What is the cardinality of the set?

The set in question is $S=\Bbb{N}×A$, where $A$ is a set with infinitely many elements and $\Bbb{N}$ is the set of natural numbers.

I think cardinality of $S$ is the same as $A$ no matter whether $A$ is countable or not. I have tried to give a bijection between them but no success.

Any help would be appreciated.

• AFAIK this result depends on AC. Assuming you have AC... if A is infinite then you can provide a mapping from $\mathbb{N}$ into $A$ (why?); then use this mapping to show that $|\mathbb{N}\times A|\leq |A\times A|$. – Steven Stadnicki Sep 19 '17 at 15:41
• @StevenStadnicki is there a quick reason that $|A \times A| = |A|$? – Omnomnomnom Sep 19 '17 at 15:43
• @Omnomnomnom Transfinite induction. Specifically, by AC it's enough to show that $\vert\alpha\times\alpha\vert=\vert\alpha\vert$ for every infinite ordinal $\alpha$, and this is proved (in ZF) by transfinite induction on $\alpha$. If we don't have choice, the first part can fail; in fact, "All infinite sets are in bijection with their own square" is equivalent to full choice!" – Noah Schweber Sep 19 '17 at 15:46
• @Omnomnomnom There's a very canonical AC argument that works by well-ordering $A$ and then using the result for ordinals (which is true by induction). (To be fair, the same argument would work relatively directly here: just order $A$ copies of $\mathbb{N}$ and then argue about the order type of the result.) – Steven Stadnicki Sep 19 '17 at 15:48

This depends, as Steven Stadnicki said, on whether we assume the axiom of choice.

First, a quick observation: it's not hard to prove by transfinite induction that if $\alpha$ is an infinite ordinal, then $\vert\mathbb{N}\times\alpha\vert=\vert\alpha\vert$. Interestingly, more is true: for all infinite ordinals $\alpha$ we have $\vert\alpha\times\alpha\vert=\vert\alpha\vert$. This is harder to prove, but not much harder.

If we have the axiom of choice, then we're done: given $A$, the axiom of choice lets us prove that there is some ordinal in bijection with $A$; now just apply the previous paragraph.

If choice fails, however, things can get weird: it's consistent with ZF (= the usual axioms of set theory, but without choice) that there are very weird infinite sets! For example, there might be amorphous sets - these are infinite sets which can't be partitioned into two infinite disjoint subsets! It's easy to show that $\mathbb{N}\times A$ can't be amorphous, regardless of what $A$ is (HINT: think about evens vs. odds ...).

A side comment: if choice fails, then "$\vert A\times A\vert=\vert A\vert$" must fail for some infinite set $A$: the statement "for all infinite sets $A$, $\vert A\times A\vert=\vert A\vert$" is equivalent to the axiom of choice (this was proved by Zermelo). Meanwhile, the statement "$\vert A\times\mathbb{N}\vert=\vert A\vert$" is strictly weaker than the axiom of choice, although it isn't provable in ZF alone.

• This has me wondering now whether even in the absence of AC we can say that either $\left|A\times\mathbb{N}\right|=\left|A\right|$ or $\left|A\times\mathbb{N}\right|=\left|\mathbb{N}\right|$. My gut instinct says no, but I don't know choiceless models well enough to be very confident in that. – Steven Stadnicki Sep 19 '17 at 17:57
• @StevenStadnicki No. If $\vert A\times\mathbb{N}\vert=\vert\mathbb{N}\vert$, then $A$ is countable (that is, has an injection into $\mathbb{N}$); this means that if $A$ is (say) amorphous, $\vert A\times\mathbb{N}\vert$ is neither $\vert A\vert$ or $\vert\mathbb{N}\vert$. – Noah Schweber Sep 19 '17 at 19:18
• Ahhh, of course - we can just look at the image of e.g. $A\times \{0\}$ for an injection. And $|A\times\mathbb{N}|$ can't be $A$ (which I had been thinking was possible) becasue $\mathbb{N}$ injects into it, right? – Steven Stadnicki Sep 19 '17 at 19:26
• @StevenStadnicki Exactly. Actually, that's overkill: just think about $A\times\{0\}$ vs. $A\times(\mathbb{N}\setminus\{0\})$. – Noah Schweber Sep 19 '17 at 19:36