/sci/ - Science & Math

https://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate

y u no mg or sqt?

>>11882386
You can find a list of set theories e.g. here

https://en.wikipedia.org/wiki/List_of_first-order_theories#Set_theories

I'd say it's even a scratch to say that ZFC is "used". It's studied by set theories for sure.
But in practice, any predicate (class) that's not evidently inconsistent will be by freely used by mathematicians :P

There's a bunch of good answers in this thread already, but here some remarks.

You can take this truth table as just the definition of the implication and roll with it.
The fact that this implication (the material implication) is a bit odd is an acknowledged situation.
Here's one reference for a few it's consequences that put people at unease
https://en.wikipedia.org/wiki/Paradoxes_of_material_implication
and there's some more.

My favorite is the following tautology of first-order logic
∃y. (B(y) ⟹ ∀z. B(z))
In words (with semantic for B), this reads:
>(If there exist anything at all then) There is a thing, such that if that thing is a bird, then everything is a bird.

Proof: Let's grant that there's these two options: Either everything under consideration is a bird, or, on the opposite, there is something which is not a bird. In the first case, it's true that everything is a bird by assumption, and the statement is true. Now, in the second option, if there something that's not a bird, then if that thing were a bird, we'd have a contradiction and anything following (in particular this statement)

What happens here is also what happens in your original question:
>Why is "P implies Q" true when P is false and Q is true?
It's true by definition, and what motivates it is the principle of explosion: If you can prove falsehood, then you can prove anything.
The mathematical motivation - and the reason why this weird material implication => still gets us far in mathematics, is that explosion comes for free in arithmetic:
If 0=1 (the most basic falsehood), then 0=0+0+0=1+1+1=3 etc.
If you have 0=1, then you have n=m for any n and m.

Note that there are alternatives to =>, i.e. alternative logics.
e.g.
https://en.wikipedia.org/wiki/Relevance_logic
Such things are more studied in the philo department than in math ones.

At one point I tried to motivate this whole game here

https://youtu.be/eeLa9tIhFMs

>>11842344
There's a bunch of good answers in this thread already, but here some remarks.

You can take this truth table as just the definition of the implication and roll with it.
The fact that this implication (the material implication) is a bit odd is an acknowledged situation.
Here's one reference for a few it's consequences that put people at unease
https://en.wikipedia.org/wiki/Paradoxes_of_material_implication
and there's some more.

My favorite is the following tautology of first-order logic
[math] \exists y.\,\left( B(y) \implies \forall z.\,B(z) \right) [/math]
In words (with semantic for B), this reads:
>(If there exist anything at all then) There is a thing, such that if that thing is a bird, then everything is a bird.

Proof: Let's grant that there's these two options: Either everything under consideration is a bird, or, on the opposite, there is something which is not a bird. In the first case, it's true that everything is a bird by assumption, and the statement is wrong. Now, in the second option, if there something that's not a bird, then if that thing were a bird, we'd have a contradiction and anything following (in particular this statement)

What happens here is also what happens in your original question:
>Why is "P implies Q" true when P is false and Q is true?
It's true by definition, and what motivates it is the principle of explosion: If you can prove falsehood, then you can prove anything.
The mathematical motivation - and the reason why this weird material implication => still gets us far in mathematics, is that explosion comes for free in arithmetic:
If 0=1 (the most basic falsehood), then 0=0+0+0=1+1+1=3 etc.
If you have 0=1, then you have n=m for any n and m.

Note that there are alternatives to =>, i.e. alternative logics.
e.g.
https://en.wikipedia.org/wiki/Relevance_logic
Such things are more studied in the philo department than in math ones.

At one point I tried to motivate this whole game here

https://youtu.be/eeLa9tIhFMs

>>11842344
There's a bunch of good answers in this thread already, but here some remarks.
You can take this truth table as just the definition of the implication and roll with it.
The fact that this implication (the material implication) is a bit odd is an acknowledged situation.
Here's one reference for a few it's consequences that put people at unease
https://en.wikipedia.org/wiki/Paradoxes_of_material_implication
and there's some more.
My favorite is the following tautology of first-order logic
[math] \exist y.\,\left( B(y) \implies \forall z.\,B(z) \right) [/math]
In words (with semantic for B), this reads:
>(If there exist anything at all then) There is a thing, such that if that thing is a bird, then everything is a bird.

Proof: Let's grant that there's these two options: Either everything under consideration is a bird, or, on the opposite, there is something which is not a bird. In the first case, it's true that everything is a bird by assumption, and the statement is wrong. Now, in the second option, if there something that's not a bird, then if that thing were a bird, we'd have a contradiction and anything following (in particular this statement)

What happens here is also what happens in your original question:
>Why is "P implies Q" true when P is false and Q is true?
It's true by definition, and what motivates it is the principle of explosion: If you can prove falsehood, then you can prove anything.
The mathematical motivation - and the reason why this weird material implication => still gets us far in mathematics, is that explosion comes for free in arithmetic:

Note that there are alternatives to =>, i.e. alternative logics.
e.g.
https://en.wikipedia.org/wiki/Relevance_logic
Such things are more studied in the philo department than in math ones.
If 0=1 (the most basic falsehood), then 0=0+0+0=1+1+1=3 etc.
If you have 0=1, then you have n=m for any n and m.

At one point I tried to motivate this whole game here

https://youtu.be/eeLa9tIhFMs