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# /sci/ - Science & Math

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It's often argued that the set of all integers is infinite because you can always increase the greatest number by 1, so it follows that the set is infinite. But couldn't you just as well conclude that this means that N is not well-defined?

 >> Anonymous Mon Jan 28 10:03:47 2019 No.10335355 >>10335352>But couldn't you just as well conclude that this means that N is not well-defined?What about it is not well defined
 >> Anonymous Mon Jan 28 10:08:09 2019 No.10335368 >>10335355You get a contradiction if you try to figure out how many elements are there
 >> Anonymous Mon Jan 28 10:11:42 2019 No.10335378 >>10335368No you dont, you get the result that the set is not finite.
 >> Anonymous Mon Jan 28 10:14:35 2019 No.10335382 >>10335368>>10335378Let me rephrase that.If you mean some proof like "assume that the set of (positive) integers is finitelet x be the greatest positive integerhowever x + 1 > xthat is a contradiction"All that is saying is that if you assume that there is a largest positive integer, you get a contradiction, therefore there is no largest positive integer, and thus the integers are not finite.
 >> Anonymous Mon Jan 28 11:01:51 2019 No.10335517 >>10335382If I have an empty set and an operation of adding one element to that set, I can construct every finite set. But I won't ever be able to construct an infinite set! So you have to assume separately that an infinite set exists. You can prove that a natural number is a well defined object only if an infinite set exists, which you have to arbitrarily assume
 >> Anonymous Mon Jan 28 11:04:08 2019 No.10335529 >>10335352It's countably infinite.
 >> Anonymous Mon Jan 28 11:07:48 2019 No.10335542 >>10335352N is well-defined because it is taken as an axiom that it is well-defined.>zf is a religionNo it's not. The point of set theory isn't to satisfy whatever physically intuitive properties you want. The point of set theory is to be able to contain models of smaller theories. To that end, disallowing infinite sets is shooting yourself in the foot, since you need infinite sets to model any "unbounded" or "potentially infinite" systems like the natural numbers.
 >> Anonymous Mon Jan 28 11:15:15 2019 No.10335568 >>10335352>>10335517>they don't know about the infinite set axiom
 >> Anonymous Mon Jan 28 11:28:34 2019 No.10335597 >>10335517>You can prove that a natural number is a well defined object only if an infinite set exists, which you have to arbitrarily assumeWhich we do
 >> Anonymous Mon Jan 28 11:43:38 2019 No.10335633 Wildberger has the right idea.
 >> Anonymous Mon Jan 28 11:53:39 2019 No.10335655 >>10335633No he doesnt
 >> Anonymous Mon Jan 28 12:46:18 2019 No.10335786 >>10335542why can't we take as an axiom that a set that contains all sets that don't contain themselves is well defined?
 >> Anonymous Mon Jan 28 12:51:54 2019 No.10335808 >>10335786Because it contradicts the other axioms
 >> Anonymous Mon Jan 28 12:59:55 2019 No.10335825 >>10335786Imagine a library that contains all possible books. Now put them in order, any order you like. Now imagine a book whose first page is different from the first page of the first book. Whose second page is different from the second page of the second book. And so on. This book would be different from all books in the library, which is supposed to contain all books. You see the problem?
 >> Anonymous Mon Jan 28 13:02:12 2019 No.10335830 >>10335825imagine a library that contains an infinite amount of books. see the problem?
 >> Anonymous Mon Jan 28 13:03:54 2019 No.10335832 >>10335830No, I can imagine that.
 >> Anonymous Mon Jan 28 13:08:13 2019 No.10335840 >>10335786it's not consistent with other axioms. it leads to contradiction.
 >> Anonymous Mon Jan 28 13:53:51 2019 No.10335945 >>10335825why are you assuming the set of books in that library is countable, retard? you dont need a set that contains all sets that dont contain themselves to exist for the set of all strings of 26 characters to exist
 >> Anonymous Mon Jan 28 14:08:37 2019 No.10335990 >>10335945You do if there's no cap on characters
 >> Anonymous Mon Jan 28 14:10:17 2019 No.10335993 >>10335352How else should we define it?
 >> Anonymous Mon Jan 28 20:47:23 2019 No.10337140 >>10335990No you don't, retard
 >> Anonymous Mon Jan 28 23:14:57 2019 No.10337408 >>10335825This only works for order type ω.
 >> Anonymous Mon Jan 28 23:26:56 2019 No.10337434 >>10335352infinity is a standin for your own personal limit you brainlet
 >> Anonymous Mon Jan 28 23:28:20 2019 No.10337436 File: 50 KB, 488x398, Religion math.png [View same] [iqdb] [saucenao] [google] [report] >>10335352*tips fedora*
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