Posted by:
HB
(
)
Date: February 16, 2024 01:03PM
Which is greater, ∞ + ∞ or ∞ x ∞?
I can answer at least part of the first one.
♾️ plus ♾️ is the same ♾️
COMMENT: What you mean, I think, is that (♾️ + ♾️) = ♾️ if the infinities in question have the same cardinality. But what about the results if the cardinalities are different?
For example, the cardinalities of the set of natural numbers as opposed to the cardinality of the set of real numbers. In that case, your conclusion is not true, as you know. Consider:
First, for those non-mathematicians (like me) who might be interested, "cardinality" in this context represents the number of elements within a set. In a finite set, the number is simply the natural number (1,2,3, . . . etc.) you get by counting the elements. The problem arises with respect to the cardinality of infinite sets, such as the set of natural numbers.
It is tempting to state that all infinite sets have the same cardinality, namely infinity (this seemed to be implied by your conclusion above), but Cantor proved that this is not so; specifically, he proved that the cardinality of the (infinite) set of natural numbers is less than the cardinality of the (infinite) set of real numbers. Once this is shown, you have a relation such that |N|<|R|. (Here the straight brackets are intended to denote the cardinality of the sets N (natural numbers) and R (real numbers) respectively.) This non-equality relation, coupled with associative, commutative, and additive identity properties, can be used as the basis for arithmetical calculations and conclusions, for example as follows:
|N|+ 0 = 0 + |N| = |N| (additive identity)
(|N|+|R|) + p = |N|+ (|R|+p) (associative property)
|N|+|R|=|R| + |N| (commutative property)
By the above, you now have:
If (|N| < |R|), then ((|N| + p) < (|R| + p); and
(p + |N|) < p + |R|)
Now, assuming the axiom of choice, you have:
|N| + |R| = {N,R} (the set containing the union of the sets N and R; and thus, since |N| < |R|, ({N}+{N}) < ({N}+{R}), and in this case ♾️ + ♾️ does not equal ♾️.
It is a little more complicated than this, but you get the idea.
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I am pretty sure that the product of the smallest ♾️ (the integers, or any set of equivalent size) is the same level of infinity.
COMMENT: The above comments apply here too. As it turns out, |N| x |R| also = {N,R} As such, addition of infinite cardinal numbers produces the same result as addition, which is not surprising because multiplication is really just the sum of additions (2x3 = 2+2+2)
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I don’t know if the same holds true for larger infinities, like the real numbers. Interesting question, but not interesting enough that I am going to try to find the answer. :)
COMMENT: See discussion above.
CAVEAT: As noted, I am not a mathematician, much less a set theoretician, so am open to comments and corrections.