/sci/ - Science & Math

>>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

>>10335355
You get a contradiction if you try to figure out how many elements are there

>>10335368
No you dont, you get the result that the set is not finite.

>>10335368
>>10335378
Let me rephrase that.
If you mean some proof like
"assume that the set of (positive) integers is finite
let x be the greatest positive integer
however x + 1 > x
that 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.

>>10335382
If 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

>>10335352
It's countably infinite.

>>10335352
N is well-defined because it is taken as an axiom that it is well-defined.
>zf is a religion
No 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.

>>10335352
>>10335517
>they don't know about the infinite set axiom

>>10335517
>You can prove that a natural number is a well defined object only if an infinite set exists, which you have to arbitrarily assume
Which we do

Wildberger has the right idea.

>>10335633
No he doesnt

>>10335542
why can't we take as an axiom that a set that contains all sets that don't contain themselves is well defined?

>>10335786
Because it contradicts the other axioms

>>10335786
Imagine 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?

>>10335825
imagine a library that contains an infinite amount of books. see the problem?

>>10335830
No, I can imagine that.

>>10335786
it's not consistent with other axioms. it leads to contradiction.

>>10335825
why 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

>>10335945
You do if there's no cap on characters

>>10335352
How else should we define it?

>>10335990
No you don't, retard

>>10335825
This only works for order type ω.

>>10335352
infinity is a standin for your own personal limit you brainlet

>>10335352
*tips fedora*