Talk:Von Neumann universe

message: To 198.228.228.153:
It seems to me that ZF set theory is only a testing ground for mathematical theories. If you can define a model for a theory within ZF, you get all the advantages of the last 100 years' research into consistency, completeness, independence, and so forth. ZF has two big advantages. It has been very well researched, and it is capable of providing models for most mathematical theories. It's solid and it's big. If you can build a model for a theory inside ZF, it's probably as self-consistent and "valid" as you can get at this point in history.
When you have demonstrated that you can build a model for a theory within ZF, you can throw away that interpretation and return to the theory without any particular ZF-based model. For example, you might first build a model for the non-negative integers as 0=∅, 1={∅}, and so forth. In this framework, you can validate your axioms, and then you can just forget the particular representation. Similarly, one represents the real numbers as Cauchy sequences or Dedekind cuts to verify that it's all self-consistent and has the right sorts of properties. Then you throw away the representation when it has served its validation purpose.
If you don't believe that the representation is thrown away, ask 10 typical mathematicians if they believe that the propositions 1\in2, 1\subsetneq2, 1\cup3=3, and \cap\, 9=0 are true? Chances are that 9 out of 10 of them won't think these statements have any meaning at all. That's because most mathematicians are not interested in ZF models. They know that someone has validated the theories they use, and they just get on with their mathematical lives. Similarly, no one really believes that the number \pi is an equivalence class of Cauchy sequences of rational numbers.
The original plan of Peano, Frege, Whitehead and Russell, Hilbert, and others, was to show that mathematics can be modelled within some kind of axiomatic system where consistency can be guaranteed in some way. That plan didn't 100% succeed, but a huge amount of clarity was brought to the issue during the attempt. Some disturbing surprises turned up too, but overall, ZF does seem fairly solid, and it has survived for 100 years, roughly speaking. So it's probably okay. And Gödel showed that AC and GCH are consistent with ZF, and Paul Cohen showed that AC and GCH are independent of ZF. Since 1963, the most important questions have been resolved to most people's satisfaction. The remaining little bits and pieces of the picture are still studied in model theory, but all in all, ZF (with or without AC and/or GCH etc.) looks like it's safe to use. Most importantly, it's a framework where you can define models for theories, and can be fairly confident that if your theory can be embedded/immersed/interpreted in ZF set theory (e.g. in the von Neumann universe), then probably your theory has no consistency problems, assuming that the interpretation map from the theory to the model is validly defined.
In other words, what I'm trying to say is that mathematics does not reside inside ZF. So the atoms don't matter. All that matters is whether you can find a map of some kind from the objects of a theory into ZF. It doesn't matter if a potato is modelled as the set ω and a tomato as the set ω × ω. If the theory is valid when mapped into ZF, you've got the benefit of some kind of standard validation. It's a kind of benchmark which everybody feels they understand well enough. And if ZF does have serious problems which are not yet discovered, we're all in trouble. You'll probably get a prize for discovering any contradictions in ZF. So it's win-win. Either ZF is safe to use, or you get fame and a big prize.
Conclusion: ZF is just a testing ground for theories. Mathematics doesn't reside inside ZF.--Alan U. Kennington (talk) 13:41, 30 June 2014 (UTC)
thread_title:  The role of ZF and the von Neumann universe in mathematics 

message: thumb|300px Isn't von Neumann universe some sort of bounding volume hierarchy? --Pasixxxx (talk) 18:19, 3 December 2011 (UTC)
replies: text: No. JRSpriggs (talk) 00:52, 5 December 2011 (UTC)
thread_title:  Could this image be used as a example? 

message: This needs a redirect from "V-Hierarchy". I couldn't find it on my own without that.--208.72.139.94 (talk) 04:46, 23 February 2014 (UTC)
replies: text: Well, you're welcome to create it if you like.  You might have to register an account to do it.  You probably should also do a little searching to see if there's an alternative meaning for the term, so as not to send people to the wrong place.  --Trovatore (talk) 07:11, 23 February 2014 (UTC)
thread_title:  Redirect from "V-Hierarchy"  

message: I have added a "stub subsection" for the existential status of V. I think (hopefully) that this is more or less the same as the topic of the "philosophical perspectives" section which I (so rudely) removed. I have tried to add some references for it.
If this new "stub subsection" is unacceptable, please modify or delete it.
--Alan U. Kennington (talk) 12:58, 13 June 2014 (UTC)
thread_title: Existential status of ''V''