Since ordinals are the ultimate recursion tool, any such function will ultimately grow faster than whatever recursive definition you can cook up by hand.
For instance even the slow growing hierarchy g_alpha, which grows very very slowly catches up to the fast hierarchy at extremely large ordinals.
Some examples:
- f_0 behaves like addition, f_1 like multiplication, f_2 like exponentiation, f_3 like tetration, f_w like Ackerman
- Graham number is bounded by f_{w+1}(64)
- Goodstein function behaves like f_{epsilon_0} (this show how much faster it grows than Ackerman). Similar for the Kirby-Paris Hydra
- n-Iterated goodstein behaves live f_{\phi(n-2,0)} where \phi is Veblen's function
- the tree function from Kruskal's tree theorem
behaves like f_{\psi(Omega^Omega^omega}, ie by the small Veblen ordinal. And the graph function from the Robertson-Seymour theorem behaves like \psi(Omega_omega).
By contrast the slow growing hierarchy grows extremely slowly, for instance g_{epsilon_0} only grows like tetration. But it still catches up to the fast growing hierarchy at very large ordinals.
Like I said, I can't really follow this stuff, but maybe the ultimate would be the Busy Beaver function? Is there such a thing as a proof of how many functions can be between something computable and Busy Beaver, given some sort of constraints?
Yes, the Busy Beaver function grows faster than the functions I gave, basically it behaves like f_{w^1_CK} (although this does not really make sense because w^1_CK, the Church Kleen ordinal, is not recursive).
By "ultimate generalisation" I was speaking about a family of computable functions that ultimately grow faster than other functions you could cook up by hand: you put all the recursivity in the ordinals, they are there for that, and logicians are very good at constructing very large (recursive) ordinals.
If you just want a very very fast growing function (but that is still computable, unlike busy beaver), you can do the a light busy beaver version: Beaver(N)=the maximal output of all turing machines T with N state where there is a proof in ZFC of length <=N that T terminates.
This function grows much faster than the other functions I described (basically it is a f_{ordinal consistency of ZFC}.
If you are interested in this kind of ideas, there is this reddit post that develops these ideas:
https://www.reddit.com/r/math/comments/283298/how_to_compute...
Yes, Emmy Noether was a pure genius (one of the greatest mathematician of all time), and was never named full professor (even though she had Hilbert support).
It would be nice to know the priority of blocking elements between firefox default anti-tracker list, ublock origin, and privacy badger.
When I see a tracker hit ublock origin does it mean that it bypassed firefox anti-tracker blocking, or is it the reverse?
Having three anti-trackers installed is also a bit inconvenient when this breaks a site, I have to disable each one successively to try to make it work again...
The Fibonacci example is interesting. If you want arbitrary precision, then the naive recursive method is exponential in $n$ to compute $F_n$. The loop method (recursive method with cache) is in $O(n^2)$. Indeed you only have $n$ steps in the loop, but at step $k$ you add $F_k$ and $F_{k+1}$ which have $O(k)$ bits so the total costs is $O(n^2)$. Finally the matrix/formal \sqrt{5} algorithm costs $O(n log n)$.
Indeed, using fast exponentiations, you only need $O(log n) matrix multiplications. But the dominant cost is actually at the last step, where you multiply two numbers of size $O(n)$ bits for a total cost of $O(n log n)$.
Speaking of mathematics in programming: I find it remarkable that the recent convergence of async programming into the use of promises and async/await (in python, js, rust...) is a weak form of the continuation monad. And the continuation monad is a form of the 'not not' Lawrere topology!
Only speaking about arithmetic rather than full set theory: in (full) second order arithmetic theory all models of the natural number N are isomorphic (in a unique way). The big drawback is that this theory is completely undecidable, so we usually interpret it in first order logic set theory where it becomes incomplete. By Godel, you cannot have an arithmetic theory with decidable proofs and completeness.
In set theory one has to be careful that you can have models which are not transitive models. So Löwenheim-Skolem implies that if you have a model, you have a countable model. But you may not have a transitive countable model. However you do have a transitive version of Löwenheim-Skolem, by Mostowski collapsing lemma: "there exists a transitive model" <=> there exists a countable transitive model".
More precisely the effective topos is a kind of realizability topos with a nno where every function is computable. Such a (non trivial) topos cannot be classical, otherwise the halting problem would be decidable. However the effective topos can only deal with calculability, not complexity. For complexity linear logic (without exponentials) and symmetric monoidal categories are better suited (but I don't know if there exists a model where every function is in P for instance).
Oh yes, it is very much alive. For instance any topos (with a nno) is a model of intuitionist set theory. And since topos are a very natural type of categories (a topos= acategory which has finite limits and a power object), intuitionist logic is important in category theory.
Two examples: in algebraic geometry, mathematicians use schemes (introduced by Grothendieck), which have allowed to make tremendous progress. But from the point of view of intuitionist logic, a scheme is just a ring, and a quasi-coherent sheave is just a module over this ring (in more details they are ring objects and module objects in the big Zariski topos).
