Cantor's Diagonal Argument( Levels of Infinity )

Question

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$:$

$|N|$$<$$|P(N)|$$<$$|P(P(N))|$$<$$\cdots$

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$

Answer 1

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:

• "$A$ and $B$ have the same size" means that there is a bijection between $A$ and $B$.

• "$A$ is at least as big as $B$" means that there is an injection from $B$ into $A$.

• Satisfying, we can prove (without Choice, even) that - using the above definitions - if $A$ is at least as big as $B$ and $B$ is at least as big as $A$ then $A$ and $B$ have the same size.

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.

Answer 2

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.

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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."