4 ms·

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 d

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?

klodolph 7y ago
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.
- wolfgke 7y ago
  > 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.
  - pdonis 7y ago
    To prove that a field is complete, your proof must hold for any Cauchy sequence, not just the ones that meet some constraint you impose.
  - klodolph 7y ago
    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 surprised that by using different definitions, you come to different conclusions.
    - avmich 7y ago
      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...
    - wolfgke 7y ago
      > 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?
- ultrafilter 7y ago
  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.
  - OscarCunningham 7y ago
    This needs axioms beyond ZFC though. Even assuming ZFC is consistent isn't enough to know that there's a minimal countable transitive model.
wololongong 7y ago
You cannot write a general Cauchy sequence of useful reals with a finite number of symbols. Hence you cannot in general express its limit with a finite number of symbols.

klodolph 7y ago

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.

wolfgke 7y ago
> 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.
- pdonis 7y ago
  To prove that a field is complete, your proof must hold for any Cauchy sequence, not just the ones that meet some constraint you impose.
- klodolph 7y ago
  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 surprised that by using different definitions, you come to different conclusions.
  - avmich 7y ago
    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...
  - wolfgke 7y ago
    > 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?
ultrafilter 7y ago
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.
- OscarCunningham 7y ago
  This needs axioms beyond ZFC though. Even assuming ZFC is consistent isn't enough to know that there's a minimal countable transitive model.

wololongong 7y ago

You cannot write a general Cauchy sequence of useful reals with a finite number of symbols. Hence you cannot in general express its limit with a finite number of symbols.