Tag yourself i'm "degenerate"

>implying you need math to other than research

Normalfag reporting
>what even is the axiom of choice?

look at all these citations

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>""R""
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>""E""
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>""A""
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>""L""
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>""S""

All math consists of logical deductions based on things we know are true. How do we know they are true? More deductions. So where does it bottom out? At the axioms. These are things we accept to be true without proving. We've gotten really good at having just a few of these. Fewer than 10 actually. Statements such as "there exists a set with infinitely many elements"

The Axiom of Choice is one such statement. It's controversial because there are many statements logically equivalent to it, some of which seem completely obvious, and some of which seem like they may not be true.

>i can arbitrarily select from an uncountably infinite set

ok, I accept that axiom.
when I define a continuous function f mapping x -> y
I want to be able to choose x on the uncountably infinite domain.
I don't think this is even hard to swallow.

I found one from the set of all real numbers
5.2537 arbitrarily

I'm just an ultra finitist on Veeky Forums and a normal when I go to class.

Does that mean I have two identities or do they average out to creep?

identities are unique, stupid

"controversial" boolesrings.org/asafk/2014/anti-anti-banach-tarski-arguments/

>boolesrings.org/asafk/2014/anti-anti-banach-tarski-arguments/

>first proof involves measure theory

pig disgusting user, you should be ashamed.

Could you explain why a non-measurable set exists according to the first line of that guys proof? I always thought that was equivalent to choice.

The first line doesn't show that a non-measurable set exists. It says that a non-measurable set *would* exist, if we had [math]\aleph_1 \leq 2^{\aleph_0}[/math]. But since we're assuming that there *isn't* a measurable set, we can conclude that therefore this inequality doesn't hold. (contrapositive)

The only word I know on that picture is truth tables.

>mfw people know semantics without knowing deduction

what is intuitionistic logic?

simple version: it's like classical logic but without the Law of Excluded Middle (LEM) and equivalen statements. LEM is the axiom/rule that says, if you have any statement P, you are allowed to assume that "either P holds, or not-P holds" is true

intuitionistic logic doesn't assert that LEM is *false*, it just doesn't include it as a rule you can use.

many specific cases work, too - for example, most intuitionistic systems let you assume "either n = m, or n != m" for any natural numbers n and m. you just don't get carte blanche to automatically conclude it for any possible statement

simplified-to-the-point-of-strawman example of why you might want this: if you believe that a proof of "P or Q" should always either include a proof of P or of Q, LEM is unsound, because it lets you prove things like "either the riemann hypothesis is correct, or it's incorrect" without specifying a proof of either of those things

but you could write a program that automatically generates a proof that n = m or n !=m for any n or m you feed it, so it's always the case that you can produce either a proof of n = m or a proof of n !=m, no matter what n and m are - therefore that particular limited version is fine

>lem is not false

the axiom of choice says you can make infinitely many choices, not just one

you can prove "finite choice" directly from ZF

but what if statement P is only sometimes correct

>Study Philosophy

Im "God"

Past the point of unredeemable. What do?

not that guy, but if by "sometimes" you mean "at some point in time it is true, but other points in time it is untrue", then this is not a problem in mathematics or logic, but one of philosophy (namely philosophy of time: en.wikipedia.org/wiki/A-series_and_B-series). If on the other hand you mean "sometimes" as in, "P" stands for say "All natural numbers n satisfy: [math]\frac{n^2}{n} = n [/math], and this statement is only sometimes true in that it is true for some numbers (all non-zero ones), but is false for some numbers (when n = 0), then this is simply an issue of the nature of the proposition labelled "P". In this case, P is false because it is not the case that for all numbers n, [math]\frac{n^2}{n} = n [/math]. But how do we capture the fact that for some n it is true? We extend "propositional logic" (which is logic using simple propositions like the one I just stated), to "predicate logic", which is logic using "predicate symbols". So our statement can instead be: P(n) (where P is the predicate "being the case that: [math]\frac{n^2}{n}[/math]"), in that case P(0) is false, whereas P(1), P(2), etc. are all true.

Ultrafinitism is just being a realist. I figured that out as a physical infant.

weird

i'm the user they warned you about, Veeky Forums
i study substructural logic but assume the metalogic is classical

>published mathematician
>normal

REEEEEEEEEEEEEEEEEE

im degenerate cuz lem with continuum hypothesis is ???

tru

Everyone remotely serious should be weird. Even if you're completely comfortable accepting the axiom of choice you should acknowledge when you're invoking it and not just fling it around without even being aware that you are.

Anything lower than weird and you're falling into constructivist retardation

constructivism is cool and good, actually,

Could someone give me a metatheoretical justification of WHY THE FUCK SHOULD I ACCEPT THE LAW OF THE EXCLUDED MIDDLE pls.
I still use it though, because it's comfortable and makes things easier, but I feel dirty and awkward afterwards :(

Why? The axiom of choice is very natural. I like to acknowledge when I'm using it, but really it's not because we need to "assume" it, just to let the reader know why the step works.

Yeah, I mean like, taking one element per set out of a family which cardinality is FUCKING WOODIN is conceivable no problem
>Tfw I do it every morning to get rid of sleepiness

Where's "no negative numbers"?

Saying how you are choosing your elements is sufficient, I don't cite the axiom of extensionality whenever I do a double inclusion argument, even thought I technically using it. Similarly when I take unions I don't cite that its an axiom of ZFC in my proof.

I guess I am weird cause I know some model theory. Which a strangely specific branch of math logic. I personally think recursion theorists are weird, and proof theorists creepy.

Anyone below creep is lost for math desu

There should be a level with "Refuses to do math outside von Neumann ordinals"

Am I hardcore just because I use sequent calculus whenever I can?

Yes sequent calculus suck dicks

actually sequent calculus most cleanly demonstrates the curry-howard isomorphism

explain voevodsky