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Suppose we assume that the Continuum Hypothesis is false. Then there must be a subset of the real numbers that has the cardinality of Aleph 1. What is an example of such a subset?
Office_Shredder said:If you could construct such a set then the continuum hypothesis wouldn't be undecidable
Office_Shredder said:The continuum hypothesis is neither false nor true under the ZF axioms. If you could construct an uncountable set with cardinality smaller than c, then the continuum hypothesis would obviously be false.
You might think that if it's impossible to construct such a set, they must not exist. But with only finitely many characters and statements of finite length, there are only countably many things we can say. So you can't construct every subset of the real numbers. So a subset whose cardinality is smaller than c might exist but we're unable to actually construct it
Office_Shredder said:That doesn't mean you can explicitly construct it. Similar to how the axiom of choice is used to state the existence of things that otherwise would not be constructible
Office_Shredder said:If you can construct an uncountable set with cardinality smaller than c, then there exists a set with cardinality smaller than c. The continuum hypothesis says no such set exists, so it must be that the continuum hypothesis is false.
yossell said:To be fair to the OP, that's not literally what was said in the first post. All that was asked was for an example of a subset of the real numbers that had cardinality Aleph 1. But a subset S of the real numbers, even a proper subset, doesn't necessarily have a lower cardinality than the real numbers. There would only be a problem with the unprovability of the continuum hypothesis if you could also show that subset S had a cardinality less than the continuum, but greater than aleph zero.
Having said that, whether or not you assume the continuum hypothesis, there is a subset of the real numbers that has cardinality Aleph_1. (We just can't prove that there is such a subset has a cardinality less that the continuum). So it's not clear exactly what the original poster had in mind with the question.
lavinia said:If we assume that the Continuum Hypothesis is false then there must be a proper subset of the line of the plane that has cardinality Aleph 1.
lavinia said:Suppose we assume that the Continuum Hypothesis is false. Then there must be a subset of the real numbers that has the cardinality of Aleph 1. What is an example of such a subset?
Suppose we assume that the Continuum Hypothesis is false. Then there must be a subset of the real numbers that has the cardinality of Aleph 1, and such that there is no bijection between the subset and the set of real numbers. What is an example of such a subset?
yossell said:To be fair to the OP, that's not literally what was said in the first post. All that was asked was for an example of a subset of the real numbers that had cardinality Aleph 1. But a subset S of the real numbers, even a proper subset, doesn't necessarily have a lower cardinality than the real numbers.
lavinia said:we assume that the Continuum Hypothesis is false
chronon said:If the continuum hypothesis is false and S is a subset of the reals of cardinality aleph 1 then you can deduce immediately that there is not a bijection between S and the reals.
George Jones said:these are quite different things.
lavinia said:These responses are all interesting but the question still stands unanswered.
CRGreathouse said:See post #2. No such explicit example can be constructed.
lavinia said:The properties of such a set could potentially be described without an explicit construction. One might appeal to the Well Ordering Principle to obtain a mapping of the reals onto one of the uncountable ordinals whose cardinality is possible for the continuum.
From there one might be able to say something about a subset of cardinality Aleph1 e.g. maybe that it must have Lebesque measure zero.
Jarle said:The continuum hypothesis is undecidable even in ZFC, so you could not construct (nor "construct" using the well ordering principle) such a subset. It is undecidable; it has been shown that it cannot be proved nor disproved in ZFC. The bridge cannot be built from the axioms to the statement.
yossell said:What's the claim, CR? I'm not following you. That you can't construct an example? Or that you can `construct' an example, but it gives you no insight? Perhaps Lavinia could be the judge of whether she finds insightful or not.
Whether or not the continuum hypothesis is true, one can give examples of aleph_1 subsets of the real numbers. Just take a mapping from the reals between 0 and 1 onto the ordinals; then consider that subset of these reals which, by this function, are mapped onto aleph_1.
Of course, I agree that this is an uninteresting example, for many reasons. One reason is that this subset may be nothing more than the interval (01) again. Another reason is that the construction goes through whether or not continuum hypothesis holds, and so may not have been what Lavinia wanted when it was explicitly assumed that CH failed. I agree that, at least for me, there's no insight here - at least for me.
Lavinia seemed to make it clear that he/she wasn't asking for an explicit constructible example - whatever that means (do you think the reals are constructible?). But (IMO) Lavinia's right that it's just not true that a description of the relevant set somehow contradicts the undecidability of the CH.
micromass said:Let's presume that the continuum hypothesis is false. Then there exists a set X of cardinality [tex]\aleph_1[/tex] in [tex]\mathbb{R}[/tex].
yossell said:What's the claim, CR? I'm not following you. That you can't construct an example? Or that you can `construct' an example, but it gives you no insight?
yossell said:there exists a set of cardinality [tex]\aleph_1[/tex] in [tex]\mathbb{R} whether the continuum hypothesis is true or false
CRGreathouse said:That you can 'construct' but cannot construct.
Correct. I don't know why you keep bringing this up, though. I think it was evident to everyone participating on this thread before you mentioned it
yossell said:Thanks. I brought it up because the argument seemed to be `suppose CH is false. Then there is a subset of the reals which is of cardinality aleph-1'. But the supposition does no real work in getting the conclusion. So I was puzzled by the role of the supposition in people's thought.
CRGreathouse said:In post #25, for example, the assumption is needed for the "know nothing about it" claim. With CH, we can explicitly construct sets of cardinality aleph_1; without, we cannot.
lavinia said:I think that this thread has lost its seam. The original question that I asked is not being discussed.
yossell said:I'm really perplexed. Could you help me by explaining your notion of explicit constructibility?
micromass said:Do you also consider the law of excluded middle as inconstructive??
micromass said:Do you also consider the law of excluded middle as inconstructive??