Cantor's Diagonal Argument( Levels of Infinity )

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 $$\infty>\infty$$

• 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?) – littleO Jun 5 at 4:09
• 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”. – Arturo Magidin Jun 5 at 4:11
• – 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.

• 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 $\displaystyle\sum_{\square=\square}^\square\square$ – 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."

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