The Universe/Set Theory Conflict?

[Muon12]

My experience and knowledge of Set Theory is pretty lacking, so I apologize if I don't seem very assertive while discussing this topic. Anyhow, this is a big picture question that I have found to be, among other things, difficult to analyze: If the number of sets containing subsets is infinite, then how can a universe exist, since in claiming that such a set exists, we are claiming that it is the final set, thereby contradicting Set Theory? I don't know how to approach this question, nor am I even sure that it is valid since Set Theory tends to deal with the abstract, not the physical.

Still though, if mathematics and logic can be simplified into sets, which many believe (proven?) and mathematics is one of our greatest tools used in describing the physical behavior and patterns found within space-time, can we ignore a question that carries with it such heavy implications? You start with the empty set, [0] which is a subset to all sets, then you move on to the set containing the empty set, [1] and from there it goes on and on, with each additional set containing all of the previous sets.

If numbers are sets, and the mechanisms of physics are carried out in conjunction with numbers (mathematics), then is our ideal version of the "universe" as a set containing all other sets of matter and energy, one that is not itself contained by a greater set, incorrect?


Oh, and just a side thought: could this be used as an argument in support of the Multiverse Theory?

 

[vagabond]

For I know, there does not exist a set that contains everything.
Suppose A contains everything. I think A is not in A, so there is something not contained in A. Is my argument correct?

 

[Data]

how can a universe exist, since in claiming that such a set exists, we are claiming that it is the final set

Please state what you mean a little more clearly. What is a "final set?" What is a universe? As well, if you're trying apply this to the world, you'll have to establish that your definition for universe is consistent with the way that the universe exists in reality (is the real universe actually a set? In what sense?).

 

[honestrosewater]

For I know, there does not exist a set that contains everything.
Suppose A contains everything. I think A is not in A, so there is something not contained in A. Is my argument correct?

It depends on whether or not A is a thing. ;) But, yes, not placing restrictions on sets leads to problems, and they've been dealt with by distinguishing between classes and sets or by limiting the "size" of sets.

If the number of sets containing subsets is infinite, then how can a universe exist, since in claiming that such a set exists, we are claiming that it is the final set, thereby contradicting Set Theory?

Ditto Data. Be careful to not confuse membership and subsets. For instance, the empty set is a subset of every set, however the empty set is not a member of every set. Also, in applying set theory to the physical world, remember that sets and classes need to be definite; Given some set or class S, every object either definitely is or definitely is not a member of S.

 

[samp]

That's why we choose universes smaller than "everything". Thinking about the set of all sets just leads to a headache.

 

[Muon12]

Posted by Data

Please state what you mean a little more clearly. What is a "final set?" What is a universe? As well, if you're trying apply this to the world, you'll have to establish that your definition for universe is consistent with the way that the universe exists in reality (is the real universe actually a set? In what sense?)

I suppose I could elaborate a bit further. A universe in the context of this question -(If the number of sets containing subsets is infinite, then how can a universe exist, since in claiming that such a set exists, we are claiming that it is the final set, thereby contradicting Set Theory?)- would mean "The Final Set: A set that is not a subset of a greater set; one that is not contained but that contains everything." The universe as most think of it and as I am relating the set question to can be described as "the unified structure of all things contained within the cosmos."

While the first possible definition is more exclusive to my question, I wonder if it could be found to have equal meaning to the 2nd, more common definition of our "universe". The question posed is not one of my own, by the way. I first heard it in a Logic Class, so providing you with more depth on this particular subject might be difficult for me. Though I hopefully clarified the question a bit in my response.

 

[matt grime]

I think you ought to learn about the mathematics of modern set theory. We declare certain things to be sets *within some model*. The collection of all the sets in that model form a proper class within that model, though they form a set within some larger model. That is nothing a priori comes with the label set - they must be shown to be sets within some model of some set theory, though which theory or model, if any , is not necessarily easy to answer.