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I just read a book’s section about general sums and products of cardinal numbers along with Koenig’s theorem. I made the following summary. The topic is not important to me but I wanna mention it in my notebook, so I try to dense it into natural language. Can you check if my summary gets the gist about it?

You can not only add/multiply cardinal numbers pair-wise or finitely often, just like you can generalize the operations of union or intersection of sets for arbitrary quantities (of sets). Especially it holds that if a < b (a,b are cardinal numbers) and if a gets added up with cardinal numbers smaller or equal to a as often as b gets multipled with cardinal numbers bigger or equal to b then the resulting product is bigger than the resulting sum (Koenig‘s theorem). It looks reasonable even at first glance because a product is designed as multiple addings, so that a product grows faster than a sum in general.

  • Another attempt at wording: The sum of the terms of a countable sequence of cardinals is strictly less than the product of the terms of a countable sequence of cardinals if each former cardinal is strictly less than the corresponding latter cardinal. – Dave L. Renfro Oct 30 '22 at 21:28
  • @ Dave L. Renfro So you‘d be ok with my text overall? –  Oct 31 '22 at 11:42
  • I think for your own notes the important thing is that when YOU read the text, the relevant aspects become apparent to YOU. Of course, at some point you'll need to pay attention to how others understand your text, but if you are new to writing mathematical statements, then I recommend mostly worrying about whether it makes sense to you, especially if you read it a few months from now after having somewhat forgotten what the exact statement is. But if you want some criticism . . . "as often as" is probably not a good way of wording that part. (continued) – Dave L. Renfro Nov 01 '22 at 01:17
  • Incidentally, the last sentence is a bit beside the point. In fact, in cardinal arithmetic there are very few results that provide strict inequality. The result about power sets and Koenig's theorem are almost the only such non-trivial results in cardinal arithmetic. For example, see my comments to this question. Thus, the relevant issue is not that Koenig's theorem seems reasonable, but that it gives strict inequality when so many cardinal arithmetic results only give non-strict inequality (even with strict inequality assumptions). – Dave L. Renfro Nov 01 '22 at 01:17

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