Thanks! Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets. Can you help me on where I can find the proof that R is a set of size Aleph2 ? @Hayden: Sure, I could have put loads more information in about the continuum, but the OP seems confused enough! Thanks. Asking for help, clarification, or responding to other answers. What is this hole above the intake of engines of Mil helicopters? So most of those $\aleph_2$ functions are very ragged, and their graphs do not look like anything you would call a "curve". The symbols do not mean what you think. Two PhD programs simultaneously in different countries. $$\omega_1 = \text{the set of countable ordinals}$$ We would still have to assume $ 2^{\aleph_1} = \aleph_2 $. You can also get $\aleph_2$ as the number of functions $f:\mathbb R\to\{0,1\}$ which is the same as the number of subsets of $\mathbb R$, since a subset $X$ can be coded by a function $f$ such that $f(x)=1$ if $x\in X$ and $f(x)=0$ if $x\notin X$. 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. @Nick: I watch so many TV shows and movies, that my slang cannot be not fine. It cannot be determined from ZFC (Zermelo–Fraenkel set theory with the axiom of choice) where this number fits exactly in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis, CH, is equivalent to the identity site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Understanding the density operator in quantum mechanics for a joint system, Understanding the mechanics of a satyr's Mirthful Leaps trait. The continuum, i.e. The continuum isn't provably a set of cardinality $\aleph_1$ unless you assume the continuum hypothesis. Understanding how an index is converted to a logarithm. Now if we consider an ordinal as a set $T$ such that $(T,\in)$ is a well ordered set, and whenever $x\in T$, then $x\subseteq T$ (for example $\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}$ are all ordinals). Simple example for a set with cardinality of $\aleph_2$. In that case, the set of all functions $f:\mathbb R\to\mathbb R$ is a good example for cardinality $\aleph_2$. :). Then $\aleph_1$ is defined as the cardinality of $X/\equiv$, the set of equivalence classes of well-orders of $\Bbb N$. Nothing wrong with that; the GCH is a perfectly cromulent foundation for mathematics, even if the set-theory experts around here are more interested in various exotic alternatives. This is all still new territory for me. You can prove, using set theory, that there are an infinite number of axioms of ZFC. Suppose some theory T has countably many axioms, how many models of $T$ are there of cardinality $\aleph_1$,$\aleph_2$,$\aleph_{\omega_1}$? In Star Trek TNG Episode 11 "The Big Goodbye", why would the people inside of the holodeck "vanish" if the program aborts? @Nick: It was an amalgamation of both, I suppose. @HighGPA Since a continuous function on $\mathbb R$ is determined by its values on $\mathbb Q$ (the set of all rational numbers), the number of continuous functions $f:\mathbb R\to\mathbb R$ is no greater than the number of functions from $\mathbb Q$ to $\mathbb R$ which is $$|\mathbb R^\mathbb Q|=|\mathbb R|^{|\mathbb Q|}=(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0\cdot\aleph_0}=2^{\aleph_0}$$ which is equal to $\aleph_1$ since we're assuming the continuum hypothesis. Meaning of the Term "Heavy Metals" in CofA? In fact under some set theoretic axioms we can prove that $\Bbb R$ is a set of size $\aleph_2$. Are there any infinites not from a powerset of the natural numbers? Can we omit "with" in the expression glow with (something)? Proof that the set of all possible curves is of cardinality $\aleph_2$? rev 2020.11.24.38066, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Aleph 2, of Cantor's infinite sets X0

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