I don't think the proof without axiom of choice is suitable for non-mathematicians. I will put it in a post since I cannot find the proof in any literature. You can try to simplify it if you think it is worth your time.
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Thank you, I'm looking forward to your post! 😃
Here is the post https://steemit.com/mathematics/@mathowl/countable-infinity-is-the-smallest-cardinal-infinity-even-without-assuming-the-axiom-of-choice-warning-technicalities-await