Timeline for In $L$, does there exist a definable non-principal ultrafilter on $\mathbb{N}$

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Jan 23, 2014 at 16:54	comment	added	Andreas Blass		@MikeBattaglia If $V=L$, you can build a basis for the functions $\mathbb R\to\mathbb R$ by similar methods to those in the answers here, but you'd have to use $L$'s well-ordering not of the reals but of the subsets of the reals, so the complexity goes way up. I agree with you that most of the functions in such a Hamel basis would be unimaginable. Note, for example, that the basis would have cardinality $2^{2^{\aleph_0}}$, whereas the number of continuous or even Borel functions is only $2^{\aleph_0}$.
Jan 23, 2014 at 7:43	comment	added	Mike Battaglia		Regarding the Hamel basis over $\mathbb{Q}$ - very interesting! One thing I've been very curious about is what a basis for the vector space of functions from $\mathbb{R} \to \mathbb{R}$ might look like. Is this constructible in $L$ as well? I can't begin to imagine what these functions would look like.
Jan 21, 2014 at 17:00	history	answered	Andreas Blass	CC BY-SA 3.0