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I posted a question (which is now deleted) about whether the negation of the statement, $X$ is countable, is that $X$ is uncountable, i.e., there is an injection from $\mathbb{N}$ to $X$ but no bijection, needs the axiom of choice. However, it was closed for being a duplicate to the question by Sosha.

But the problem is that I fail to find a direct connection between the questions. But he closed without any comment, and I have no chance to get a further answer for it. So my question is about the rationale for how the questions are closely connected. I tried to take the contrapositive of the claim 'Every infinite set has a countably infinite subset':

If every subset of $X$ is not countably infinite, $X$ is not infinite.

Is this correct? The answers in Sosha's question state that AC is needed. OK. We know that not being infinite means $X$ is finite, i.e., there is an initial segment $I$ of $\mathbb{N}$ such that there is a bijection from $I$ onto $X$. Then, how could it answer my question? Please point out what I am missing here.

Hermis14
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1 Answers1

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There are models where there exist infinite sets that are Dedekind-finite. Therefore your simplified cardinality trichotomy proposition (by replacing one set with the natural numbers) cannot be proved in ZF.

Alberto Takase
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  • Thank you very much for your answer, but I feel something's wrong in this place. Some users want questioners with no moderate experiences in the fields to enlighten themselves by closing questions by a single vote. Do you think that the original question in the link is really a duplicate? I love to learn math and enjoy asking, and answering questions. But excessive intervening makes it frustrating. – Hermis14 Feb 06 '22 at 16:57
  • I think the original (now deleted) question started off simple, but then you tried to redefine "uncountable" to something non-standard and reword your question to something that seemed weird (to me at first glance). This may have incentivized people to check for duplicates so that future readers get redirected away from your question. (just my opinion without any evidence) – Alberto Takase Feb 07 '22 at 06:30
  • I didn't know that it's not a standard definition of being infinite but your comment's convincing. Thank you :) – Hermis14 Feb 07 '22 at 06:52