Countability of the irrational numbers
One of the excersises in my book says "Is the set of all irrational numbers countable? Justify your response"
Here is what I said (For the sake of writing let the set of irrationals):"
1. First let us establish that is an infintely countable set. This is readily seen since there is a 1-1 mapping of onto so , thus the integers are countable.
2. Lemma: If is a countable set and if the set of all n-tuples where , then is countable.
3. So realizing that the rationals may be expressed as . So now since if we express every rational number as we can show that the rationals may be expressed as a set of ordered pairs (2-tuples). We have that the rationals may be expressed as an analogue of #2 with expressed as , and since , a countable set, it follows that the rationals are countable.
4. Lemma: The set of all Real numbers is uncountable
5. Lemma: The union of any number of infinitely countable sets is countable.
6. Assume that is countable, That would imply that is countable. A contradiction.
Therefore is uncountable "
Does that look alright?