Can you explain this more?If you could construct such a set then the continuum hypothesis wouldn't be undecidable
I am a little confused here. If the unit interval had a cardinality greater than the first uncountable ordinal then it would still have to have a proper subset that has the cardinality of the first uncountable ordinal just as it has countably infinite subsets.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
You said that if you could construct such a set the continuum hypothesis would obviously be false. Why is that?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
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.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.
My question was simple - I thought. 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. Constructibility had nothing to do with the question. All I wanted to know was what such a subset might look like and maybe even an example. By example I did not mean construct per se but more some defining rule.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.
But, strictly speaking, whether or not the continuum hypothesis is false, there is a proper subset of the real line that has cardinality aleph 1. For even under the assumption that the continuum hypothesis is true, and the real line has cardinality aleph 1, there are still proper subsets of the real line that have the same cardinality as the real line (for instance, the points between 0 and 1/2), and thus (under the assumption) have cardinality aleph 1.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.
you meant something likeSuppose 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?
Is this what you meant? yossell has pointed out that these are quite different things. This is similar to the following.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?
No, in the original post, lavinia asked for an cardinality Aleph 1 subset of the reals together with the condition thatTo 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.
and, as chronon points out in post #14,we assume that the Continuum Hypothesis is false
Thus,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.
is not correct.these are quite different things.
See post #2. No such explicit example can be constructed.These responses are all interesting but the question still stands unanswered.
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.See post #2. No such explicit example can be constructed.
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.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.
Right. You could 'construct' it from ¬CH in the same sense that you can 'construct' a well-ordering of the reals with AC, but it's not going to give you any insight.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.