/sci/ - Science, Technology, Engineering, and Mathematics

Math is wack. Americium 21/10/11(Mon)19:07 No. 17837 ID: d2a5e8

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Just as Cantor's theorem shows there is no set of all sets, there is also no set containing all truths.

Or to put it another way: it's a truth that all truths can't be collected up into a totality.

http://www.pgrim.org/articles/grim_no_set_of_all_truths.pdf

Anonymous 21/10/12(Tue)17:14 No. 17840 ID: 126958

I feel like it should be possible to construct an axiomatic system of sets where a set of all sets would not lead to a paradox.

Americium 21/10/12(Tue)23:46 No. 17841 ID: d2a5e8

>>17840
You most certainly can modify set theory to allow a for maximum set of truths, namely predicative set theories.

However, such predicative theories come with their own issues. For instance, the category of sets is no longer a topos (maybe a pretopos), and so we can't define anything much more than first-order logic.

That being said, I personally wouldn't want to work with a theory of sets that doesn't have all possible powersets and/or function sets.

Americium 21/10/13(Wed)00:01 No. 17842 ID: d2a5e8

>>17841

No wait, I might have misspoken my second part:
https://ncatlab.org/nlab/show/predicative+topos

That being said, I still stand by my point that I wouldn't want to work in any set theory without a set of functions between any two sets.

Anonymous 21/11/10(Wed)03:56 No. 17896 ID: 586542

>>17837
I don't buy this argument
consider a finite set {1,2}
then 1 is in {1} {1,2} and not in {2} {null}
and 2 is in {2} {1,2} and not in {1} {null}
The powersets are generated by the elements of the set.
This is like saying that there is no vector space basis of primes (2,0...)x1+(0,3,...)x2+(0,0,5,...)x3+... = v = (2,3,5,7,...)

Americium!Metal3G/gs 21/11/12(Fri)05:07 No. 17899 ID: d2a5e8

>>17896
Do go on.

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