ready to do some math kids?
standard disclaimer that i'm being incredibly loose with technical details here so you don't all need to become math majors to understand me.
some terminology:
(you can read these now, or skip to the interesting stuff below and come back when you realize you have no idea what i'm talking about)
sentence:
in first order logic, you can say something along the lines of
"everything has this property"
"there is something which doesn't have a certain property"
"everything is equal to a particular, special thing"
and you can imagine you can get as complex as you want, you could say that exactly three things have these two properties, and everything else has another property, and so on. i know these sentences look pretty limited but trust me, you can do some powerful math with them.
language
basically the alphabet / set of rules which you create your sentences from
when we talk about the size of a language, we talk about the number of sentences you can possibly write in that language
model
so when i was talking about sentences back there i kept on using "thing" and "property". well a model is basically *context* for a sentence. so here are models of some of those sentences above
"everything has this property" becomes "every fraction is equal to itself"
"there is something which doesn't have a certain property" becomes "there is a person who does not like trump"
"everything is equal to a particular, special thing" becomes "all ponies are equivalent to twilight"
more importantly, the model has to make sense, so if you translated
"everything has this property" as "everyone is darvince" well that's not a model anymore it's a countermodel
by the way, if we want to talk about the size of the model, that basically just means how many things the model is dealing with. so for example, if you're modeling three people having a conversation, your model might have size 3. if you're modeling the natural numbers, size infinity
deduction/proof
a special way of proving things. you should just think of this as proving things / making a logical argument but with a bunch of formalized rules.
for convenience i'll just say prove/proof to refer to deduction
consistent
you can't prove something and it's opposite. usually used to describe a bunch of sentences
cardinals
basically all you need to know here is that there are numbers, and then there are infinities.
just like there are an infinite number of numbers, and you can compare numbers, there are also an infinite number of infinities, and you can compare them, and obviously some are larger than others
so instead of calling some of them numbers and others infinities, we'll just call them all cardinals
for easy reference, i'll call the infinite cardinals infinity 1, infinity 2, etc...
set:
a group of things which has no duplicates
super simple.
ZFC:
this cool collection of exactly 9 sentences which is used by mathematicians alot. it talks about sets.
just to give you a taste of what these sentences are, the first one says that two sets are equal if everything in one set is in the other and vice versa
ok now we can start talking about the interesting stuff that you can do with math:
soundness theorem:
if you can prove something, it's actually true
this is probably the least interesting of all the things i've learned.
you basically prove it just by showing that all the formalized rules for proof i mention above are actually correct
completeness theorem:
if something is true, you can prove it
this is pretty mindblowing. i mean maybe there's something that's true but even the world's greatest logician can't prove it.
but wait, that can't happen because completeness. midn blown
turns out to prove this you need the model existence lemma just below
model existence lemma:
if a set of sentences is consistent, there's a model of it.
ok this one is a bit harder to understand. but go back to the definition of sentences i gave with all those really vague general statements
and you could ask "is there any possible context in which these sentences could make sense?"
you can imagine that i could put together a bunch of sentences so bizzare and vague that no possible context (model) would ever cause them to make sense
for example, what if i started saying "everything has property 1", and "if something has property 1 then it has property 2", and i keep on making up these rules (sentences), so that any possible model (context) you can think of doesn't make any sense
but this lemma says that there will always be a model (context) so that all the sentences make sense, which is pretty cool
you just have to make sure that your sentences are consistent (you can't prove something impossible with them)
compactness theorem:
if every finite subset of some (possible infinite) collection of sentences has a model, then all of those sentences have a model (and vice versa)
ok, so suppose you have an infinite number of sentences, and you want to know if any model will make them all happy
if you can show that every time you pick a finite number of sentences from that infinite collection of sentences, those sentences have a model
then it's equivalent to showing that there is a model which satisfies all those infinite sentences
this is pretty cool because the models for your finite sets of sentences could have nothing to do at all with the model for the infinite sentence
like the model for one finite set of sentences could be the context of politics, and the model for your infinite collection could be talking about natural numbers
skolem-lowenheim theorem:
if you have a bunch of sentences, and they make sense under a model at least as large the language you wrote those sentences in
then for any cardinal at least as large as the language, there is a model of that size which satisfies those sentences
so suppose you're some sort of deity and you're writing a rulebook (collection of sentences) for constructing universes
and you want to write the rule that "there will be infinity 5 objects in this universe"
but you can't, because skolem lowenheim says that if your rulebook allows a universe with infinity 5 objects to be created, then it must also allow universes of any other infinite size to be created
skolem's paradox:
alright, the example i gave for skolem-lowenheim was a bit off the deep end, so here's a logicish puzzle based off of that theorem which is pretty cool
with the sentences in ZFC, you can prove there infinities beyond infinity 1, so you can show there is an infinity 2,3, and so on. moreover, you can create sets with more than infinity 1 objects in them
on the other hand, skolem-lowenheim says that, there is a model (a context) for ZFC with only infinity 1 objects in it.
so this is counterintuitive, because on one hand, you only have infinity 1 objects -- but you can talk about sets (which don't have duplicates) with more than infinity 1 objects in them. which sounds kind of impossible, but there's a good solution to this puzzle which i won't give
anyway it's like 4 am here and that's why any sane person shouldn't take formal logic like i did or else you'll end up writing long posts like these of questionable value
reply with how much you understood of this, or i'll logic you out of existence
Topic: some things i learned in formal logic (Read 1599 times)
