/sci/ - Science & Math

>>10595289
Because the axioms being used (ZFC) are wrong (or at least incomplete). Those set theorists who are actually not retarded are working on improving the axiom system. You can find one example if you google "Woodin Ultimate L".

>>10595300
Gödel's incompleteness theorems proves every axiom system "wrong".

>>10595300
Woodin is a homosexual.

>>10595309
That's a kneejerk 'gotcha' reaction. Nowhere do the incompleteness theorems rule out answering all or very many "natural" questions such as CH, or AC for that matter.

>>10595289
I'm just gonna dump these here real quick

>>10595328
Sorry if I'm disrupting your conversation

>>10595332
But I just really like these images

>>10595333
Only have these four though, good night

>>10595328
>>10595332
>>10595333
>>10595334
Have this and see if you can pinpoint precisely how many levels of troll it's operating on.

>>10595395
This image is not at all funny because it's full of inconsistencies. Basically, only retards could possibly be entertained by it.

>>10595407
>Basically, only /sci/-posters could possibly be entertained by it.
ftfy

>>10595289
>>10595300
>>10595309
Can you prove that the only way to improve the axioms isn't to just take CH as an axiom or not? In other words, can we even understand math to the point where we can have some axiom implying CH that isn't obviously CH?

>>10595395
Killing -1/12 people is equivalent to bringing 1/12 person in to life. In other words, an abortion. And abortion is wrong.

>>10595466
Yes, there are other more-or-less "natural" statements which imply CH or its negation. Some of them are strictly stronger. As for understanding, yes, it is a problem we may never reach a consensus on. Perhaps one day with a new way of looking at it we'll reach a weak consensus on CH, similar to the case now with the Axiom of Choice.
>>10595319
What gave it away?

>>10595395
>reimann
>not riemann

>>10595469
>bringing someone into life is equivalent abortion
What

>>10595432
You didn't change anything, what gives?

>>10595289
Pull the lever iff the trolley was heading toward [math]\aleph_0[/math].

The speed the trolley moves at is irrelevant. An amount of people larger than countable infinity will necessarily result in overcrowding, and even though their conditions are equivalent (strapped to trolley tracks and forced to die) will result in more (infinite, in fact) suffering. Lesser infinite suffering is always the moral choice.

This is not even a difficult question.

>>10595407
That's the joke retard. Glad you liked it

>>10595309
No. Only ones that can produce peano arithmetic. There are weaker arithmetics which are perfectly usable that are complete.

>>10595847
I started typing and realized Anon was actually right

>>10595328
Let the jury die, since that will instantly create a non-replenishing function, whereas the justice system is necessarily regenerative over the domain of "your defense attorney."

TL;DR: Killing the one defending you is infinite lives lost in an unjust cosmos.

>>10595945
If that qualifies as a joke, then so would any random string of characters. Just accept that your meme is retarded.

>>10595289
multy track drifting

>>10595902
do nothing because I’m a lucky little shit

>>10595951
>there are weaker arithmetics than the [math]I\Sigma_1[/math] fragment of PA that are perfectly usable

>>10596066
>so would a random string of characters
>he’s never seen a stroke post
Most stroke posts are literally just incomprehensible strings of letters and they’re pretty funny.

>>10595407
>>10596066
You're actually retarded if you can't see that the joke is that [math]\zeta{\left(0\right)} < \zeta{\left(-1\right)}[/math].

>>10595824
>1/12 of a person
>survives 1/12 of a pregnancy

>>10595496
Yeah but would you want CH true or not? As I understand it, there are some guys who would want it true, and some others who wouldn't, depending on their field of research. For example, proper forcing axiom implies [math]2^{\aleph_0}=\aleph_1[/math]

>>10598194
sorry I meant [math]2^{\aleph_0}=\aleph_2[/math]

>>10598194
>>10598195
Ideally we'd find some new and very convincing way of looking at it. PFA is only convincing if you're a total faggot so far up Cohen's dick that you think set theory exists to facilitate forcing arguments rather than the other way round. Aleph_2 would be pretty funny though.

"Ultimate L" is a bit better but still very hard to for a non-set-theorist to understand without basically becoming a set theorist from the effort

>>10595289
Try it.

>>10598226
I'm pretty sure that there is proof that all methods of determining whether or not you would want the continuum hypothesis to be true are indistinguishable from just taking it as an axiom. Just like there is proof that there is only one proof that the general set is not equinumerous to its power set, up to redundant statements.

Are the ultimate L axioms more powerful than the ZFC ones?

>>10598226
>"Ultimate L" is a bit better but still very hard to for a non-set-theorist to understand without basically becoming a set theorist from the effort
It doesn't matter because all of this autism over "should we adopt the V = Ultimate L or forcing axioms" is irrelevant to anyone who's not a set theorist anyway.
Why does there have to be a single "canonical" axiom system anyway? This boomer set theorist idea is totally dated in the age of topos theory and pluralist mathematics.

>>10600741
>all methods of determining whether or not you would want the continuum hypothesis to be true are indistinguishable from just taking it as an axiom
There probably is a result like that if you attach some extremely technical adjective to the front of every noun. But no such result can rule out a new and convincing way of thinking of something. Example: not everyone was easily convinced that the lambda calculus modeled all algorithmic computation, because obviously it's weird looking; but when Turing's machine model turned out to be equivalent, it won over 99% of people. This is creative and social process isn't something you can easily formulate a theorem about.
>>10601053
>why does there have to be a single "canonical" axioms system?
>This boomer set theorist idea is totally dated in the age of topos theory and pluralist mathematics.
This point of view is called "formalism" and it is older than boomers, and a lot of boomer-era set theorists are formalists too. But like finitism, it's a tricky position to defend: you have to believe, more or less, that [math]\mathbb{R}[/math] exists but its power set [math]\mathcal{P}(\mathbb{R})[/math] does not. Or that [math]\mathbb{N}[/math] exists, but not its power set. Or that they "exist", but are inscrutable in some strong way: hard even to get your head around.
In any case I think even a non-formalist will still like alternative axiom systems most of the time. Not topoi though, those are top fucking gay

>>10595466
Yes. To take a similar example: Zorn's Lemma, the Axiom of Choice, the Hausdorff Maximal Principle, the Well Ordering theorem, the Law of Trichotomy, and Tukey's lemma are all equivalent over the standard axioms of ZF set theory. I cant remember the exact details off the top of my head, but some of these axioms are actually stictly stronger than the others in systems of set theory that dont satisfy all the ZF axioms, even though they are equivalent when combined with the entire set of ZF axioms.Moreover, the Law of Trichotomy is actually a theorem (not an axiom) if we're only considering countable ordinals. This means that any countable model of set theory satisfying transitivity will entail the Law of Trichotomy as a theorem (although this will not be true for uncountable models of set theory, since these contain "inaccessible ordinals" that arent not within the purview, so to speak, of finitary inductive proofs).

>>10595395

>>10601087
what the fuck do you think formalism is