3:41 AM
10 hours ago, by
user21820 There are other uses of well-orderings in 'core' mathematics. If you do want every vector space to have a basis, then you do need a well-ordering of the vectors in that space. They seem to have no real-world implication, but hey it's still mathematics.
I guess... that will count as mathematically useful, given how so many functional analysis proofs are much nicer with a Hamel basis, and all of these are uncountable thus at least $\omega_1$...?
We'll I guess... Hamel functions, Vitali sets, $\omega_1$ and other uncountable well orderings are relatively tame compared to Bernstein sets, Large cardinals...
and banach tarski ( so far have seen no application of that in any proofs outside set theory)
Artem, we frequently deal with non-measurable sets in probability theory. For example, when we have a stochastic process $\xi_t$, we are concerned with the filtration (increasing sequence of $\sigma$-algebras)
$\mathcal F_t$ generated by the process at times $s \le t$. Consequently, an event such as
$\{ \xi_2 \ge 0 \}$ is
not measurable with respect to the $\sigma$-algebra $\mathcal F_1$. My point with this comment is not to answer Matt's question; rather, it's to say that you can't easily dismiss it by saying that we only work with measurable sets in probability. —
Tom LaGatta Aug 18 '10 at 18:33
that has not said whether Bernstein sets were involved (since those are absolutely non measurable, meaning that for any measure $\mu$, it is nonmeasurable)
63
I saw this video about the busy beaver function and looked at the applications section of the Wikipedia article about the busy beaver function.
I concluded that there is zero practical or even theoretical value in searching these numbers S(n) : n > 5. Now in the video the person says, that peopl...
It seemed to me so many impredicative objects not only have no real life counterpart, but are also mathematically unimportant
And that link is not a concrete application, because all it has done is adding footnotes (notice all those theorems presented all have the word "Bernstein set" in them, meaning that these theorem are only about Bernstein sets, thus are useless when there is no Bernstein sets)
so I don't get how that is a "concrete application" when all it has done is adding footnotes
and in this one, while it seemed nice that we can put algebraic structures on top of Bernstein sets, the application section is all about partitioning $\Bbb{R}^n$ with these sets, but why are we interested in these partitions, it tells us nothing other than Bernstein sets can be a partition of $\Bbb{R}^n$?
My point being (I don't know if I have searched deeply enough but yesterday I did crawl through all google results, MSE and MO using the search term "Bernstein set" and "Bernstein set" applications)
In theory, if we nuke every single paper that talks about Bernstein set, mathematics will not suffer a crippling loss and came crashing down
(and we cannot really ignore this implication, because Bernstein sets are the norm of nonmeasurable sets, thus it basically means that almost all nonmeasurable sets (one of the most significant byproduct of the axiom of choice) are mathematically unimportant)
Meanwhile in the ZF end, infinite Dedekind finite sets and amorphous sets are also not very optimistic either in their mathematical importance:
Here it seems to suggest $\Delta$ Dedekind cardinals can be used to talk about prime ideals, but it seemed to be focusing more on the algebraic structure on $\Delta$ itself than relations to other algebras, thus still kind of stuck inside set theory by being a footnote
