# Difference between consensus number n and consensus number infinite

In book Concurrent Programming: Algorithms, Principles, and Foundations of Michel Raynal, in Section 16.5.1, Theorem 75 says

Compare&swap objects have infinite consensus number.

and in Section 16.5.2, Lemma 40 says

The mem-to-mem-swap object type has consensus number n in a system of n processes.

Then, in Section 16.1, author puts in table both Compare&swap and men-to-mem-swap as objects with consensus number infinite.

So, what is the difference between objects with consensus number n and objects with consensus number infinite?

$$n$$-register assignment, where a process can atomically write to multiple registers, allows up to $$2n-2$$ processes to synchronize wait-free if there are $$n$$ registers. More processes require more registers. Therefore the consensus number of an objet consisting of $$n$$ registers that can be assigned atomically is $$2n-2$$.
Lemma 40 states that a mem-and-swap object allows the wait-free synchronization of all the processes in a system of $$n$$ concurrent processes, for any number $$n$$. This means that for any number $$n$$, the consensus number of mem-and-swap is at least $$n$$. (Stating that the consensus number is $$n$$ in a system of $$n$$ processes is a bit of a misnomer since the definition of the consensus number doesn't actually depend on the total number of processes.) Since consensus number of mem-and-swap is at least $$n$$ for any $$n$$, it's infinite.