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This question is from Introduction to Mathematical Logic

If the set of symbols of a consistent generalized theory $\mathbf K$ has cardinality $\aleph_\alpha$:, then $\mathbf K$ has a model of cardinality $\aleph_\alpha$·

As a high-school student , I have never encountered even $\aleph_0$ in my life.much less I know about $\aleph_\alpha$. The only thing I know is that they are cardinal numbers.

So, I think I have to back of a bit and learn a thing or two about cardinals.Can someone give a reference about this cardinal and ordinal numbers so I can do this kind of proofs involving cardinal numbers?

Asaf Karagila
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Kripke Platek
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    These are called aleph numbers. Maybe Jech - Set Theory is a good reference. – ureui Jun 06 '21 at 06:32
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    See Textbooks on set theory. However, since you've never seen $\aleph_0$ before, you don't want to dive into a book that has a lot of focus on axiomatics and formalizations, which most of those books do. Instead, you first need an introduction to basic cardinality ideas. Two examples are Introduction to the Theory of Sets by Joseph Breuer (Chapters 2 and 3) and Basic Set Theory by Shen/Vereshchagin (Chapter 1). – Dave L. Renfro Jun 06 '21 at 06:59
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    @ureui: Jech has two books on set theory. One is graduate level and entirely inappropriate for a high school student who has never seen $\aleph_0$ before, and the other, while at the upper undergraduate / beginning graduate level, is much too lengthy and focused on axiomatics for someone at the OP's level to get an overview of cardinal numbers from as well as possibly being several years beyond the OP's present mathematical maturity level (which is difficult to judge; Mendelson's book is rather advanced, but much of the earlier material is fairly elementary). – Dave L. Renfro Jun 06 '21 at 07:04
  • @DaveL.Renfro Thanks for your response. To be honest , I would have never asked for a reference if I did not encounter cardinals like $\aleph_{\alpha}$ before the chapter of axiomatic set theory in Mendelson. But because now I have to learn them because they are introduced this early in the book , I am forced (!?) to ask for a reference. I guess I am asking for something that gives me a general overview of cardinal numbers so that I can atleast Do and understand proofs involving them. – Kripke Platek Jun 06 '21 at 07:12
  • Probably you (1) want to get an introduction to the idea of countable sets and uncountable sets, then (2) work a bit with specific cardinals such as $\aleph_0$ and $2^{\aleph_0} = c$ and $2^c$ and their arithmetic, then (3) know what the (transfinite) sequence $\aleph_0,$ $\aleph_1,$ $\aleph_2, \ldots, \aleph_{\alpha}, \ldots$ is (at least in a general overview sense), and see if that suffices. The parts of the two books I suggested are mostly for (1) and (2). (continued) – Dave L. Renfro Jun 06 '21 at 07:35
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    Especially for (3) (but also covers (1) and (2)), I strongly recommend Chapter 2 (Chapter 1 not needed) of Set Theory and Metric Spaces by Irving Kaplansky. Many people are likely to recommend Naive Set Theory by Paul Halmos, but in my opinion this is not a book all that helpful for someone wanting to learn the basics about cardinality of sets and cardinal numbers, but instead is for someone who mostly knows this material from other courses (e.g. real analysis) and wants to see a more formalized treatment. – Dave L. Renfro Jun 06 '21 at 07:35
  • By the way, Mendelson's text (first third or so, maybe less) was a text for a course I took back in 1982, and while I was then well aware of $\aleph_{\alpha}$ cardinals (indeed, even the fact that $\aleph_{\alpha} = \alpha$ is possible as an equality for ordinals, which is truly mind-blowing when you understand what it means), I thought Mendelson's insertion of this amount of set theory into the meta-language of a supposedly introductory mathematical logic book made it very non-user-friendly for its intended readers. – Dave L. Renfro Jun 06 '21 at 07:45
  • @DaveL.Renfro I did watch a video about $\aleph_0$ , $\aleph_1$ , ... , $\aleph_\alpha$ , ... in a Vsauce video How to Count Past Infinity .When I first watched the video , I thought "ok , that was cool". Now after 2 years , I encounter them again in Mendelson (The book I am currently studying logic from.Probaly will take 3 years to finish it , but I will do it) . Also , I am now curious about how were you first introduced and made well aware of $\aleph_\alpha$ cardinals ? – Kripke Platek Jun 06 '21 at 08:00
  • I first learned about different notions of infinity from Gamow's book (Chapter II) and Asimov's book (near the end) and Kasner/Newman's book and other books found in public libraries (this being around 1971, age 12-13, well before the internet). I learned what I called (1) and (2) from various undergraduate texts I saw my last year in high school in a somewhat nearby university library in 1976-77 (continued) – Dave L. Renfro Jun 06 '21 at 08:34
  • (also Fall 1976 Anderson/Hall's book and Spring 1977 Kasriel's book, which were texts for two of the many college courses I took while in high school). I learned what I called (3) from Chapter 2 of the Kaplansky book I previously cited, a book that I read much of during my last semester of high school (Spring 1977) when I was also taking that topology course using Kasriel's book. BTW, I need to leave for a few hours after posting this comment. – Dave L. Renfro Jun 06 '21 at 08:34
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    Of course the various suggestions Dave L. Renfro has given are good (I trust - I'm not familiar with them), but I'd like to point out that you do not need to understand aleph notation to understand Mendelson's statement. It just says that there is a model of $\bf K$ with the same cardinality as the set of symbols of $\bf K$. That is, you can put the model in a one-to-one correspondence with the symbols of $\bf K$ without any elements left over on either side. – Paul Sinclair Jun 06 '21 at 15:20

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To take this off the unanswered list, I would like to elaborate Paul Sinclair's comment. Outside of higher set theory or logic, there is almost never a need to deal with actual cardinals. This is because every statement of the form "$S$ has cardinality $k$" is equivalent to "$S$ bijects with $k$". And in ordinary mathematics, we almost always have no need to know what kind of set this $k$ is. In particular, the statement you quoted is equivalent to:

Every consistent generalized theory $T$ with infinite countable symbol set $S$ has a model that bijects with $S$.

This can be proven without any notion of cardinality (or similar) at all. The main reason to invoke the notion of cardinals is if we want to have a canonical representative for each collection of sets that bijects with one another, just like we have each $k∈ℕ$ as a canonical representative for the collection of all sets of size $k$. Even then, for the purpose here (i.e. countable sets) we could very well use $ℕ$ as our natural representative...

user21820
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