The Axiom of Choice Is Wrong (2007)
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The Axiom of Choice Is Wrong (2007)
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Re: The Axiom of Choice Is Wrong (2007)
#2Mathologer recently did a video on the puzzles mentioned in this article: https://www.youtube.com/watch?v=aDOP0XynAzA
Re: The Axiom of Choice Is Wrong (2007)
#3As time goes by, I increasingly view things like uncountable infinities and the axiom of choice as "a fun maths game", rather than having any intrinsic truth or falsity. Other's view may differ.
Here is an interesting thing I've never seen anyone write down (I should do it myself) -- we don't need uncountable infinities.
* How many natural numbers are there? Countable
* How many rational numbers are there? Countable
* How many numbers are solution to a polynomial? Countable
* How many numbers are the output of any turing machine (including programs that run forever, producing an infinite decimal)? Countable
* How many numbers are the answer to any maths problem anyone can write down (where the problem has at most a countable number of answers)? Countable.
If the set of all numbers any can express in any sensible way, and the solution to any problem any could ever have, is countable, why do we need the other uncountables?
Re: The Axiom of Choice Is Wrong (2007)
#4As time goes by, I increasingly view things like uncountable infinities and the axiom of choice as "a fun maths game", rather than having any intrinsic truth or falsity. Other's view may differ. Here is an interesting thing I've never seen anyone write down (I should do it myself) -- we don't need uncountable infinities. * How many natural numbers are there? Countable * How many rational numbers are there? Countable…
How many real numbers are there?
Re: The Axiom of Choice Is Wrong (2007)
#5I'd assume that even if in every case the number of incorrect guesses is finite, the expected number of people that fail to guess their color is infinite.
Am I right about this?
Re: The Axiom of Choice Is Wrong (2007)
#6As time goes by, I increasingly view things like uncountable infinities and the axiom of choice as "a fun maths game", rather than having any intrinsic truth or falsity. Other's view may differ. Here is an interesting thing I've never seen anyone write down (I should do it myself) -- we don't need uncountable infinities. * How many natural numbers are there? Countable * How many rational numbers are there? Countable…
This is discussed heavily in topics around constructing the reals and constructivism. Wikipedia has a short section about using the computables instead of the reals: https://en.m.wikipedia.org/wiki/Computable_number
It is pretty fascinating to think about: that almost all real numbers are not computable ("almost all" of course meaning "all but a countable set"). And yet you almost certainly will never run into a noncomputable real except in computer science when you're specifically defining one to illustrate something about computability (like Chaitin's constant).
Re: The Axiom of Choice Is Wrong (2007)
#7As time goes by, I increasingly view things like uncountable infinities and the axiom of choice as "a fun maths game", rather than having any intrinsic truth or falsity. Other's view may differ. Here is an interesting thing I've never seen anyone write down (I should do it myself) -- we don't need uncountable infinities. * How many natural numbers are there? Countable * How many rational numbers are there? Countable…
> As time goes by, I increasingly view things like uncountable infinities and the axiom of choice as "a fun maths game", rather than having any intrinsic truth or falsity.
Is a hammer true or false? I don't know, but it's good for hitting nails.
Re: The Axiom of Choice Is Wrong (2007)
#8As time goes by, I increasingly view things like uncountable infinities and the axiom of choice as "a fun maths game", rather than having any intrinsic truth or falsity. Other's view may differ. Here is an interesting thing I've never seen anyone write down (I should do it myself) -- we don't need uncountable infinities. * How many natural numbers are there? Countable * How many rational numbers are there? Countable…
How many sets of natural numbers are there? An uncountable number.
Of course, you could throw out some axioms so that you create an axiom system in which I can't define such a thing as "a subset of the natural numbers", but such a world doesn't feel "real" (in a platonic sense) to me.
Re: The Axiom of Choice Is Wrong (2007)
#9As time goes by, I increasingly view things like uncountable infinities and the axiom of choice as "a fun maths game", rather than having any intrinsic truth or falsity. Other's view may differ. Here is an interesting thing I've never seen anyone write down (I should do it myself) -- we don't need uncountable infinities. * How many natural numbers are there? Countable * How many rational numbers are there? Countable…
How many sets of natural numbers are there? An uncountable number. Of course, you could throw out some axioms so that you create an axiom system in which I can't define such a thing as "a subset of the natural numbers", but such a world doesn't feel "real" (in a platonic sense) to me.
In this case, what you "throw out" are uncomputable (or non-recursive, depending on terminology you like) sets; i.e. sets for which the membership function is not decidable. Yes, there are an uncountable number of these sets, but they can't be defined in any useful way.
Re: The Axiom of Choice Is Wrong (2007)
#10As time goes by, I increasingly view things like uncountable infinities and the axiom of choice as "a fun maths game", rather than having any intrinsic truth or falsity. Other's view may differ. Here is an interesting thing I've never seen anyone write down (I should do it myself) -- we don't need uncountable infinities. * How many natural numbers are there? Countable * How many rational numbers are there? Countable…
You could have answered all of your questions with "finitely many", because, after all, we can each only perform a finite number of actions in the world.
In general, the infinite hierarchy of infinite sets "exists" because we can define it.