/sci/ - Science & Math

>>12518356
I'm not sure what the difference between explicit and arbitrary would be here, but no that's not the short version of it.
I try to be shorter the second time around: You take the derivative of "L(x,v,t)" with respect to v at the value q(t). Whether v=q'(t) is determined by x=q(t) does not play a role in the derivative here, since you take the derivative of L in which x and v are independent parameters (independent dimensions). So the derivative ∂L(x,v,t)/∂v, is blind to whatever is going on in x (even if for whatever reason you'd eventually evaluate x at v). The fact that this expression is finally evaluated x=q and v=q', which are evaluated at some values t is not relevant, and thus the functional form of q doesn't matter.

To understand which derivatives in the Euler-Lagrange equations are taken, you might want to follow d'Alamberts principle (at the risk of getting confused by yet another notion of derivative or variation, [math]\delta [/math]), or you look at spatial calculus of variation examples and have faith that something that makes sense for lengths also makes sense for a kinetic-minus-potential energy quantity.

>>12018097
Doesn't sound ultra to me tbqh. I think consistency is a vastly overrated feature. If I let out the formalist in me, an inconsistent theory is sure as good as a consistent one. I one and can provide a sequent ending in the string of letters "[math] \neg(A\land \neg A) [/math]" and in one I can't. Shouldn't bother an ultra-finitist. Literally the only difference between an inconsistent theory and a consistent one, apart from what they can prove, is that the majority of people don't find inconsistent theories interesting. That's but a meta-meta-mathematical property.

>>11727635
Why do you write "No" here?

I think you didn't understand what I wrote there.
Your add function will be 2-ary, each of it's argument being a lambda term. What the add function returns is another lambda term (another function which takes arguments)

>summand1->summand2->x->y->z
Yes, your add will take summand1, summand2 and return a function of type x->y->z.
The return value (the number "summand1 plus summand2") is another python function, call it A. As such, the add function doesn't need 4 arguments. The add function doesn't need the argument slots that A would have.

All inputs and return values are always functions.

>>11614768
>how inherent is the connection between synthetic geometry and euclidean geometry? do trig functions hold up well in spaces other than euclidean?
The first and second part of the question somewhat don't go together / ask about very different things.
To answer the first question as it stands, there was of course lots of work on synthetically axiomatizing geometry, see e.g.
https://en.wikipedia.org/wiki/Hilbert%27s_axioms
or
https://en.wikipedia.org/wiki/Tarski%27s_axioms

Tarski generally has a lot of funky stuff if you look into it, e.g.
https://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals

>>11616474
To cite the Wikipedia page of AoC
>In some model, there is a set that can be partitioned into strictly more equivalence classes than the original set has elements, and a function whose domain is strictly smaller than its range. In fact, this is the case in all known models.

>Any ideas on how to prove this
I'm not aware of the proofs of this (although probably you find one in some of the AoC books, there's some popular ones).

>This sounds absurd
I doubt it's problematic, it's probably an artifact of Cantors cardinality definition.
For two generic sets X and R, you have that |R|<|X| if you can inject R into X but there's no bijection. Now if the objects are weird enough so that you provably can't find a bijection without non-constructive means, then you're already there. I could imagine the proof of the relation goes like that.

Note that there's other size-comparison relations, e.g.
https://en.wikipedia.org/wiki/Subcountability

Also, note that ZF (with or without ZFC) can't even prove that |Y|>|X| implies |P(Y)|<|P(X)| or give any hint on how |R| compares to the cardinalities of the ordinals (apart from being bigger than |\omega_1|)