> So here are a few things that don’t cause controversy:
> * If we have one set, we can definitely pick an element from it.
Can we? How?
The point of a set is that it doesn't distinguish between its elements in any way. It's not ordered, there is no special element, etc. So how can you pick one element from a set?
If you pick a total ordering over the elements in the set, then you can easily pick the smallest, or largest. But that's not picking an element from a set, it's picking an element from a specific ordering of that set! You can't do that without coming up with an actual ordering!
Is picking an element from a single finite set something you can build using ZF? Or is it another, less controversial, axiom? What do constructivists think about this? Is there a flavour of mathematics where this axiom is not assumed? What are the consequences of that?
> * If we have one set, we can definitely pick an element from it.
Can we? How?
The point of a set is that it doesn't distinguish between its elements in any way. It's not ordered, there is no special element, etc. So how can you pick one element from a set?
If you pick a total ordering over the elements in the set, then you can easily pick the smallest, or largest. But that's not picking an element from a set, it's picking an element from a specific ordering of that set! You can't do that without coming up with an actual ordering!
Is picking an element from a single finite set something you can build using ZF? Or is it another, less controversial, axiom? What do constructivists think about this? Is there a flavour of mathematics where this axiom is not assumed? What are the consequences of that?