4 ms·

> 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 co

> 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 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_(mathematics)&oldid=942687770 https://en.wikipedia.org/w/index.php?title=Scheme_(mathemati...). TLDR: Mathematics is often the art of "defining your problems away".

klodolph 7y ago
> Redefining the rules if something does not work is how you do mathematics works all the time: 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.
pdonis 7y ago
> 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?

klodolph 7y ago

> Redefining the rules if something does not work is how you do mathematics works all the time: 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.

pdonis 7y ago

> 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?