continuum is aleph 2

If you only count continuous functions, there are only $\aleph_1$ of those. Asking for help, clarification, or responding to other answers. If you think of "the continuum as an example for cardinality $\aleph_1$" then I'm guessing that you (or your professor) are (at least implicitly) assuming the so-called Generalized Continuum Hypothesis. “Question closed” notifications experiment results and graduation, MAINTENANCE WARNING: Possible downtime early morning Dec 2/4/9 UTC (8:30PM…, bound on the cardinality of the continuum? Or perhaps you meant to say $ \mathcal P ( \mathcal P ( \mathbb N ) ) $. Similarly, if we consider $A$ to be a set of size $\aleph_1$, then $\aleph_2$ is defined as the cardinality of the set $\{R\subseteq A\times A\mid (A,R)\text{ is a well order}\}/\equiv$, where $\equiv$ as before is the order-isomorphic relation. This talk will discuss some partial results and test questions. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. @Nick: I watch so many TV shows and movies, that my slang cannot be not fine. Namely, the least size of an uncountable set. Thanks. :). Can there be an $\aleph_2$ or even greater? @HighGPA If $f:\mathbb R\to\mathbb R$ is continuous, then for any real number $x_0$ there is a sequence of rational numbers $r_n$ such that $x_0=\lim_{n\to\infty}r_n$ and therefore, by continuity, $f(x_0)=\lim_{n\to\infty}f(r_n).$, @HighGPA If $f,g:\mathbb R\to\mathbb R$ are continuous functions, then $h=f-g$ is a continuous function, and so $\{x:h(x)=0\}$ is a closed set. In fact under some set theoretic axioms we can prove that $\Bbb R$ is a set of size $\aleph_2$. :P, That is true, just thought it might be worth mentioning. ...so for example And unless you're rejecting the law of excluded middle, it means that it's fine. Silly of me. @AsafKaragila Thank you vey much! In fact the reason why we cannot e ectively pro- Is the word ноябрь or its forms ever abbreviated in Russian language? The definition of $\aleph_2$ is the least uncountable cardinal larger than $\aleph_1$. Any other "geometrical" example friendlier to visualize? But that doesn't mean that you need set theory in order to say what the axioms of ZFC are. Of course, as Clive pointed out, it is provably unprovable that $|\Bbb R|=\aleph_1$ from the standard axioms of set theory (namely $\sf ZFC$), so it is also unprovable that $\mathcal P(\Bbb R)$ has size $\aleph_2$. Assuming the generalized continuum hypothesis, $\aleph_2=2^{\aleph_1}=2^{2^{\aleph_0}}=\cal |P(P(\Bbb N))|=|P(\Bbb R)|$. You need set theory (or something similar) to do (1), but not (2). the cardinality of $\mathbb{R}$ is $2^{\aleph_0}$, which is not provably equal to $\aleph_{\alpha}$ for any fixed value of $\alpha$. You are using standard terminology incorrectly. I hope not. Is oniichan also used to refer to a big sister? In Star Trek TNG Episode 11 "The Big Goodbye", why would the people inside of the holodeck "vanish" if the program aborts? The most canonical set of cardinality $\aleph_{\alpha}$ (for any ordinal number $\alpha$) is the ordinal $\omega_{\alpha}$. It only takes a minute to sign up. Making statements based on opinion; back them up with references or personal experience. Suppose some theory T has countably many axioms, how many models of $T$ are there of cardinality $\aleph_1$,$\aleph_2$,$\aleph_{\omega_1}$? Using this definitions, we have really standard ways to define $\aleph_1$. MathJax reference. The cardinality of the set of real numbers (cardinality of the continuum) is $${\displaystyle 2^{\aleph _{0}}}$$. :-). Can someone be saved if they willingly live in sin? has cardinality $\aleph_2$. Canada, Commercial and Industrial Mathematics Program, Centre for Quantitative Analysis and Modelling, Dean's Distinguished Visiting Professorship, CAIMS-Fields Industrial Mathematics Prize, Mathematics-in-Industry Case Studies Journal. Though it is provably unequal to some values, like $\aleph_\omega$. However, it is important to realize that we are considering arbitrary functions. The symbols do not mean what you think. That is, $${\displaystyle {\mathfrak {c}}}$$ is strictly greater than the cardinality of the natural numbers, $${\displaystyle \aleph _{0}}$$: The Continuum Hypothesis, the Perfect Set Property, and the Open Coloring Axiom A common philosophical justi cation for CH is that we cannot e ec-tively demonstrate the existence of a subset of R of cardinality strictly between jNjand jRj. Aleph 2, of Cantor's infinite sets X0
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