/sci/ - Science & Math

>>12230883
I'd weaken that a bit by saying "formalizable system"

>>12230892
That's generally a fair case you make there, but one must also consider that pre-1930 math was a world where mathematicans didn't conceive that fully formal arithmetic would leave open some universally quantified statements. They didn't think mathematical questions, to be answered in a indistputible way, need a hard syntatic foundation.

And the issue is "solved" by Cohen relative to some framework and that's in a way the only agreeable answer you can get.
Otherwise, I can "solve" it right here:
There is a sequence of injections between infinite sets
[math] {\mathbb N} \to \{0,1\}^{\mathbb N} \to {\mathcal P}{\mathbb N} \to {\mathbb R} [/math]

But for a hand-on notion of "function", there is no injection
[math] {\mathbb R} \to \{0,1\}^{\mathbb N} [/math]

So [math] \{0,1\}^{\mathbb N} [/math] violates the continuum hypothesis and we're done.

The
>hand-on notion of "function"
is of course relative to your system of choice, just like the question people discussed above, asking whether function-extensionality holds.

The question
>Are two functions equal if all their return values are equal
can't be settled for all systems at once, and many of them can by consistent but have different answers.
CH is the same kind of question as the "function" extensionality question.
There's many good notions of functions, just like there are many notions of the reals.
If you fix a system, you can get an answer. For ZFC set theory, CH is answered in that sense.

Gonna start reading this today. If you're bored on this Saturday, read along

https://www.esat.kuleuven.be/cosic/publications/thesis-2.pdf

You're ansking a why question and it will just lead to other why questions.

The einstein equations are
G=k·T
where G is the Einstien tensor, T is the (model specific) energy/pressure density tensor and k is some constant. G is basically the second derivative of the metric tensor g, what whatever.

If T is such and such, then G is such and such. The question "How does it affect" spacetime is moot, given that they are just related by an equality sign.

Consider
a=(1/m)·F
where a is acceleration, m is mall and F is force. Now "How does force affect acceleration"? Just like so.

You could take a reductionist perspective and say: Well the deeper theory is quantum mechanics, and F=ma is just an emergent law that's just some mean values of some wave functions, and that's the how.
(A similar line of thought will work for gravity too, except we don't have a good theory of quantum gravitiy yet.)

But is that an explanation of how? You'll just look at the defining equations and quantum mechanics, say
d(psi)/dt = k·H psi
(Schrödinger equation)
and ask "how does the fundamental model/the Hamiltonian operator H change the wave function psi in time?". Well, like so.

>>8018691
Yeah I know. Actually I made that Idris thread a few day ago. Too bad nobody wants to ride that train with me.

Even Microsoft has one or two such languages, a few places on the right of C in the alphabet:

https://en.wikipedia.org/wiki/F*_%28programming_language%29