4

I think the answer is no. I tried to find finite subsets which has a total order under subset relation such as the following:

$\{\{a\},\{a,b\},\{a,b,c\},\{a,b,c,d\}\}$

Then I noticed that two sets with the same size cannot be present on this subset because they cannot be subsets of each other. Since there can be at most $1$ set with size $1$ and size $2$ and size $n$, I think such a subset would be at most countable. But I am not sure if my reasoning is correct. Please help me correct it if it is wrong. And if it is true, can you help me give a more formal proof?

Nika
  • 727
user666150
  • 612
  • 1
    Your reasoning fails for infinite sets, as they all have equal cardinalities, but there can be strict inclusions between them. Indeed, such uncountable subsets exist: https://math.stackexchange.com/q/1182145/127263 – Wojowu Nov 27 '19 at 20:39
  • @MatthewDaly But a chain is totally ordered by inclusion? – Mark Kamsma Nov 27 '19 at 21:11

0 Answers0