In synthetic differential geometry, one use a certain intuitionist topos to formalize intuitive aspects of reasoning with infinitesimal quantities (without relying on delta and epsilons). Note that non standard analysis also allows to work with infinitemismal quantities (by using ultraproducts), but has the same power has standard analysis (and in particular is classical). Synthetic differential geometry is intuitionnist, so is less powerful, but on the other hands its theorems are automatically valid for more models (complex models, formal smooth schemes...). In SDG, your reals R are a field with an (actually several) element epsilon !=0 (the infinitesimals) and such that epsilon^2=0 (that's why this only work in intuitionist logic, in classical logic R being a field would imply that epsilon=0). In this model every function f: R->R is differentiable, and we have f(x+epsilon)=f(x)+f'(x) epsilon.
In proof theory (like coq) one work with Martin Lof type theory, which is the type theory of topoi (and so is intuitionist). A recent development is to work with homotopy type theory instead, the type theory of \infty-topos. This is more powerful (and can be used as fundations for all of maths), and solve real problems, like correctly defining equality (equality is actually a tricky concept, for instance it may not be the case in the prover's theory that "for any two functions f and g such that forall x, f(x)=g(x); then f=g", but this is obviously a feature we want.) On the other hand it is quite a bit trickier to define (we need (infty,1)-categories).
On the other hand, constructive set theory is still rather a niche part of mathematics (this may change with the recent development of HOTT). It is hard not to use the LEM (and I am not good at it, I am not an expert on this subject. Expert say it gets easier over time to get used to not use LEM).
For instance, in the intuitionist reals, from "x >= 0 and x != 0" you cannot conclude that "x != 0". To prove x>0 you would need to exhibit a natural number n such that x>=1/n (so in other words you need a constructive proof that x>0).
But this is not always possible, you have models of R where there is an x != 0 which is smaller than all the 1/n.
There is however one nice feature of intuitionist theory: every geometric sentence which is provable in classical theory is actually true in intuitionist theory. So if you have a geometric sentence (https://ncatlab.org/nlab/show/geometric+theory) you can use LEM in your proofs, the proof will still be valid intuitionally!
No it is not zero CO2 output in 2050 but about 65% reduction only, or 80% if a 25% reduction of electricity consumption is achieved. This is 1) too late, 2) not even on par with France's electricity C02 production.
And a nuclear accident would not involve closing the whole country, it would involve an exclusion zone of 10-20 km (Fukushima exclusion zone is 20km, but it include zones where the residue radioactivity is <50 mSievert, way less than some natural radioactive places), a whole lot of economic damage (but negligible compared to climate change), and between 0 and (at worst) a few thousand deaths.
When you put it this way, you might as well say that a major accident that takes out half of Western Europe would be neglige in comparison.
German plants are old. There's issues with a few of them. Decommissioning them seems inevitable. The question is whether building new nuclear plants, at probably about 15+ years for the first plant to start operations, is really a better option than investing in renewables?
Again, a major accident would not take out half of Western Europe, this is fear mongering. A major accident would be severe, with a large exclusion zone and a lot of economic damage, but sill kill less than coal kill in Germany each year. And Germany was still constructing new coal power station in 2018!
I am not saying that Germany should invest in nuclear instead of renewable, but in nuclear along with renewable. And yes 15 years is a long time, but still a lot less long than 30 years (and Germany lost 10 years by being stubborn about nuclear).
And other countries that want to do a energy transition should not make the same mistakes as Germany. And political parties in France that want to close nuclear plants are environmentally severely misguided.
Yes long term we should get rid of it (for fusion), but in short term the priority is climate change. And nuclear fision is a tremendous asset for climate change. There are still huge challenge of course, and we are unfortunately not at all on course to tackle them, but depriving ourselves of nuclear is only going to make things worse.
Since ordinals are the ultimate recursion tool, any such function will ultimately grow faster than whatever recursive definition you can cook up by hand.
For instance even the slow growing hierarchy g_alpha, which grows very very slowly catches up to the fast hierarchy at extremely large ordinals.
Some examples:
- f_0 behaves like addition, f_1 like multiplication, f_2 like exponentiation, f_3 like tetration, f_w like Ackerman
- Graham number is bounded by f_{w+1}(64)
- Goodstein function behaves like f_{epsilon_0} (this show how much faster it grows than Ackerman). Similar for the Kirby-Paris Hydra
- n-Iterated goodstein behaves live f_{\phi(n-2,0)} where \phi is Veblen's function
- the tree function from Kruskal's tree theorem behaves like f_{\psi(Omega^Omega^omega}, ie by the small Veblen ordinal. And the graph function from the Robertson-Seymour theorem behaves like \psi(Omega_omega).
By contrast the slow growing hierarchy grows extremely slowly, for instance g_{epsilon_0} only grows like tetration. But it still catches up to the fast growing hierarchy at very large ordinals.