@user170039 IEP is an interesting source of ideas. However there problems in some of their articles. I was curious and clicked on one of the listed links and the example given is fallacious.

It says:

> Consider a collection of objects. The collection has some size, the number of objects in the collection.

This already is only valid for finite collections. Otherwise one has to define precisely what is meant by size.

> Now consider all the ways that these objects could be recombined. For instance, if we are considering the collection {a, b}, then we have four possible recombinations: just a, just b, both a and b, or neither a nor b. In general, if a collection has κ members, it has 2^κ recombinations.

This is not clearly valid for infinite collections for the same reason. But there is a valid way to reason about subcollections of a collection S. Simply look at the functions from S to bool, which I shall denote by func(S,bool). We shall not need to define the 'size' of func(S,bool).

> It is a theorem from the nineteenth century that, even if the collections in question are infinitely large, still κ < 2^κ, that is, the number of recombinations is always strictly larger than the number of objects in the original collection. This is Georg Cantor's theorem.

Again, this is invalid without a proper definition of "size". I'll challenge anyone to provide a philosophically justified definition of size where this holds. In contrast, I can in fact philosophically justify a definition of size and show that the claim does not hold!

Despite that, Cantor's diagonalization argument does give a philosophically justified conclusion, namely that there is no surjection from S to func(S,bool).

As I've explained before to a number of people (not sure whether including you), the proof is even a computable one in the BHK sense, where if you can certify that f is a surjection from S to func(S,bool) then I can transform that certificate in to a certificate of a falsehood.

So what breaks after that? Well, no surjection from S to func(S,bool) simply does not imply no injection from func(S,bool) to S.

It does in ZF, but the notion of size via bijection is bad enough and ZF is far worse, philosophically speaking.

So no go.

So suppose we do have a universal type obj. Then Cantor's argument will show that there is no surjection from obj onto func(obj,bool). But as I said above, that does not imply that func(obj,bool) does not inject into obj. In fact, it obviously does via the identity map, by definition of obj!

What goes wrong? To obtain a surjection from obj onto func(obj,bool), you would need to create something like ( obj x ↦ x∈func(obj,bool) ? x : ( obj y ↦ true ) ). This looks like a surjection, but it's actually not. Notice that it would be a function only if we have decidable membership in func(obj,bool), namely that every object is either accepted or rejected by func(obj,bool). Otherwise there is no reason we should assume this will be a function, not to say surjection.

On the other hand, the notion of size is correctly captured by the notion of injection. Indeed func(obj,bool) is no bigger than obj, as we should have given the intended meaning of obj.

Then what is captured by surjection? I always say it is roughly complexity.

func(obj,bool), despite being smaller than obj, is more complex. You can intuitively see that obj has a trivial structure; it accepts everything.

@user170039 I did not look to see who wrote the page, but I don't have the time right now for a protracted discussion about it. I just mentioned it as it is an issue I have noticed is often ignored, which seems to be because people rarely consider how much ZFC they assume (without justification) when talking about set-theoretic things like ZFC ordinals and cardinals.

What makes that particular article especially incoherent is that it is explicitly talking about a non-classical logic, but its stated example is fallacious because the paradox only works in classical ZF. This means that you do not actually have any paradox to entertain if you don't have classical logic in the first place, so the article fails to provide motivation for paraconsistent logic.