Comment by bmitc

The non-sandwich analogy is called Hilbert’s hotel.

Saying that two sets have the same cardinality is equivalent to them having a bijection between them.

So the claim is that the natural numbers and the natural numbers plus a sandwich have the same cardinality. This can be proved by the bijection:

    0 -> sandwich
    1 -> 0
    2 -> 1
    3 -> 2
      .
      .
      .
    n -> n-1
      .
      .
      .

There is actually more though! If you had an infinite but countable amount of sandwiches (that is a sandwich for every natural number), that plus the natural numbers still has the same cardinality as just the natural numbers. There, the bijection is

    0 -> sandwich_0
    1 -> 0
    2 -> sandwich_1
    3 -> 1
    4 -> sandwich_2
    5 -> 2
      .
      .
      .

No natural number or sandwich is left out by this mapping.