Re: Set of irrationals closed under sum

In article <3m18mcF14hotnU1@xxxxxxxxxxxxxx>,
José Carlos Santos <jcsantos@xxxxxxxx> wrote:
>Hi all,
>
>Let x be an irrational number and let A = {x,2x,3x,...}. Then A is countable
>set of irrational
>numbers such that A + A is a subset of A.
>
>My question is: is there an *uncountable* set A of irrational numbers such
>that A + A is a
>subset of A?

Sure. Let B be a Hamel basis for R over Q. Let S be any uncountable
subset of B which does not include a rational number (there is at most
one rational number in B), and let A be the Z-span of S.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.

Relevant Pages

  • Re: "It is easy to see...."
    ... all this stuff cold, but it's been too many years... ... what I accept as reality." ... Arturo Magidin ... Prev by Date: ...
    (sci.math)
  • Re: Dedekind Domain
    ... Sorry I can't be more detailed; I'm away from the office, and plan to ... what I accept as reality." ... Arturo Magidin ... Prev by Date: ...
    (sci.math)
  • Re: Split or Irreducible
    ... If u is a root for f, ... what I accept as reality." ... Arturo Magidin ... Prev by Date: ...
    (sci.math)
  • Re: A few more GRE questions
    ... Hmmm... ... what I accept as reality." ... Arturo Magidin ... Prev by Date: ...
    (sci.math)
  • Re: what am i missing here?
    ... equal to pi/2. ... what I accept as reality." ... Arturo Magidin ... Prev by Date: ...
    (sci.math)