I have this axiom which states the completeness property of a set $A$:
Suppose that $A$ is a set. Every non-empty bounded above subset of $A$ has a least upper bound.
But then my prof told me that this Completeness axiom causes computability issues (later on he talked about computable numbers -- so I have been thinking that he actually referred to that). The thing is, I don't really get what he means by that. What's the connection between that statement above with computability? Hope one of you can help? Cheers!
The main link is related to failures of the completeness theorem if we read "real" as "computable real".
A Specker sequence is a bounded, increasing, computable sequence of rationals whose limit is not computable. If we view the sequence as a set of rationals, it is a decidable, bounded, nonempty set of rationals whose supremum is not computable.