics is often the art of "defining your problems away".
The author claims that this set is countable but not sure if that is true. My argument is based on Cantor's theorem [1], which states that the power set has cardinality strictly greater than the set. In order for the set of symbols to be finite field it must grow therefore since rational is infinitely countable from Cantor it must hold that "useful reals" is uncountable. [1] https://en.wikipedia.org/wiki/Cantor%27s_t…
(Note, by the way, that there's any number of other fields one could put inbetween; such as the field of algebraic reals, or computable reals, or the fraction field of the ring of periods...)
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…
But yes, Chaitin's constant is an example of a number that is definable but not computable.
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.
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't be precisely described, a contradiction.
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.
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…
Exploring the consequences of alternative definitions is fantastic, let’s do more of that.
Redefining the terms that somebody else uses in a conversation sucks royally. Everybody hates it when people do that. Don’t be that guy.