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Charles C.Pinter - Set theory

Let $a,b,c$ be any cardinal numbers.

Give a counterexample to the rule: $$a+b = a+c \implies b=c$$

Does there exist a counterexample?

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$$\begin{align}&1.\quad\aleph_0=\aleph_0+0=\aleph_0+1\\&2.\quad 0\neq 1\end{align}$$

Where $\aleph_0$ is the cardinality of countably infinite sets, e.g. the non-negative integers, $\mathbb N$.

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  • $\begingroup$ This is certainly the very simplest counterexample. $\endgroup$ – Michael Hardy May 11 '12 at 23:59
  • $\begingroup$ or any infinite cardinal $\endgroup$ – Greg Martin May 12 '12 at 0:03
  • $\begingroup$ @Greg: Assuming the axiom of choice, yes. $\endgroup$ – Asaf Karagila May 12 '12 at 10:06
  • $\begingroup$ @AsafKaragila: aha, equivalently the well-ordering principle - interesting. Can one construct a counterexample if the axiom of choice isn't assumed? $\endgroup$ – Greg Martin May 12 '12 at 19:41
  • $\begingroup$ @Greg: Indeed. We say that A is a Dedekind-finite set if whenever $B\subsetneq A$, $|B|<|A|$. Equivalently this is to say that $|A|<|A|+1$. Every finite set is Dedekind-finite, and assuming the axiom of choice the opposite is also true: Dedekind-finite sets are finite sets. However it is consistent that without the axiom of choice there may be infinite Dedekind-finite sets. These sets may be used to construct such counterexamples. $\endgroup$ – Asaf Karagila May 12 '12 at 21:48
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Here's one: $|\Bbb N| + |\{1\}| =|\Bbb N| + |\{2, 3\}| $, but $|\{1\}| \ne|\{2,3\}|$.

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You will probably learn later that for any infinite cardinal $a$ the equality $$a+a=a\cdot a=a$$ holds. This implies that for any infinite cardinal $a$ you have a counterexample $a+a=a+0$.

More generally, if $b$, $c$ are infinite cardinals then $$b+c=b\cdot c=\max\{b,c\}.$$

The proof of these general result is not that simple, it requires axiom of choice. See e.g. the following questions: About a paper of Zermelo and How to prove that from "Every infinite cardinal satisfies $a^2=a$" we can prove that $b+c=bc$ for any two infinite cardinals $b,c$?

However, showing the above result for some special cases, like $a=\aleph_0$ or $a=2^{\aleph_0}$ is not that difficult and it might be a useful exercise for someone learning basics of set theory and cardinal arithmetic.

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