/sci/ - Science & Math

>>14956241
I saved this pic on the 13th Sept. 2006 from fotocommunity.de

>>12211810
It's true in classical first order logic at least, yes.

For any predicate ϕ,
(∃x.x=x) ⟹ ∃y. (ϕ(y) ⟹ ∀z.ϕ(z))

In words,
"If there exists anything at all, then there exists a thing such that if it has the property ϕ, then everything has the property ϕ"

PROOF of the logical schema above:
Assume something exists.
Then for any predicate, it either holds for all terms in the universe of discourse, or there is something for which it doesn't hold.
In the first case, the property holds for everything and we're done.
In the second case, take the thing for which the property does not hold. Then assuming the property holds for it, we get a contradiction and everything follows from explosion. In particular, it follows that the property holds for everything.
QED

Example:
"There is a thing, such that if that thing is a bird, then everything is a bird."
PROOF (same as above, just more specific case)
Not everything is a bird. We know that your mom isn't a bird. So if we assume that your mom is a bird, we can prove everything.

Of course, this proof uses LEM for an existence statement, uses explosion and uses material implication, so it's a non-relevant (in the sense of relevance logic) as you can get.
You can check out the Paradoxes of Material Implication Wikipedia article for more examples along those lines.

It's the same thing with
>There exists a student, such that if he passes this term, everyone will pass.
If you use explosion on the student that fails the exam, you can formally derive stuff like "everybody passes", but the proof is not relevant in the relevance-logic sense.

>>12211810
It's true in classical first order logic at least, yes.

For any predicate [math]\phi[/math],
[math] (\exists x. x=x) \implies \exists y. (\phi(y)\implies \forall z.\phi(z)) [/math]

In words,
"If there exists anything at all, then there exists a thing such that if it has the property [math]\phi[/math], then everything has the property [math]\phi[/math]"

Example:
"There is a thing, such that if that table is a bird, then everything is a bird."
PROOF (for the example, you can skip this and go on to the more formal proof below)
Not everything is a bird. We know that your mom isn't a bird. So if we assume that your mom is a bird, we can prove everything.

PROOF of the logical schema above:
Assume something exists.
Then for any predicate, it either holds for all terms in the universe of discourse, or there is something for which it doesn't hold.
In the first case, the property holds for everything and we're done.
In the second case, take the thing for which the property does not hold. Then assuming the property holds for it, we get a contradiction and everything follows from explosion. In particular, it follows that the property holds for everything.
QED

Of course, this proof uses LEM for an existence statement, uses explosion and uses material implication, so it's a non-relevant (in the sense of relevance logic) as you can get.
You can check out the Paradoxes of Material Implication Wikipedia article for more examples along those lines.


It's the same thing with
>There exists a student, such that if he passes this term, everyone will pass.
If you use explosion on the student that fails the exam, you can formally derive stuff like "everybody passes", but the proof is not relevant in the relevance-logic sense.

>>12211810
It's true in classical first order logic at least, yes.

For any predicate [math]\phi[/math],
[math] (\exists x. x=x) \implies \exists y. (\phi(y)\implies \forall z.\phi(z)) [/math]

In words,
"If there exists anything at all, then there exists a thing such that if it has the property [math]\phi[/math], then everything has the property [math]\phi[/math]"

Example:
"There is a table, such that if that table is a bird, then everything is a bird."

PROOF of the logical propositon:
Assume something exists.
Then for any predicate, it either holds for all terms in the universe of discourse, or there is something for which it doesn't hold.
In the first case, the property holds for everything and we're done.
In the second case, take the thing for which the property does not hold. Then assuming the property holds for it, we get a contradiction and everything follows from explosion. In particular, it follows that the property holds for everything.
QED

Of course, this proof uses LEM for an existence statement, uses explosion and uses material implication, so it's a non-relevant (in the sense of relevance logic) as you can get.
You can check out the Paradoxes of Material Implication Wikipedia article for more examples along those lines.

But to sum it up again in terms of the bird example: We know that your mom isn't a bird. So if we assume that your mom is a bird, we can prove everything.

It's the same thing with
>There exists a student, such that if he passes this term, everyone will pass.
If you use explosion on the student that fails the exam, you can formally derive stuff like "everybody passes", but the proof is not relevant in the relevance-logic sense.

>>12028348
>Like, 8 feels prime-ish cause its a p^p
really makes you fink

>>12028440
The sequences x and y are equivalent if the fraction of position where they are different tends towards zero?

>>12028585
>finitely
I donno, but I think I can try to think of two sequences that are different in infinitely many places, but more sparsely so than linear and not finite.
So if I pick a prime [math]p[/math], then two sequences that are forced to be different at the indices k given by the floor of [math]\log(i\cdot p)[/math] for all i could already be equivalent by this definition. I guess this is asking whether there's only few ways in which I can make essentially different sequences still be equivalent in such a way?
That's not a ways to solve the question, but the set of sequences is such a shit mathematical object that I don't even dare to make a guess. Let's say I guess that |X/∼|=|X| can be disproven in a strong enough theory.