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It is possible to prove that $a=b \Longleftrightarrow a+m= b+m ,\,\,\, \, \{a,b,c\} \subset \mathbb{R}$ , or this is just an axiom?

I'm curious if there is any demonstration of this simple statement.

Thank you.

Gerry Myerson
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Voyager
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  • What is the definition of $\Bbb{R}$ and $+$? – Hanul Jeon Feb 20 '14 at 08:08
  • @tetori Set of the Real numbers and addition – Voyager Feb 20 '14 at 08:09
  • How to define the set of real number? Dedekind cut or Cauchy sequence? – Hanul Jeon Feb 20 '14 at 08:10
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    @Voyager The point is that when you ask whether something is provable or just an axiom, you have to specify exactly what your formal system is - you may 'know' what the reals and addition look like, but there are several distinct ways of axiomatizing them and your question can have different answers depending on the axiomatization. – Steven Stadnicki Feb 20 '14 at 08:11
  • @StevenStadnicki hmmm, sorry for that, i am still a high school student. – Voyager Feb 20 '14 at 08:12
  • So, in essence, you should first learn what are real numbers ;) Most of the real analysis books discuss this in the first chapter, if you're curious enough. – Marcin Łoś Feb 20 '14 at 08:18
  • @MarcinŁoś Wow, that's cool! Thanks!! – Voyager Feb 20 '14 at 08:22
  • @tetori Dedekind cut and Cauchy sequence are means of constructing the reals from the rationals. There are also direct definitions of the reals, for example as a complete ordered field. – Mark Bennet Feb 20 '14 at 08:39
  • @MarkBennet I know these methods that define the theory of reals. I just give a specific examples of constructing method of reals. – Hanul Jeon Feb 20 '14 at 10:59
  • @tetori No problem - it just looked as though your comment allowed only two possibilities. It matters for this question whether the reals are constructed from the rationals - in which case you have to prove that the reals have the various properties you want; or whether the reals are defined as a complete ordered field - in which case this question follows from the fact that they form an additive abelian group. The question then is whether an object defined by the axioms actually exists (and also whether it is unique). – Mark Bennet Feb 20 '14 at 12:04
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    Also, what is $c$? – Nishant Aug 06 '14 at 04:22

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If you seek a formal demonstration to this problem, you could use a simple necessity-sufficiency proof:

Necessity (or =>): We consider $a=b$ true.

This means that $a+m = b+m$ can be written as $a+m = a+m$ or $b+m = b+m $ . These two statements are true.

So we proved that if $a=b$ then $a+m=b+m$.

Now for sufficiency (or <=): We consider $a+m=b+m$ true.

Subtract m from both sides of $=$ and get $a=b$.

We proved that if $a+m=b+m$, then $a=b$

The demonstration has been made from left to right and from right to left, which means that the original statement is true.

Vlad Cristescu
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