Have you heard of Wittgenstein's ruler?
Unless you have confidence in the ruler’s reliability, if you use a ruler to measure a table you may also be using the table to measure the ruler.
In this scenario you aren't the ruler. You are the table. You are as dumb as a doorknob.
Any non-idiot who actually understands how formal systems work knows that AC is not an exclusive set-theoretic principle.
Every formal system has an AC-equivalent.
Here is the Univalent version
But I shouldn't even have to give you any links because it is on the same fucking wiki page YOU told ME to read. Oops?
I thought you said you are studying Category theory? And you've probably been at this way longer than I have. So how come you don't know this?
Some of the references here go as far back as 1974.
I guess you didn't do your homework before calling me out, huh?
Of all the variations of AC, my most favourite is the intuitionistic/constructivist one (could you have guessed?). It is so human. So to-the-point!
It's simply the function F that was desired. In plain English: Tell me what you want this function TO DO and I will construct it for you!
In intuitionistic theories of type theory (especially higher-type arithmetic), many forms of the axiom of choice are permitted. For example, the axiom AC11 can be paraphrased to say that for any relation R on the set of real numbers, if you have proved that for each real number x there is a real number y such that R(x,y) holds, then there is actually a function F such that R(x,F(x)) holds for all real numbers. Similar choice principles are accepted for all finite types. The motivation for accepting these seemingly nonconstructive principles is the intuitionistic understanding of the proof that "for each real number x there is a real number y such that R(x,y) holds". According to the BHK interpretation, this proof itself is essentially the function F that is desired.
Could you perhaps tell us a little bit more about those sets. What do they contain?
What TYPES of elements are you choosing out of these non-empty sets? Axioms? Integers? Inference rules? Reals? Strings? Booleans? Monads? All of the above?
What you call "choosing an element from a collection of nonempty sets" (such convoluted language!), a statistician simply calls "random sampling".
And a human simply calls it "choosing stuff". You have a bunch of baskets full of things - you stick your hand and grab ANYTHING.
I am not talking about set theory. I am talking about the set of ALL FORMAL SYSTEMS, of which ZFC is just one particular example.wtf wrote: ↑Thu May 23, 2019 10:35 pmIt's equivalent to some familiar mathematical statements such as: every vector space has a basis; every surjective function has a right inverse (also called a section; and that any set can be well-ordered. It has a long history going back to Zermelo in the early 1900's.
Its importance in my responses to Pete are that it's known to be independent of the other axioms of set theory. So we can have set theory with choice and without it. So Pete's claim that truth follows from assuming axioms is wrong. It's falsified by the very well-known example of the axiom of choice. That's the only reason I mentioned it.
ALL formal systems have axioms.
And in every formal system a different set of foundational axioms would produce a different set of theorems (you know this, right?).
Which should be obvious as fuck because that's how the butterfly effect works!
So. How did you CHOOSE your axioms?
From the bins full of AXIOM-Types which ones did you pull out?
When you encounter somebody who seems to be speaking "bullshit". Never forget the alternative hypothesis
Because from where I am looking, other than Mathematics you actually know nothing about formal systems.
You have mistaken yourself for an expert. Like most academics.
My own definitions?
I am using the standard English dictionary definition of "choice" and I am demonstrating to you that when a person undertakes to construct any formal system from first principles, they are faced with CHOOSING their foundational axioms.
That you have CHOSEN to apply this principle exclusively to the narrow world of ZFC is well. Your problem.
Yep. You sure put your foot in your mouth on that one
I am going to repeat myself, and make it clear that this time I mean it as an insult.
You think you know Mathematics, but you don't even know how to USE it.