/sci/ - Science & Math

>>12057303
RH is equivalent to e.g.

[math] \sigma(n) \le H_n + {\mathrm e}^{H_n} \log(H_n) [/math]

where [math] H_n = \sum_{k=1}^n \frac{1}{k} [/math] and sigma is the sum of divisors function.

And if exp and log bother you, you can further break it down to characteristic functions relating to prime factors.
Here's a long list
https://mathoverflow.net/questions/39944/collection-of-equivalent-forms-of-riemann-hypothesis

https://www.maths.ed.ac.uk/~cbarwick/papers/future.pdf

>>11693613
>the
is there only one?

>>11693607
>>Literature (pretty good)
what does that mean

>>11702957
It'll be fine. Learn combinatorics.

>>11703831
[math] 84 = \sum_{n\in\{1,2,3\}} 2^{2n}[/math] ?

>>11527275
In ZF, it's just denotes the cardinality of a limit ordinal (and then also it's finite successors).
Sometimes its put in correspondence with the collection of all items of a given size (not just represented by the ordinal).

>>11527040
What makes sense is
[math] \aleph_0 = |\omega_0| < 2^{| \omega_0 |} = | {\mathbb R} | [/math]
where || is cardinality and [math] \omega_0 [/math] the first infinite counting numbers.

Also, it's consistent to fix [math] | {\mathbb R} | [/math] to the cardinality of a lot of the ordinals above
[math] \aleph_0 = |\omega_0| < 2^{| \omega_0 |} [/math]
and then e.g.
[math] \omega_0 \in \omega_i [/math]

>>11527275
In ZF, it's just denotes the cardinality of a limit ordinal (and then also it's finite sucessors)
Sometimes its put in correspondence with the collection of all items of a given size (not just represented by the ordinal).

>>11527040
What makes sense is
[math] \aleph_0 = |\omega_0| < |2^{| \omega_0 |}| = | {\mathbb R} | [/math]
where || is cardinality and [math] \omega_0 [/math] the first infinite counting numbers.

Also, it's consistent to fix [math] | {\mathbb R} | [/math] to the cardinality of a lot of the ordinals above [math] \aleph_0 = |\omega_0| < |2^{| \omega_0 | [/math] and then e.g.
[math] \omega_0 \in \omega_i [/math]

>>11469413
Gödel's incompletness theorem says that
Theorem (Gödel, PA):
>there exists no (I) proof (II) that shows all (III) proofs are not inconsistencies.
The proofs are encoded as numbers, so "all" in "all proofs" (III) is a PA universal quantifier (\forall, in first order logic (FOL) although funny enough, sidenote, Peanos first paper was SOL). And the proof (II) is one of those proofs/numbers.
This is arithmetization and one approach to answering Hilbert.

But note that this approach naturally is arguably more than what Hilbert asked for (in any case, this is the line of argument about what's to be proven here by the author.). Hilbert wants consistency proofs of his math from modest assumption. Gödel here approaches is by showing PA can't show self-consistency (there's no II proof) and, thus (e.g. if you have a set theory like ZF that can model PA) PA surely can't ever punch upwards and say something about ZF that we'd believe.

Since PA has non-standard models (e.g. extensions beyond all 1,2,3,...), you can think of a PA quantifier as one that can live in all those universes, it doesn't speak about a categorical world of numbers but is forced to speak about them all, and this is where using it to speak about all proofs makes problems - the forall of this theory is too weak to prove that is-not-inconsistency statement about its numbers.

Now what the author shows is
Theorem (Artemov, provability logic about PA):
>for all proofs, you can find a proof that the proof is not one of inconsistency.
Not that this formulation doesn't start with (I), i.e. it's not formally a non-existence statement of an arithmetized proof of a (self-reflecting) statement. In fact iirc the forall in the authors version isn't the same as in the body of the statement and that helps and is why it works.
Instead of arithmetizing proofs and showing that consistency is rejected, he inserts a modal provability operator and proofs "constructive consistency"

>>11434885
You misunderstood what I'm saying, but whatever. I'm also not the guy you started the conversation with fyi.
Indeed, LEM is also just a formal statement. What I'm saying is that
>It's vacuously true that current king of France is bald and not bald
sounds like you make a statement about the world by application of formal logic. I wouldn't leave the formal realm when discussing such things, even if it's just to get a point across.

Sure, in FOL, you can also prove
>there's a thing such that if it is a bird, everything is a bird
but plain FOL doesn't give the nicest theory of implication and Australian/Relevance logic has too complicated semantics for it to ever leave the Philosophy departments, sadly