Predicativity reading todo

 don't really want to take SRH down this road ... we don't want definitions we want to understand why but anyway:

WHAT IS PREDICATIVISM?
https://www.math.wustl.edu/~nweaver/what.pdf

https://planetmath.org/predicativism

https://en.wikipedia.org/wiki/Impredicativity

https://www3.nd.edu/~tbays/papers/burgess.pdf

Can we generate some predicative statements that we can accept?

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Broadly there is a problem with building houses on their own roofs. In Constructivism you need foundations to build on. And that creates the need for hierarchy rules that force systems into layers.

Now no one can escape Cantor's powerful use of this to create the hierarchy of Infinities that is do vast as to defy definition. Its not that numbers of infinite, but that the number of number types are infinite and so on in an ascending hierarchy that is beyond infinity. It is boundless because any bound greater than 2^current cardinality cannot fit within the current new number and create a new type. Diagonalisation and the Power Set blows apart any set.

So "totality" in set theory is a problem as the combinations of set elements becomes a predicate that lies outside the set.

But we can also create sets not by proving the existence of new things but just filtering things without any concern for whether they exist.

Certainly all the Self-Reference proofs operate by supposing an element and then showing that if it were the exist it would create an contradiction. So we prove it does not exist.

Or perhaps we can just say that if anything were to be selected by the definition it would create a contradiction, so we  blame the definition without needing any actual existence. This was Frege's contribution. ∃x | H(x) could be a problem if x is a Horned-Rabbit. This is actually an absurdity used by Buddha in India in 500BC. What are we saying is absurd about a Horned Rabbit if it doesn't exist? When ∃x | H(x) is false we are saying that there is no x such that H(x) without anything actually existing.

Likewise we could make statement above that make definitions without getting involved in actual existence.

Could a self-referential sentence be written that does not actually imply that it exists?

We run into problems in that surely we can always implied existence in self-referential sentences. If a sentence exists, and is self-referential then it must be referring to an existence. But anyway just pushing the boat out in this Predicativity issue that always been hanging around SRH.

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Given the set X = {1,2}

∃x ∀y| x ∈ X & y ∈ X & x <= y

So we're saying there is a smallest value. While predicative it's not a problem because 1 has other definitions like S0 the successor of 0. So we have a thing "1" which exists outside this expression. All we are saying is that in the set X which contains "1" and "2" 1 < 2. Obviously in fact "1" is beyond the scope of this definition because we're using its magnitude to select it.

In Ramsey's example of "tallest man in the room" we can argue the same that the man has a height which is defined outside the definition. Ramsey is arguing that the "man" exists beyond the definition but surely we can also say that to apply the predicate H(x) (Height of x) we are going beyond the definition.

We don't need to be constructivist, problems only occur when we are using predicates that attribute qualities to elements that are defined by the set. Any predicate that attributes qualities to elements, that is itself dependent on the set is the only way to cause a problem surely?

"Sets that do not contain themselves" is such a problem.

What is the predicate?

{ x : x ∉ x }

So the ONLY way to find out whether x belongs to the set is to look at x! Now we are in trouble.

This was visualised well in the story of the librarian who decided to catalogue the books in the library. They noticed that and while most books did not reference themselves, some book did referenced themselves. They decided to write two catalogues for these books. The Catalogue of books that do reference themselves and the much larger catalogues of books that do not. Having written the books the librarian adds them to the shelves, only then realising that they are not catalogues yet. For the Catalogue of books that do reference themselves they realise adding it to that catalogue is fine. But then adding the Catalogue of books that do not reference themselves to that catalogue causes a problem, because not it does reference itself. So the librarian moves it to the other catalogue, but not it does not reference itself. This book fits into neither catalogue, and yet a book must either reference itself or not reference itself.

You might say the problem of course is that the book is not complete until all the references are in, and so it is not referencing a finished book, or an actually existing book yet.

But if the librarian sticks to catalogues of finished books then the problem still remains. Because once finished they catalogues can then be added to themselves, which means they are not finished yet.

The actual problem is that the Predicate namely "x ∉ x" depends only on x itself. How do you know whether a book belongs in the catalogue? You must scan the book for itself. How do you know what "itself" is you must look at the book. The itself is not defined by anything "outside" the book. Once you have picked that book up the "itself" is not defined. Its like the set of all things that you are holding at the time. The scope is limited entirely to the context of the operation there is no "objectivity."

Note computer languages have "this" or "self" references that when dereferenced point to the current object. All the objects are in objective space like books in a library and have exact addresses or names. Each self variable name is then replaced with the relative address of that object at run time.

We can't escape the paradox then by binding the "self" with the name of the book. That produces a list of questions about whether each book in the library with title T contains the text T. We assume that we've already titled the two catalogues of "Books that Contain" and "Don't Contain" themselves.

That maps back into computers. Suppose we have a base class with a method PrintNotSelfReferential(). We then have a class hierarchy derived from that and a load of instantiated object in memory. Now all object in memory must either dereference themselves or not. So we make a new class derived from the base with a method to go through memory and call PrintNotSelfReferential() on every object that does not dereference itself. It gets to itself and with no reference to itself it calls its own PrintNotSelfReferential(). In so doing it references itself and so executes PrintNotSelfReferential() on a method that ends up dereferencing itself. If the method had a validation step to check the output and redo if wrong you then have a loop and it never halts. The classic "I am false."

So there's no real way out of this. Once something is defined ONLY in  terms of itself ere be dragons!

But why?

I meant to say at the top, what of fixed points in this?

Is it meaningful to ask for the fixed point of a set like :  

{ x : x ∉ x }

#TODO