Question:


  let $\alpha$ be a infinite cardinal , show that:
  $$\alpha=\aleph_{0}\alpha$$
  where $\aleph_{0}$ is the cardinality of the natural numbers is denoted aleph-null


this equation is from a book,and I can't prove it,and the author say this is important equation,and can't prove it.Thank you for you help


Question: Does adding a comma before "or" change the meaning of a phrase?
Answer: This is not an Oxford comma because it is not the serial comma customarily used before a coördinating conjunction at the end of a series. 

It is instead an example of paired commas used as a weak or less obtrusive form of parentheses.

These are all equivalent:


A pronoun is a word that is used instead of a noun, or noun phrase, for example he, it, hers, me, them, etc.
A pronoun is a word that is used instead of a noun (or noun phrase), for example he, it, hers, me, them, etc.
A pronoun is a word that is used instead of a noun — or noun phrase — for example he, it, hers, me, them, etc.


Such “parenthifying” punctuation is something that could be omitted without great loss.  I don’t think you can use the comma there, because “or noun phrase” cannot be freely omitted without changing the overall meaning.  

Also, the indefinite article seems intended to distribute across noun and noun phrase in that sentence, and if you break it up with a comma, you lose that.

My own preference would be to write that this way:


A pronoun is a word used in place of a noun or noun phrase; for example, he, it, hers, me, mine, and them are all pronouns.


Incidentally, this is the pro- that means “for”, so a pronoun is a word used for a noun (or noun phrase).