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RE: Infinity and Beyond - Part 3
I don't understand this
Countable infinity has a nice extra for us though. It is the smallest infinity. So finding a surjection from the natural numbers to the target set is sufficient to show countability and even equal size to the set of the natural numbers. Why is it the smallest? Well, take any infinite subset of the natural numbers and you can always find a bijection onto the natural numbers, i. e. they are the same size, ergo there is no infinite set smaller than the natural numbers.
What do you mean by infinite subset? It seems like you are taking a subset with the same cardinality....
I see your point. I think I'll have to do some more research on this. It is rather late here though, so I'll get back to you tomorrow.
The easiest way to prove it is by using axiom of choice. It is a bit harder to prove it without using axiom of choice.
Well a place where the axiom of choice does not hold is pretty scary. Hope you don't have nightmares ;)
Ok, so I have come to the point where I don't see how you can prove it without assuming the axiom of choice (AC). I think I will add that to the text with a link to the axiom. And I couldn't find one.
If you can find or construct a proof that does not require AC I would very much appreciate it, as I would like to avoid mentioning AC in fear of it confusing people. The target audience of this post are non-mathematicians after all.
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.
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