50 1. Logic and foundations translates E1() + x1,...,Ek() + xk fail to cover [0,2) we leave this as an exercise to the reader. Remark 1.12.8. The key distinction here between the low and high di- mensional cases is that the free group a,b is not amenable, whereas Zk is amenable. See [Ta2010, §2.2, §2.8] for further discussion. 1.12.5. Summary. The above discussion suggests that it is possible to retain much of the essential mathematical content of set theory without the need for explicitly dealing with large sets (such as uncountable sets), but there is a significant price to pay in doing so, namely that one has to deal with sets on a “virtual” or “incomplete” basis, rather than with the “completed infinities” that one is accustomed to in the standard modern framework of mathematics. Conceptually, this marks quite a different ap- proach to mathematical objects, and assertions about such objects such assertions are not simply true or false, but instead require a certain compu- tational cost to be paid before their truth can be ascertained. This approach makes the mathematical reasoning process look rather strange compared to how it is usually presented, but I believe it is still a worthwhile exercise to try to translate mathematical arguments into this computational frame- work, as it illustrates how some parts of mathematics are in some sense “more infinitary” than others, in that they require a more infinite amount of computational power in order to model in this fashion. It also illustrates why we adopt the conveniences of infinite set theory in the first place while it is technically possible to do mathematics without infinite sets, it can be significantly more tedious and painful to do so.

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