Suppose for each $j$ there exists $a_jin I_jsetminus mathfrak{p}$. Let $a = a_1…a_n$. Then $ain I$, but $anotin mathfrak{p}$.

]]>“We have taken something so big (is big even the right word?) and stuffed it into a small, tangible little package so that we can carry it around in our finite brains and feel like we know something.” This part made me think about using forcing to collapse cardinals.

Nice article.

Cheers

]]>My early experiences with the infinity go back to my trying to count even more as a child. “What comes after million, billion, trillion, …?” I would ask my uncle. Later a cousin of mine teased my mind with a paradox: Draw an upright triangle and a segment connecting the middle-points of the two of its sides. Now to every point on this segment corresponds a unique point on the base determined by the line crossing through the tip of the triangle and that point. Geometrically, it is obvious that this map is bijective: connect any point on the base to the tip and you’ll get a point on the segment. So, there are just as many points on the smaller segment as on the larger base! How could that be possible.

Then later I read in a book about the paradox of an arrow never reaching the target because first it will have to travel half of the distance left, then half of the remaining distance… There will always be a nonzero half of the distance left to travel, thus at the very end, ultimately, it BARELY reaches it. So, no question of ever reaching going one more foot ahead.

The first mathematical definition of infinity that I really liked and was stunned by its simplicity was that “a set is infinite (in cardinality) iff it has a bijection with one of its subsets.” Then of course came Cantor’s amazing uncountable binary set.

Two incidences that made me reconsider my understanding of infinity were: 1) you can rearrange the terms of a series and end up with a different sum! What the hell?! 2) It is a trivial task to show that [0,1] and (0,1] have the same cardinal number. Thus, a bijection exists between them. Task: Find one! I had this on my mind for some months until I came up with one 🙂 Once you find it you see how simple it is! Good luck with your own bijection.

Endnote: There is an infinity related to values in real line, as in limits of functions, and there is another one related to cardinality of sets.

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