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I believe $\infty-\infty$ is indeterminate, but is $\aleph_0-\aleph_0$ zero? In Altar Ego's response to What is the result of $\infty - \infty$? those two infinities must've had different ordinality or cardinality or whatever measurements of size.

Aleph Null minus itself could be interpreted as the size of the set of all natural numbers minus the size of the set of odd numbers, for example. There's a one to one correspondence between each set since they have the same cardinality, it doesn't matter that one set seems "denser". To me intuitively, no matter the size any exact number minus itself should be zero. Is this an exception and if so why?

What about other arithmetic like Aleph Null divided by itself? (I know, infinity isn't a number but)

Gerry Myerson
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Pineapple Fish
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    How do you define subtraction of cardinals? what is $-\aleph_0$? we define $|X|$ as the least ordinal with bijective to $X$, so with $\aleph_0$ you mean $\omega$? and with $-$ you mean $\setminus$? if so $\aleph_0-\aleph_0=\omega\setminus\omega=\emptyset :=0$. If not you need to be more clear on what is $\aleph_0-\aleph_0$, however if we don't have choice $\aleph_0$ is a proper class so even $\setminus$ makes not sense – ℋolo Jun 24 '18 at 00:37
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    Cardinal arithmetic (including subtraction) is discussed (briefly) at https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic – why not have a look, and report back to us, Benjamin? – Gerry Myerson Jun 24 '18 at 00:43
  • @GerryMyerson Ok. I don't have time now but as soon as I can, definitely by tomorrow. Thanks! – Pineapple Fish Jun 24 '18 at 00:46
  • @GerryMyerson Actually I should look up some more notation too since I'm just playing with this part of math for fun right now. I'll read the details when I can. – Pineapple Fish Jun 24 '18 at 00:53

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