In this post I will answer a question of @targodan :

*Can you prove that countable infinity is the smallest cardinal infinity even without assuming the Axiom of Choice?*

This question is a response to the  [Infinity and beyond part 3 post](https://steemit.com/steemstem/@targodan/infinity-and-beyond-part-3#@targodan/re-mathowl-re-targodan-re-mathowl-re-targodan-infinity-and-beyond-part-3-20180221t082957620z). In that post I claimed that the answer is yes. Unfortunately, I was unable to locate any literature which supported my claim so I had to write this post.

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![alef0.png](https://steemitimages.com/DQmVaYizkkAGSMAoatTXLUJfsD9xga8RtJZPWhZTZ6eULjP/alef0.png)
**WARNING**
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Before I begin I will have to warn you that this post is aimed at Steemians who completed a set-theory course. A basic understanding of the Axiom of Choice and cardinalities is crucial to appreciate this post.

## Preliminaries

Denote by ![ql_83191e4ad2936d232bf2f306e0f134a7_l3.png](https://steemitimages.com/DQmTe9HgZQKgt1Sx4SfYYWpZ4EJvxc9zTudW5fZqEg2CkgL/ql_83191e4ad2936d232bf2f306e0f134a7_l3.png)  the integers. The cardinality of a set A will be denoted by |A|. We then define the cardinality ![n00.png](https://steemitimages.com/DQmV9jkmvkiL3knhgYuihXNaoHCyj7idC1MH61dMHTcTV8g/n00.png).  Throughout this post we will assume Zermelo-Fraenkel set theory without the Axiom of Choice. 

## Main result and proof



We will show the following:
 
**Theorem**      ![aleph0.png](https://steemitimages.com/DQmV3exTQN4VWRu3edWvkHSedWA2Jf6EViZrYPnsmtTs9bJ/aleph0.png)    *is the minimal infinite cardinal.* 


*Proof*.  Let A be a set with minimal infinite cardinality. This implies that |A| ≤ ![aleph0.png](https://steemitimages.com/DQmV3exTQN4VWRu3edWvkHSedWA2Jf6EViZrYPnsmtTs9bJ/aleph0.png) . The aim of the rest of this proof will be to show that   |A| ≥ ![aleph0.png](https://steemitimages.com/DQmV3exTQN4VWRu3edWvkHSedWA2Jf6EViZrYPnsmtTs9bJ/aleph0.png)   since if we can show this then we have that    ![aleph0.png](https://steemitimages.com/DQmV3exTQN4VWRu3edWvkHSedWA2Jf6EViZrYPnsmtTs9bJ/aleph0.png) = |A|  which proves the theorem.  

Take ![ainA.png](https://steemitimages.com/DQmTiAJfqeZEE6ZhoxXNer8qFY6sH8XE1mxFatm4HEBp99z/ainA.png) and define ![hatA.png](https://steemitimages.com/DQmPuvvJJh6JSDV7HRTrkZSe9c5u4V2f8vb8WdwhMVX9eYq/hatA.png). Then it straightforwardly follows that ![AAAAAA.png](https://steemitimages.com/DQmcMS5HZHqKZZSm2iLkUCU1PkGYhyqwEi9NMeQo4LrLntV/AAAAAA.png). This implies that there exists a bijection ![ff.png](https://steemitimages.com/DQmVftakgTkXgpoYetPTNgZgPAkpUZssUTktWP9BWW7ZgKF/ff.png).   Observe that by the definition of ![hatA.png](https://steemitimages.com/DQmc8CmCoodbHfdLL9exAqSZzGZGhWij44QRhZ8W5AW3kts/hatA.png) we have that ![ffff.png](https://steemitimages.com/DQmPY7WsXvny3boF7gM2vAJgjgdxaGV1Y4e8L9Qc5WTkjqe/ffff.png) if and only if n=0. Then using that f is injective we also get that ![fffm.png](https://steemitimages.com/DQmdwMsiByXi2Y6c9uankwiaMLbFB6WMh1ic4sjSHHuD53m/fffm.png) if and only if n=m. We then define the function  ![Gn.png](https://steemitimages.com/DQmXiRNvKsVGi6Y3s3q6F9DCprx6fKXmQBHfp7KPRhGCKnJ/Gn.png) which is injective with respect to its range which follows from the previous sentence. We define the set ![BB.png](https://steemitimages.com/DQmV6xMn3RTXuZ928WS2jjWGiMMdzJeHC4yQL3kBzsbmKAV/BB.png). Observe that |B| = ![aleph0.png](https://steemitimages.com/DQmV3exTQN4VWRu3edWvkHSedWA2Jf6EViZrYPnsmtTs9bJ/aleph0.png)  and that B is a subset of A. Now it follows that |A| ≥ ![aleph0.png](https://steemitimages.com/DQmV3exTQN4VWRu3edWvkHSedWA2Jf6EViZrYPnsmtTs9bJ/aleph0.png). This completes the proof.  ![ql_f06e81357e1af52100514b44d8845ca3_l3.png](https://steemitimages.com/DQmTpSGwBfQkR1DMHPxZeLhr6vDcK9VwxihFj94Kx2pzL16/ql_f06e81357e1af52100514b44d8845ca3_l3.png)


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## Source
Top photo made with [inkscape](https://inkscape.org/en/). All equations written using [quicklatex](https://inkscape.org/en/). They are both free-ware!  

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## Thank you!

Thanks for being so kind to read my post. You are awesome! Please follow me if you enjoyed it. If you have any questions just post them below and I will answer them. Or if you might have a nice topic you want me to cover also let me know below. :o)

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## Owl tax
![pexels-photo-106686.jpeg](https://steemitimages.com/DQmayFU36G3H6WjZGEoXkq9SQ9zquHJR6GyReMz3xqoskAX/pexels-photo-106686.jpeg)
[Photo by Mark Broadhurst](https://static.pexels.com/photos/106686/pexels-photo-106686.jpeg)

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