Question: Constructing a Subset of the Real Numbers with Cardinality Aleph 1

[micromass]

I think that this thread has lost its seam. The original question that I asked is not being discussed.

On the contrary, I think these are valid responses to your question. The point is that the answers might just no be so simple as you would like.

You ask, with CH being false, what a set of \aleph_1 might look like in \mathbb{R}. The answer is sadly, that we don't know. CH just gives you the answer that such a set exists, without telling you what it looks like or what the properties are.
So if you just accept ZFC+(CH), then your answer is unanswerable.

The closest answer I can give you, is that you can construct a model of ZFC where CH does not hold. You do this through "forcing", this is a very difficult technique. With forcing, you can build an entire universe of sets. In this universe lies a set of real numbers which has cardinality e.g. \aleph_2 and in this universe, you can explicitly find a set of cardinality \aleph_1. But this example of a set will be dependent on the universe you built.

So in general, you cannot construct a set of \aleph_1. However, you can still ask what such a set looks like. In general, there is nothing you can say about such a set. But with additional axioms, there are some things you can say. For example, Martins axiom will tell you that the set has Lebesgue measure 0 (so it will be a Cantor-like set) and is of first category in \mathbb{R}.

 

[CRGreathouse]

I'm really perplexed. Could you help me by explaining your notion of explicit constructibility?

It's pretty simple -- whenever you have an object which is 'created' only through an axiom which says that something exists without giving any information about it, it's not explicitly constructed. Thus AC and not-CH give rise to non-constructive arguments.

I'm perplexed that you'd be confused by this; it's pretty clear that sets of cardinality aleph_1 under not-CH and nonprinciple ultrafilters under AC are nonconstructive in that we can't say anything about a given object to distinguish it from any other.

Example: By not-CH, I choose x as a subset of R with cardinality aleph_1. By not-CH I choose y as a subset of R with cardinality aleph_1. Does x = y? You can't say -- you don't know anything about x and y themselves.

This is unlike subsets of R with cardinality aleph_1 under CH, where I can explicitly display x = R and y = R \ Z, for example.

 

[micromass]

Do you also consider the law of excluded middle as inconstructive??

 

[CRGreathouse]

Do you also consider the law of excluded middle as inconstructive??

No, but I'm not sure if it would matter if I did as that's not germane to this thread.

 

[yossell]

Do you also consider the law of excluded middle as inconstructive??

Good question! CRG's worries about lack of constructibililty sound to me similar to the worries the intuitionists and opponents of Cantor's methods of introducing infinite sets without an explicit construction.

 

[yossell]

It's pretty simple -- whenever you have an object which is 'created' only through an axiom which says that something exists without giving any information about it, it's not explicitly constructed. Thus AC and not-CH give rise to non-constructive arguments.

meh - this concept of`any information' is just as unclear and vague to me. The aleph-one sets are such that they can be 1-1 mapped onto the set of all well-orderings of the natural numbers. That's a kind of information.

it's pretty clear that sets of cardinality aleph_1 under not-CH and nonprinciple ultrafilters under AC are nonconstructive in that we can't say anything about a given object to distinguish it from any other.

Of course there are different sets of cardinality aleph_1: and there's a subset of (01) of this cardinality; there's a subset of (12) of this cardinality. We can say things to distinguish the two. I agree that these may be uninteresting differences: there may be some class of predicates we're interested in, and we're wondering whether we can show that there are different aleph_1 sets that differ with regard to these predicates. But until you tell me what those predicate are, I don't see this notion of information you have.

Example: By not-CH, I choose x as a subset of R with cardinality aleph_1. By not-CH I choose y as a subset of R with cardinality aleph_1. Does x = y? You can't say -- you don't know anything about x and y themselves.

Despite the use of the word `choose', this has nothing to do with the axiom of choice. Sure - let x be a solution of an equation, let y be a solution of an equation. Does x = y? Who knows? It was never specified. So?

This is unlike subsets of R with cardinality aleph_1 under CH, where I can explicitly display x = R and y = R \ Z, for example.

So the real line IS. in your book, explicitly constructed or displayed? The powerset operation does count as a way of giving an explicit display? I mean - that might be fine: you're allowed to define constructible how you like. I'm not seeing what concept you feel you have here, though.

 

[CRGreathouse]

CRG's worries about lack of constructibililty

I have no worries about constructibility! I'm addressing the question of the OP, which is very much about constructibility.

 

[CRGreathouse]

meh - this concept of`any information' is just as unclear and vague to me.

That's fine by me. You'd do better to ask the OP what he/she meant by "example", regardless -- I'm just trying to address that question.

If you feel that you can give an explicit example, by all means ignore my objections and do so.