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Clearly, uncountable cardinals exist. How do we know that $\aleph_1$ is the smallest one? Is there a proof that there is no cardinal strictly between $\aleph_0$ and $\aleph_1$.

Andrés E. Caicedo
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Luke
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    Please review the definition of the words you are using. The question is trivial if you understand what the symbols mean. – Andrés E. Caicedo Mar 16 '20 at 14:43
  • @AndrésE.Caicedo : What is the definition of $N_0$ and $N_1$? (Neither the above question nor the linked question defines those) – Michael Mar 16 '20 at 14:47
  • I did not know the name "aleph." I do remember various definitions of such degrees of infinity as being "the smallest [thing] that satisfies [conditions]..." which seems to pre-assume such a smallest exists. A commenter above hinted that there was a "trivial" way to understand these issues. – Michael Mar 16 '20 at 15:33
  • @Michael the point is that $\aleph_1$ is basically defined as the smallest uncountable cardinal. Thus, the question needs some precision what is actually meant or unclear. For if one accepts as a given the existence of $\aleph_1$ the smallest uncountable cardinal, then the question is trivial in that clearly there is no uncountable cardinal smaller than the smallest uncountable cardinal. – quid Mar 16 '20 at 17:07
  • @quid : Eventually the terms “cardinal” and “ordinal” are supposed to help talk about sets. If for two sets $X$ and $Y$ we can compare their sizes (such as $|X|\leq |Y|$ or $|X|<|Y|$) by existence (or non-existence) of injective functions from one set to the other, it makes sense to ask if there is a “smallest” set that is “larger” than the natural numbers. When additional terminology is introduced, it is not always easy to distinguish between proving things associated with that terminology, versus using that terminology to help craft an argument that answers the fundamental questions. – Michael Mar 16 '20 at 18:16
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    @Michael I think the original comment is arguably a bit terse. However, the point is that if $A$ is meant to denote the smallest object with some property, then the literal question why $A$ is the smallest object with some property is trivial at least if one know that $A$ is meant to denote the smallest object with some property. Of course one might not know this. But the advice given in the initial comment was precisely to look up what $A$ means for then the question goes away. – quid Mar 16 '20 at 18:47
  • @quid : Thanks, I understand your point. Of course we can try to define $A$ as the smallest real number in the open interval $(0,1)$ and we see that we cannot simply define things into existence. The advice about looking up a definition is good but, in this case, there can be multiple (equivalent?) definitions found in different places, stated using additional terms like "cardinal," "ordinal," "transfinite induction," written by various people, some of whom may be (circularly?) referring to others, some of whom refuse to state things that are "widely known." – Michael Mar 16 '20 at 20:43
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    @Michael yes, I agree in a way, and I don't think the first comment was ideal. Likely one should have taken the title as indicating the actual question. That said, the point of the comment was also to incite OP to state what their definition is. – quid Mar 16 '20 at 21:03

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