While reading an article on Cantor's Diagonal Argument, I read that there are infinite numbers of levels of Infinity.

The Reason they gave for above claim– Let $N$ be set of all positive integers. Then$:$


Thus starting with infinite set $N$, the power set operation iteratively produces new infinite set with strictly larger cardinality that all their predecessors.

$Doubt$$:$ From above information, can we state that some infinities are greater than others or $\infty$$>$$\infty$

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    $\begingroup$ There are different sizes of infinity (speaking loosely), but the expression $\infty > \infty$ appears to be meaningless. (What does the symbol $\infty$ denote in that expression?) $\endgroup$ – littleO Jun 5 at 4:09
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    $\begingroup$ For purposes of cardinality (which is what Cantor’s theorem is about), the symbol $\infty$ is meaningless. You use other symbols to denote cardinalities. Look up the aleph numbers, the beth numbers, and more generally, “cardinals”. $\endgroup$ – Arturo Magidin Jun 5 at 4:11
  • $\begingroup$ Here: en.wikipedia.org/wiki/Cardinal_number $\endgroup$ – Yanior Weg Jun 6 at 17:40

One important thing to keep in mind is that naive ideas about infinity are not yet mathematics. In order to ask whether a statement like "$\infty>\infty$" is true, we need to first define our terms precisely.

An essential aspect of this is that different fields may find it useful to attach different precise definitions to the same naive notion: e.g. "$\infty$" in calculus has a different connotation than "$\infty$" in set theory.

Cantor is working from the perspective of (what would become) set theory. He's interested in cardinality - that is, the sizes of sets as measured by maps:

In this context one might naively write "$\infty$" for the size of an infinite set, but this assumes that that's unambiguous: that any two infinite sets have the same size in the relevant sense, that is, that . Cantor's theorem shows that that is (perhaps surprisingly) false, and so it's not that the expression "$\infty>\infty$" is true or false in the context of set theory but rather that the symbol "$\infty$" isn't even well-defined in this context so the expression isn't even well-posed.

  • $\begingroup$ This observation that $\infty$ is just a context-dependent symbol and not for a fixed underlying concept of infinity even goes so far that $\displaystyle\sum_{\square=\square}^\infty\square$ cannot be viewed as a special instance of $\displaystyle\sum_{\square=\square}^\square\square$ $\endgroup$ – Hagen von Eitzen Jun 5 at 5:06

For sets $A$ and $B$ we make these definitions:

  • $|A|\leq |B|$ : There is an injective function $f:A\rightarrow B$.

  • $|A| < |B|$ : There is an injective function $f:A\rightarrow B$ but no injective function $g:B\rightarrow A$.

  • $|A| = |B|$ : There is a bijective function $f:A\rightarrow B$.

From these definitions, the following (nontrivial) facts can be proven:

i) If $|A|\leq |B|$ and $|B|\leq |A|$ then $|A|=|B|$.

ii) If $|A| \leq |B|$ and $|B|\leq |C|$ then $|A|\leq |C|$.

These facts (and similar ones) allow us to manipulate set inequalities in intuitive ways.

As shorthand, we sometimes write $|A|<\infty$ to mean "$A$ is a finite set." We sometimes write $|A|=\infty$ to mean "$A$ is an infinite set." These statements are not intended to be used in the above injection/bijection set inequalities because they are not precise enough: The equation $|A|=\infty$ does not tell us that there is a bijection from set $A$ to set $\infty$ because it does not specify what is meant by "set $\infty$." As you know, there are different infinite sets, and some are "bigger" than others.

So we certainly cannot conclude that if $|A|=\infty$ and $|B|=\infty$ then $|A|=|B|$ (that would be incorrectly claiming that if $A$ and $B$ are infinite sets then there is a bijection between them). Also, if $|A|=\infty$ and $|B|=\infty$ but $|A|<|B|$, it does not make sense to conclude $\infty < \infty$ (as the inequality $\infty < \infty$ does not have any meaning). Rather, the statement $|A|=\infty$ and $|B|=\infty$ but $|A|<|B|$ just means that $A$ and $B$ are both infinite sets, there is an injection from $A$ to $B$, but no reverse injection. To avoid confusion, you could simply avoid using expressions like $|A|=\infty$ when dealing with set inequalities, rather just say "$A$ is an infinite set."

  • $\begingroup$ Can you tell me about a bijective function which mapes from set of primes to set of naturals $\endgroup$ – user75659 Jun 7 at 7:58
  • $\begingroup$ What about it? Just list the primes and that gives you a injection. $\endgroup$ – Michael Jun 7 at 23:52

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