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

[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.